comparison of limit equilibrium and uncoupled continuummaximum hydraulic conductivity of the...

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Slope Failure of Embankment Dam under Extreme Flooding Conditions:Comparison of Limit Equilibrium and Continuum Models

Kevin S. Richards1 and Krishna R. Reddy2

1Federal Energy Regulatory Commission, 230 South Dearborn Street, Suite 3130, Chicago, Illinois 60604, USA; PH (312) 596-4447; FAX (312) 596-4460; email: [email protected] 2University of Illinois Chicago, Department of Civil and Materials Engineering, 2095 Engineering Research Facility, 842 West Taylor Street, Chicago, Illinois 60607, USA; PH (312) 996-4755; FAX (312) 996-2426; email: [email protected]

Abstract

During the spring of 2003, the downstream slope of an embankment dam began to move during an extreme flooding event. The current study was implemented to evaluate the slope stability of the embankmentand as a case-study to compare the limit equilibrium and continuum modeling methods. Initially, stability analyses were performed using conservative shear strengthof soils estimated from both boring log data and an infinite slope back analysis. The Mohr-Coulomb failure criterion was employed in the computer simulations. The computer program UTEXAS4 and Spencer method of solution was used to evaluate the limit equilibrium factors of safety andthe critical failure surface. The computer program FLAC4.0 was used to evaluate the stress/strain behavior of the slope and determine factors of safety using the strength reduction method. Both these limit equilibrium and continuum modelsyielded slip surfaces that were close in geometry and location to the actual slip surface observed in the field. The factor of safety for the pre-failure embankment geometry was lower utilizing the limit equilibrium approach than that estimated by the continuum approach; however, the continuum approach appeared to yield more accurate results when using the best estimate of the shear strength. Back-analyses performed based on the observed failure surface in the field after flooding event provided estimates of shear strength of the embankment soil.The back calculated shear strength utilizing the continuum model fell within acceptable ranges as determined from the boring log data, with the continuum model yielding the more conservative (lower) back calculated shear strength. Back calculated shear strengthusing the limit equilibrium method was higher and its use may result in unconservative designs.

Introduction

Stability of earth dams under extreme natural events such as flooding and earthquakes is often a major concern to geotechnical professionals in the United States and worldwide. This paper presents the results of a study conducted to determine stability of an embankment dam under extreme flooding conditions. The dam is a 408.4 m long 13.7 m high earth embankment dam located in the United

Proceedings of the Geo-Frontiers 2005 Conference, ASCE, Austin, Texas, January 2005

States (Figure 1). The dam was constructed with no compaction of the downstream slope by dumping poorly graded medium-fine sand into place from a rail trestle that ran along the crest of the dam. The method of construction left the downstream slope near the natural angle of repose. The upstream slope was flattened by hydraulically distributing the dumped material during the initial construction. The embankment is currently configured with approximately 2H:1V downstream slope and a flatter upstream slope with a 15.2 m wide crest. Minimal foundation treatment was done prior to placement of the embankment materials as evidenced by the presence of occasional residual soil along the base of the dam.

The daman upper alluviualluvium, whichthe upper alluvibelow the dam.

The emband SP-SM soils predominately oindicate the santhe stratigraphya

m


100

2

Figure 1. Aerial photo of the dam.

is underlain by alluvial granular soils. The alluvium is divided into m with similar characteristics as the embankment fill; and a lower

is also poorly graded sand but with a larger percentage of fines than um. The bedrock surface is slate, found at depths from 7.5m to 24 m

ankment and Upper Alluvium materials are classified as uniform SPin accordance with the Unified Soil Classification System composed f medium sand that contain on average 5% fines. SPT values d is loose sand. Figure 2 shows a cross section of the dam including nd the normal and extreme phreatic levels.

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385

390

395

400

405

410

415

-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40

Figure 2. Cross-section of the damprior to slope failure.

Given the gradations and uniformity coefficients, we estimated that the maximum hydraulic conductivity of the embankment soil is 8.5 x 10-2 cm/sec. An embankment with such a high hydraulic conductivity would have a relatively flat phreatic surface and high seepage exit point on the downstream slope. Phreatic levels in the piezometers screened within the embankment confirm this characteristic. Phreatic levels measured within the foundation indicate generally lower potential than those measured for shallower intervals within the embankment fill.

Analysis Methodology

Stability Analysis Methods. This study utilized two different methods to analyze slope stability: limit equilibrium method and continuum method. These two methods wereselected to allow for comparison of each method and to provide better insight into the overall behavior of the dam and the variables that may be controlling factors. The study was conducted in three stages; 1) evaluate slope stability of the dam with a limit equilibrium approach based on an estimate of the shear strength parameters developed from soil index properties; 2) similarly determine the factor of safety for slope stability using the continuum method with the same shear strength parameters, and 3) back calculate shear strengthbased on the observed failure surface in the field under extreme flooding event using both the limit equilibrium approach and continuum approach for comparison. The results of the analyses helped to assess the performance of each analysis method.

The program UTEXAS4 developed by Wright (1999) was used to assess the limit equilibrium stability of the dam. The limit equilibrium method used in this evaluation employs the method of slices and Spencer (1967) solution to estimate a factor of safety for the slope. The Spencer’s solution method is considered a rigorous solution method in that it requires force and moment balance before a solution is considered valid.The computer program FLAC4.0 developed by the Itasca Consulting Group (2002) was used for the continuum analysis of slope stability. FLAC (which stands for Fast Lagrangian Analysis of Continua) is a model that computes the stresses and resultant strains in continua by a finite difference method. It uses an explicit solution method. Hence, the usual problems of numerical

Slate

Lower Alluvium

Upper AlluviumFill

Dam Breach Flood

Normal Pool


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instability commonly encountered with implicit finite difference codes are not as pervasive with FLAC. FLAC also has the ability to model groundwater flow in either coupled or uncoupled mode with mechanical analysis of deformations. Lagrangian analysis allows for distortion of the grid, so that the end state at each node is the beginning state of the next stress cycle.

There are a number of methods that could have been employed to determine the factor of safety using FLAC. The FLAC shear strength reduction (SSR) method of computing a factor of safety performs a series of computations to bracket the range of possible factors of safety. Cala and Flisiak (2001) provide a good overview of the method. During SSR, the program lowers the strength (φ angle) of the soil and computes the maximum unbalanced force to determine if the slope is moving. If the force unbalance exceeds a certain value, the strength is increased and the original stresses returned to the initial value and the deformation analyses recomputed. This process continues until the force unbalance is representative of the initial movement of the slope and the φ angle for this condition is compared to the φ angle available for the soil to compute the factor of safety.

The flow model can be incorporated into the FLAC analysis as either coupled or uncoupled flow. In the coupled flow model, flow and mechanical deformations are computed in alternate time steps by default. The advantage of the coupled method is that deformation in the slope may impact the flow characteristics and vice-versa. Alternatively, the two computations can be uncoupled and a steady seepage condition can be modeled without deformation. Once the steady seepage condition is reached, the flow can be turned off and the mechanical deformation computations proceed independently. The coupled mode may be a better method for modeling situations where pore pressure dissipation plays and important role in the deformation process. However, the embankment under consideration is composed of freely draining materials. Hence, the uncoupled flow model was selected for the current study. This is more analogous to the UTEXAS4 model since phreatic levels are static during the analysis and a simple Mohr-Coulomb failure criterion is employed.

Model Discretization. Two slope geometries were considered in the model study: Case 1- conditions existing just prior to slope failure (Original Slope), and Case 2- conditions existing after placement of a reverse filter that was used to stabilize the slopefollowing the 2003 event (Repaired Slope).

The dam was modeled as two-dimensional layers of uniform isotropic material in both UTEXAS4 and FLAC, with the base of fill located at an elevationdetermined from the site exploration data. The underlying alluvium was divided into an Upper and Lower Alluvium due to the minor variation in unit weight between the Upper Alluvium and the embankment fill. The two layers of alluvium were also assumed to be uniform and isotropic. Due to the fact that the existing borings do not penetrate to bedrock through the section analyzed, bedrock depth was estimated from borings taken near the dam abutments. Bedrock was assumed to be roughly horizontal and the other contacts between strata were selected from boring logs. The


default program subroutine for vertical discretization was allowed to perform the vertical discretization in UTEXAS4. The UTEXAS4 model runs were all performed with the same discretized model for the pre-failure conditions and only the top slope surface was modified for the remediated slope condition. These two cases were discretized separately in the FLAC model. Figures 3 and 4 illustrate the discretization used in the FLAC model for the Case 1 and Case 2 slope geometries, respectively.

Figure 3. P

Figure 4. Pos

Only the pre-failure calculated shear strength. underlying material, the reThis variation does not impno impact on the back calculatto the repairs. As it turnedembankment fill in the Fpredominately on the respoe

Boundary Conditionsbedrock at an elevation of 387strong characteristic and ththis boundary. In FLAC, thboth x and y directions. Nomodeled as a no-flow boundboundaries fell beyond the and therefore circle searchthrough these boundaries.

m


15

re-failure slope grid configuration.

m
15
5

t-failure slope with the reverse filter.

slope geometry was used in comparison of backDue the similarity in mechanical properties with the verse filter was not differentiated in the FLAC model. act the analyses conducted for this comparison, as it has ed φ angles or to the factors of safety for the slope prior out, the critical failure surfaces are all confined to the

LAC model. Therefore, all comparisons are based ns of the embankment fill.

. The lower boundary was assumed to be the slate

.10 m. In UTEXAS4, this boundary is assigned a very erefore failure circles were precluded from passing below e 387.10 m elevation was assigned a fixed condition for strains were allowed along this boundary and it was

ary for the decoupled flow model. The left and right circle search quadrant employed in the UTEXAS4 model es for minimum factor of safety were not allowed to pass By default, UTEXAS4 only searched the left half of the

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slope that corresponds to the downstream slope. In FLAC, the left and right boundaries were fixed in the x direction only.

Groundwater boundary conditions were modeled with a piezometric line in UTEXAS4. The piezometric linewas based on actual piezometer readings for the spring 2002 flood and was estimated as described below for the 2003 dam break condition. Groundwater boundary conditions for the flow portion of the FLAC model assumed a pore pressure distribution equivalent to the piezometric levels measured during the 2002 spring flood and estimated from piezometer levels measured the day after the dam break flood of 2003. Since the reservoir pool was 0.3 m higher during the dam break event as compared to the day after, the piezometric levels were adjusted 0.3 m higher to represent the day of the peak flood.

The FLAC model phreatic levels were developed with the uncoupled flow model by assigning tail water and head water pore pressures at the far edges of the model. The resulting phreatic levels closely approximate conditions during the day of the dam breach. Utilizing the water table method rather than the uncoupled flow routine and revising unit weights to represent saturated and unsaturated areas, may result in a better simulation of the phreatic surface. Modeling the phreatic change in response to reservoir elevation was considered an important part of the analysis; hence, the uncoupled groundwater flow model was employed within FLAC.

Soil Parameters. The Mohr-Coulomb failure criterion was assumed for both the UTEXAS4 and FLAC models. It is possible to incorporate other failure criteria and constitutive models in FLAC model. Since UTEXAS4 slope stability is based on the Mohr-Coulomb failure criteria, for ease of comparison, the same Mohr-Coulomb criteria was selected for all layers in the FLAC model. The shear strength utilized in the initial analyses was estimated using dry density and relative density developed from site exploration data and checked by back calculating a φ angle from an infinite slope analysis. A back-analysis using the infinite slope method was performed utilizing the steep slope at the critical slope section, where a shallow failure is occurring in the embankment. A factor of safety of 0.99 was assumed and the drained shear strength of the fill was determined to be φ=31.7 .̊ This estimate agrees well with the lower bound 31˚ estimated from the relative density and dry density. Table 1 shows the soil parameters used in the analyses. In the repaired slope, the reverse filter consists of a 0.6 m thick layer of gravel sized material overlain by approximately 1.2 m of sand at the toe. The shear strength properties for the reverse filter were estimated from published values for similar granular materials.

The FLAC model also requires elastic moduli values. The bulk modulus (K) was estimated to be 492,588 kg/m2 for the embankment fill and Upper Alluvium and 726,945 kg/m2 for the Lower Alluvium. The shear modulus (G) was estimated to be 205,256 kg/m2 for the embankment fill and Upper Alluvium and 545,221kg/m2, for the Lower Alluvium. These values were estimated based on the Young’s modulus estimated based on SPT results with an assumed over-consolidation ratio of 1.0.


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Table 1. Soil parameters assumed in the analyses

Embankment Fill

Upper Alluvium

Lower Alluvium

CaseNumber

Loading Conditions

φφφφ c γ(kg/m3)



1 Spring Flood 31̊ 0 2015 31˚ 0 2068 36˚ 0 1945

2 Dam Breach Flood

31̊ 0 2015 31˚ 0 2068 36˚ 0 1945

A hydraulic conductivity value of 8.5 x 10-2 cm/sec was used for the embankment fill and for the underlying Upper Alluvium from which the fill material was derived. The Lower Alluvium was assigned a value of 1 x 10-2 cm/sec. No allowance was made for variations in horizontal or vertical hydraulic conductivity in the model.

For the back-calculation of shear strength based on the observed failure surface in the field, the φ angle was varied under the pre-failure slope condition for the dam break reservoir pool elevation. A series of runs were performed until a φangle was found which resulted in a factor of safety of slightly less than 1.0. This method of solution assumes the slope failed simultaneous with the reservoir pool peak.

Loading Conditions.The reservoir rose rapidly and peaked at an elevation of 411.16 m (all elevations are relative to 1929 Mean Sea Level) after an upstream dam breached during the spring of 2003. For comparison, the spring flood of 2002 rose to a peak reservoir elevation of approximately 409.96 m. The phreatic level at the toe of the dam corresponding to these two flood events increased a total of 1.17 m above background levels in 2002 and 1.89 m during the 2003 event.This difference of 0.72 m resulted in slope failurefor the 2003 event. In order to model these two events,two loading conditions were evaluated: Dam Breach Flood of 2003, and Normal Spring Flood of 2002. For comparison of theUTEXAS4 and FLAC back calculatedshear strengths, the dam breach flood of 2003 loading condition was used. The phreatic levels were assigned based on the piezometer measurements taken during each of the events.


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Results and Discussion

Comparison of Limit Equilibrium and Continuum Model Results . The factor of safety for the original slope subjected to a normal spring flood in the UTEXAS4 analyseswasabout 0.90 (φ=31̊ ), which is considered too low for normal spring flood conditions since the slope remained stable during previous spring flood loading. By comparison, the FLAC model yielded a factor of safety of 1.02 using a φangle of 31̊. This appears to be a more realistic number, assuming the shear strength is 31˚, in that the sloperemained stable during the normal spring flood condition(Figure 5).

FLAC (Versio n 4.00)

LEGEND

4-Dec-03 20:59step 141427

Cons. Time 1.6254E+058.259E+01 <x< 1.164E+021.320E+03 <y< 1.354E+03

Velocity vectorsMax Vector = 1.339E-09

0 2E -9

X-displacement contours 2.00E-02 3.00E-02 4.00E-02 5.00E-02 6.00E-02 7.00E-02 8.00E-02 9.00E-02

Contour interval= 1.00E-02

1.32 2

1.32 7

1.33 2

1.33 7

1.34 2

1.34 7

1.35 2

(*10^ 3)

0.85 0 0. 900 0.95 0 1. 000 1.0 50 1 .100 1.1 50(*1 0^2)

JOB TITLE : Close ViewSlope Displacements Following 2002 Spring Flood

Universityof Il linois ChicagoChicago, IL USA

Figure 5. FLAC analysis results for the 2002 spring flood (yields a factor of safety of 1.02 with φφφφ=31̊ ).

For the dam breach event, the potential failure surface in the FLAC model is similar to that of UTEXAS4, which can be seen by comparing Figures 6 and 7. These are similar failure modes in that they both represent shallow failure of the steeper portion of the downstream slope and are located in the same area. Both cases show that the failure is confined to the embankment fill, and does not involve the underlying Upper Alluvium or penetrate the crest of the dam. These results also correlate with the results of the infinite slope analyses, which show that the factor of safety of the steep part of the slope is 0.99 for a φ angle of 31.7̊. The computed factor of safety is 0.81 and 0.89 with UTEXAS4 and FLAC methods, respectively. The UTEXAS4 estimate for the factor of safety appears to be more conservative than the FLAC analysis.


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Figure 6. UTEXAS4 analysis results for the dam breach event (yields a factor of safety of 0.81with φφφφ=31̊ compared to 0.90 for the 2002 spring flood).

FLAC (Version 4.00)

LEGEND

4-Dec-03 21:58s tep 284 494

Con s. Ti me 3.7235E+057 .222E+ 01 <x< 1.1 71E+021 .319E+ 03 <y< 1.3 64E+03

Velocity vectors

Max Vecto r = 2.50 0E-09

0 5E -9

X-displacemen t contours

0.0 0E+ 00 2.50 E-0 1 5.00 E-0 1 7.50 E-0 1 1.0 0E+ 00 1.2 5E+ 00 1.5 0E+ 00 1.7 5E+ 00 2.0 0E+ 00

Contou r interval = 2.50E-0 1

1. 320

1. 325

1. 330

1. 335

1. 340

1. 345

1. 350

1. 355

1. 360

(*10^ 3)

0 .750 0.80 0 0 .850 0 .900 0.95 0 1.00 0 1. 050 1 .100 1.1 50(*10^2)

JOB TITLE : Close V iew Slo pe Disp lacemen ts Fol lowing 2003 F lood

Universit y of Illinois Chi cagoChi cago, IL US A

Figure 7. FLAC analysis results for the dam breach event (yields a factor of safety of 0.89 with φφφφ=31̊ compared to 1.02 for the 2002 spring flood).

The rapid increase in phreatic level brought about by the increase in reservoir

elevation caused increased pore pressure to develop near the toe of the dam. This was evident by the phreatic levels measured during the event and sudden occurrence of artesian conditions in some of the shallow piezometers. The increased pore pressure caused the factor of safety to decrease from 0.90 to 0.81 in the UTEXAS4model. This is about a 10% decrease from the normal spring flood condition to cause failure in the slope, which indicates the initial slope configuration must have had a factor of safety close to 1.00. The factor of safety estimated by the FLAC analyses is 0.89 during the dam breach loading condition, which represents a decrease of 12.7% from the normal spring flood condition factor of safety of 1.02. This decrease is on the same order of magnitude as observed with UTEXAS4, although the FLAC model

Approximate Failure Surface


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may be slightly more sensitive to phreatic conditions. An interesting point is that if the φ angle of 31 ̊ is used, the FLAC model shows that the slope was stable during normal pool and becomes unstable during the dam breach event, which corresponds well to what was observed in the field. The UTEXAS4 model indicates the slope is unstable for both scenarios. Comparing factors of safety between the two methods shows that UTEXAS4 (0.81) is approximately 10 % lower than FLAC (0.89) for the dam breach flood.

These two methods utilize very different approaches to estimate the factor of safety, which may explain the different results obtained in this study. UTEXAS4 is based on a limit equilibrium approach that assumes mobilization of shear strength along the entire failure surface, beginning with iteration of balanced forces for each slice in the Spencer solution method. The FLAC model determines a factor of safetybased on a defined maximum unbalanced force for a failure surface that is not limited to a simple geometry such as a circular surface, utilizing the stress/strain characteristics for the soil.

Back-calculation of φφφφ angles. Both models, UTEXAS4 and FLAC, were used to back calculate φ angle. The method employed was to assume the slope failure that occurred during the 2003 dam breach event occurred at a factor of safety slightly less than 1.0. The actual failure may have occurred at lower pool elevations than assumed in the model, but exact information about the timing of the failure and corresponding pool elevations is not known. This method may result in a back calculated φ angle that is high, but should be good for comparison purposes between the two models.

UTEXAS4 yielded a back calculated φ angle of 36.5̊ for the embankment fill subjected to a dam breach pool elevation of 411.16 m. The factor of safety under these conditions was 0.99, which was interpreted to be initiation of failure. This resulting back calculated φ angle appears to be unconservative. Based on the previous site investigations and laboratory analysis, it does not appear that the embankment would have a φ angle this large. A series of back calculations were performed using the FLAC model, ranging from the 31̊ estimated failure criteria discussed above, to the 36.5˚ estimated from UTEXAS4. The results from these runs are summarized in the table below.

Table 2. Comparison of back calculated φφφφ angles using various methods.

Factors of Safety for 2003 Dam Breach Pool Elevationsφφφφ angle Infinite Slope UTEXAS4 FLAC

31̊ 0.81 0.8931.7̊ 0.9933̊ 0.9634̊ 1.01

36.5̊ 0.99 1.12


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The interpolated back calculated φ angle for the FLAC analyses is approximately 33.5̊. The FLAC analysis predicts a friction angle 3̊ lower than UTEXAS4, or approximately 10 percent lower. In this case, the φ angle back calculated usingthe FLAC model appears to be more conservative.

Displacements. To better understand the FLAC results, a history point was plotted in the area of the scarp. The y-displacement history for this point on the slope indicates approximately 0.6 m of downward movement after failure in the model. The actual movement of the actual scarp following the 2003 dam breach event was on the order of 0.3 m. The FLAC model probably overestimated strains due to lower elastic moduli values used in the analysis. A better estimate of the elastic moduli values would have resulted deformations more in line with those observed in the field. This illustrates the sensitivity the FLAC model to elastic parameters, which can be difficult to determine for granular materials, but also highlights the ability to calibrate the model based on measured movements.

Conclusions

Limit equilibrium methods have been the most common methods employed by engineers to estimate the stability of slopes in the recent past. However, with the advent of personal computers, continuum methods such as FLAC have become a reasonable alternative. The amount of time required to set up and perform a continuum model is similar to that required to perform a limit equilibrium analysis; however, there are some advantages with the continuum approachin that it is easy to model noncircular failure surfaces. One drawback with the continuum models is that additional parameters such as elastic moduli must be known, which can be difficult to determine for granular soils. However, field measurements of actual movements can be used to help calibrate the model parameters.

For the embankment dam analyzed in this study, factors of safety determined from the continuum model, utilizing reasonable estimates of the φ angle derived from index properties of the soils, appear to correlate well with the actual behavior of the slope. The limit equilibrium method yielded lower values for the factors of safety (employing Spencer’s solution method), which could result in more conservative designs based on the limit equilibrium method for a given shear strength. However, as a tool for back calculation of shear strength, the continuum method yields more conservative values. The difference in back calculated φ angles and computed factors of safety between the two methods wasabout 10%. The sensitivity of each method to phreatic levels was also very close. It was on the order of 10% and 13% drop in the factor of safety for the higher phreatic condition for the UTEXAS4 and FLAC models, respectively.

References

Cala, M., and J. Flisiak (2001). “Slope stability analysis with FLAC and limit equilibrium methods”, in FLAC and Numerical Modeling in Geomechanics,


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Billaux et al. (eds), Swets and Zeitlinger, ISBN 90 2651 859 5.Itasca Consulting Group, Inc. (2002). “FLAC- Fast Lagrangian Analysis of

Continua”, Volumes 1-8 of program documentation, Minneapolis, Minnesota.Spencer, E. (1967). “A method of analysis of the stability of embankments assuming

parallel interslice forces”, Geotechnique, Vol. 17, No. 1, 11-26.Wright, S.G., (1999). “UTEXAS4- A computer program for slope stability

calculations”, Shinoak Software, Austin, Texas.


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