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Kinetic Energy Method for Predicting Initiation of

Backward Erosion in Earthen Dams and Levees

KEVIN S. RICHARDS1

U.S. Army Corps of Engineers, Institute for Water Resources, Risk ManagementCenter, Pittsburgh, PA, 15222

KRISHNA R. REDDY

Department of Civil and Materials Engineering, University of Illinois at Chicago,842 West Taylor Street, Chicago, IL 60607

Key Terms: Piping, Internal Erosion, Suffusion,Earthen Dams, Levees, Kinetic Energy

ABSTRACT

Current methods to evaluate the potential risk ofearthen dam and levee failures by internal erosion failto consider appropriate failure criteria for theinitiation, continuation, and progression phases andare often based on subjective assessment. Six differentinternal erosion failure modes may occur within a damor levee, its abutments, or the foundation that aretriggered or facilitated by different mechanisms andtherefore have different failure criteria. In non-cohesive soils, suffusion, backward erosion, heave, orconcentrated leak erosion (scour) are possible, al-though the physical mechanisms driving the failurecriterion for each of these are very different.Currently, no credible failure criteria exist forevaluation of the initiation of backward erosion. Thisarticle presents derivation of a specific failurecriterion for initiation of backward erosion in non-cohesive soils using the critical kinetic energy (Ekcrit)of initiation. Laboratory experiments conducted withdifferent soils showed that Ekcrit is affected by thephysical characteristics of the soil, effective stressconditions, and angle of the seepage path. Ininternally stable, non-cohesive soils (e.g., uniformmedium sands) with up to 6 percent non-plastic fines,backward erosion initiated when the Ekcrit of seepageexceeded 0.075 Joules. In non-cohesive soils with 10percent non-plastic fines, which are also prone tosuffusion, the Ekcrit for backward erosion is reducedby a factor of as much as 100. Moreover, in soil with20 percent plastic fines, initiation of backward erosiondid not occur. Concentrated leak erosion alonghydraulic-induced fractures was the dominant process

of internal erosion in soils with plastic fines. A newbackward erosion assessment methodology based onthe factor of safety is proposed for initiation ofbackward erosion that relates the laboratory-derivedEkcrit and the anticipated maximum actual kineticenergy, as measured in the field or estimated duringdesign. The advantage of this method is thatlaboratory-derived Ekcrit can take into account thenatural factors that affect initiation of backwarderosion. Several examples are presented to demon-strate the application of the factor of safety method-ology under typical field conditions.

INTRODUCTION

There are more than 85,000 dams in the UnitedStates, and more than 4,000 of these dams areconsidered deficient (ASCE, 2009). The number ofdeficient dams has been increasing at a rate of abouttwo dams per each high-hazard dam that is repaired(ASCE, 2009). Foster et al. (2000) summarized thefailure statistics of dams taken from the Interna-tional Commission on Large Dams and otherstudies and found that 46 percent of all dam failurescan be attributed to some form of internal erosion.They reported that 35 percent of reported damfailures and 59 percent of the incidents caused byinternal erosion occurred after the 5 years the damwas put in service. These statistics highlight theneed for continued surveillance for changes inseepage and potential for internal erosion in olderdams.

A number of different internal erosion failuremodes may occur within a dam or levee, itsabutments, or the foundation that are triggered orfacilitated by different mechanisms (Richards andReddy, 2007). ‘‘Internal erosion’’ is used in a generalsense to indicate any process that causes erosionwithin a soil mass. Several different internal erosionfailure modes have been defined in the literature, as1Corresponding author email: [email protected].

Environmental & Engineering Geoscience, Vol. XX, No. 1, February 2014, pp. 85–97 85

follows: (a) heave—the process of displacing a soilmass or soil particles as a result of an imbalance inseepage-induced uplift forces; (b) concentrated leakerosion (scour)—whereby erosion initiates along thewalls of a forming crack, geological or structuralcontact, or voids triggered by settlement or construc-tion defects (this is the most common failure modeand occurs in particular where conduits or otherpenetrations are present through a dam or where soilembankments are constructed on rock foundations);(c) tunneling or jugging—usually a vadose zoneprocess in natural slopes or a process that occurs inembankments or foundations composed of dispersivesoils (Sherard and Decker, 1977); (d) backwarderosion—the process of creating a pipe by progressiveerosion beginning at an exit point and workingbackwards toward the reservoir or the surface of thedam. It is triggered by intergranular (Darcy-flow)seepage and progressive erosion; (e) suffusion—aspecial case of backward erosion peculiar to gapgraded soils (also known as internally unstable soils)with a higher percentage of coarse grains that areskeletal-grain supported soils, whereby finer matrixsoil suffuses through pores between coarser skeletalgrains; and (f) suffosion—which is similar to back-ward erosion but which occurs in a gap graded soilthat is not grain supported by coarse skeletal grains.While each of these definitions describes a specificseepage-related failure mode, they might also occur insome combination or occur at different locationswithin the dam or foundation.

Foster (1999) evaluated the probability of failure ofdams by internal erosion and defined a three-stageprocess consisting of initiation, continuation, andprogression. Fell et al. (2001a,b) began to assess eachstage of internal erosion individually and as asequence of events necessary to ultimately result in adam breach. Initiation occurs when physical condi-tions have reached a critical state and soil begins toerode. The erosion may continue if self-filtration orplugging doesn’t ensue. If the erosion continues, avoid will begin to form and gradually progress. In thecase of backward erosion, the void progresses towardthe reservoir while it enlarges. In the case of internalerosion, the void gradually enlarges during progres-sion. Progression may either be permanently ortemporarily halted if the soil cannot support a roofover the void.

The U.S. Army Corps of Engineers (USACE)began to utilize risk-based approaches in the evalu-ation of geotechnical structures in 1996 (USACE,1996). The U.S. Bureau of Reclamation formed aRisk Cadre in 1998 to develop a risk analysis processfor dam safety that further defined initiation,continuation, and progression of internal erosion

and began to assign risk based on specific siteconditions that affect each of these stages (Bowles etal., 1998). These efforts have resulted in new methodsfor assessment of internal erosion failure modes basedon risk (USBR, 1999, 2000) and the recognition thatthere are deficiencies in the current state of practice.Risk assessment is also being adopted by state damsafety engineers (ASDSO/FEMA, 2001). The meth-odologies for assessing seepage-related risks dependon the prediction of initiation of backward erosion,and better tools are needed to assess the probabilityof initiation of backward erosion.

Internal erosion is a significant failure mode forolder dams, and the number of deficient dams isincreasing at an alarming rate. When a change inseepage is observed at an older dam, particularly onewithout modern defensive measures such as filters orcores, one must know how to properly evaluate andrespond to the condition. Richards and Reddy (2008,2010, and 2012) performed laboratory experiments toevaluate the initiation of backward erosion and toassess the effect of various loading and soil gradationsand plasticity. This article expands upon that workand derives a new methodology based on kineticenergy to assess backward erosion initiation potentialin existing earthen dams and levees constructed ofnon-cohesive soils (with plasticity indices below 7).

THEORETICAL BASIS FORBACKWARD EROSION

Internal erosion processes can differ, and the mainfocus of this study is the initiation of backwarderosion in non-cohesive soils. External forces, such asdrag induced by intergranular seepage, gravity, andfrictional resistance between grains, operate onparticles subjected to seepage. The relationshipbetween interparticle forces and critical shear stressesdepends on whether the particle is rolling or sliding(Bonala, 1997) when backward erosion is initiated.Sharma et al. (1992) showed that particles are releasedvia rolling rather than sliding or lifting whensubjected to hydrodynamic forces. Reddi and Bonala(1997) proposed the following limit-equilibriumexpression (derived from the work of Visser [1976];Hubbe [1984]; and Sharma et al. [1992]) for the torquebalance on a rolling particle in terms of the resistingtorque from interparticle bond strength (F 3 a) andthe driving torque (bR3tc) due to the drag forcetransmitted to the particle surface from seepage shearstress (Figure 1). The equation neglects any lift force,which is appropriate for sufficiently slow viscousshear flow in a statistically disordered porous medium(Fernandez, 1974):

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86 Environmental & Engineering Geoscience, Vol. XX, No. 1, February 2014, pp. 85–97

F~bR3

a

� �tc, ð1Þ

where F is the interparticle bond strength at initiation(N); R is particle radius (m), assuming sphericalparticles; b is a unitless proportionality coefficient(44.84); a is the radius of particle contact area (m);and tc is the critical shear stress (kPa).

The term b in the above equation is a combinationof unitless terms that have been derived for estimationof the tangential force transferred to a particle byshearing fluids derived by O’Neill (1968) and forapplication of the resultant force in a rolling particlederived by Hubbe (1984). The critical shear stress (tc)is the hydraulic-induced shear stress at which the soilparticle initially begins to roll. For the purpose of thisderivation, the critical shear stress is considered equalto the boundary shear stress in porous media, and theshear velocity that induces the shear on the soilparticle is assumed to be equal to seepage shearvelocity. Hence, in terms of seepage shear velocity,hydraulic shear stress at the soil-fluid contact iscommonly expressed as follows (Shields, 1936; Wil-cock, 1996):

t~rf v2, ð2Þ

where t 5 hydraulic shear stress (Pa); rf 5 fluiddensity (kg/m3); and v 5 seepage shear velocity (m/s).Combining Eqs. 1 and 2, it can be seen that

interparticle bond strength (F) is directly proportionalto the square of the critical seepage velocity. Hence,

F~bR3

a

� �rf v2

c , ð3Þ

where vc 5 critical seepage velocity (m/s).The mass of the soil partcle is proportional to the

cube of its radius (assuming spherical particles), thus:

Mp~rp

4

3

� �pR3, ð4Þ

where Mp 5 mass of the particle (kg); rp 5 particledensity (kg/m3); and R 5 particle radius (m).

Combining Eqs. 3 and 4, the limit equilibriumequation becomes

2p

3

� �a

b

� �F~

1

2

� �rf

rp

!Mpv2

c : ð5Þ

Based on the above equation, the interparticle forcesat the limit state are initially overcome by hydraulicforces that are directly related to the square of thecritical velocity.

The law of conservation of energy can be used tofurther assess particle movements during initiation oferosion. Energy is required to perform the initialwork for soil particle movement. Excluding friction-induced thermal changes in accordance with Bernoul-li’s Law, there are three types of hydraulic energygenerally available to do the work of initiation oferosion in a porous medium:

Etot~EizEvzEp, ð6Þ

where Etot 5 total hydraulic energy (Joules [J]); Ei 5

energy contribution due to hydraulic gradient (J); Ev 5

energy due to seepage velocity (J); and Ep 5 pressureenergy. Each of the energy components is derived fromBernoulli’s equation in terms of the elevation head,pore velocity head, and hydraulic pressure head. Ingeneral terms, Bernoulli’s equation is

rghz1

2

� �rv2zp~constant: ð7Þ

In porous media, the total system energy includesthe energy of both the liquid and solid phases. Hence, ifthere is an increase of kinetic energy in the solid phase,such as when a particle is dislodged, the sum of theenergy of the other components must decrease an equalamount to maintain constant total system energy.

Figure 1. Particle hydrodynamic torque balance (F a 5 1.4 FH

R), where the total force from the wall shear stress (FH) acting on aparticle at radius R is in torque balance with the interparticle bondforce, F, acting on the particle base. The unitless factor 1.4accounts for pressure distribution on the particle. The equilibriumis a function of the rolling mechanism, where a is the radius of thecontact area (Reddi and Bonala, 1997).

Kinetic Energy Method


The limit equilibrium equation (Eq. 5) does notinclude terms for pressure or elevation head but isstrictly related to the seepage velocity, particle mass,and ratio between fluid and particle density. Anincrease in seepage velocity above an equilibriumlimit is required to overcome interparticle forcesand to initiate particle erosion by the mechanismdiscussed.

For a unit volume of soil, which contains thevolume of one particle and the surrounding saturatedpore space, Eq. 5 may be rewritten in terms of thekinetic energy of the fluid phase, thus:

(e)2p

3

� �a

b

� �F~(e)

1

2

� �rf

rp

!Mpv2

c

or

(e)2p

3

� �a

b

� �F

Mp

~(e)1

2

� �rf

rp

!v2

c ,

which equals

(e)2p

3

� �a

b

� �F

Mp

~1

2

� �Mf

Mp

� �v2

c

or

eð Þ 2p

3

� �a

b

� �F~

1

2

� �Mf

� �v2

c , ð8Þ

where e 5 void ratio, which is equal to (Vvoids/Vsolids),where Vvoids 5 volume of voids and Vsolids 5 volumeof solids.

The kinetic energy of seepage at initiation ofbackward erosion or critical kinetic energy (Ekcrit),which is equal to the right term in Eq. 8, may beexpressed as

Ekcrit~1

2

� �Mf

� �v2

c~fF , ð9Þ

where f 5 a lumped parameter that includes thegeometric factors (e, a, and b; all non-dimensionalconstants except for a, which is in meters); F 5

interparticle resisting force (N); and Ekcrit~12

� �Mf v2

c (J).

The magnitude of the critical kinetic energy isproportional to the interparticle bond strength (F)and a geometric factor (f). The geometric factor isdependent on the soil characteristics, such as voidratio, grain-to-grain resisting force, and grain size.These parameters vary from soil to soil, so the Ekcrit isdependent on soil characteristics.

Eq. 9 implies that only a change in the kineticenergy of seepage is required to overcome theinterparticle bond strength to initiate backwarderosion. Hence, in terms of Bernoulli’s energyequation, one might expect that the increase inparticle kinetic energy required to initially removesoil particles from an at-rest condition duringbackward erosion is balanced by an equal decreasein the kinetic energy of the seepage fluid.

LABORATORY EXPERIMENTS TO STUDYBACKWARD EROSION

In order to quantify the kinetic energy requiredfor the initiation of backward erosion, several labor-atory tests were conducted on different soils usinga newly developed True Triaxial Piping Test Appa-ratus (TTPTA) (Richards and Reddy, 2010, 2012).The selected stress conditions in these experimentswere representative of typical conditions that mightbe encountered in dams of up to 9 m (30 ft) inheight.

TTPTA DESCRIPTION

The TTPTA is a rectangular box constructed of2.54-cm (1-inch)–thick aluminum plates with threeinflatable load cells that apply confining stresses tothe soil in three mutually perpendicular axes andpressurized inlet and outlet for introduction of seepagewater at controlled pore pressure and hydraulicgradient. Figure 2 shows the overall schematic of thetesting system, and Figure 3 shows the true triaxialload cell with the soil sample. A detailed description ofthe TTPTA equipment and test procedure wasprovided by Richards and Reddy (2010).

TESTING PROGRAM

Both natural and prepared mixed soils were used toassess the critical kinetic energy of backward erosion.Particle size analysis was performed (ASTM, 2007),and the soils were classified using the Unified SoilClassification System (ASTM, 2000). The grain sizedistributions of the natural soils and mixed soils areshown in Figure 4a and b, respectively. The naturalsoils are classified as SW, ML, CL-ML, and SP,representing a wide range of soil types. Mixed soilswere prepared using 80–90 percent natural, uniform,medium sand mixed with either 10 percent or 20percent kaolin (to represent low plasticity fines) or 10percent or 20 percent montmorillonite (to representhigh plasticity fines). The Plasticity Index (ASTM,2010) for the samples ranged from non-plastic to 84.

Richards and Reddy


Figure 2. True Triaxial Piping Test Apparatus (TTPTA) consists of a True-Triaxial Load Cell in which the sample is tested. Pressurizedvessels control pore pressure at the inlet and outlet, an Air Bladder Pressure Control Panel controls the confining stresses, and an Inlet-FlowControl Panel increases inflow during the test. The Scale records the mass-discharge rate, which is used to compute the critical velocity. Notshown is the instrumentation that measures pore pressures and hydraulic gradient in the Load Cell.

Figure 3. Photograph of the loaded True-Triaxial Load Cell (without lid), filled with soil prior to the test. Note the granular filter at theinlet end and the turbidimeter at the outlet end. Quick connect in photograph is used for attaching the pressurized air line used to controlthe intermediate principal (s2) confining stress. Similar connections are used for the s1 and s3 directions (not shown).



Figure 4. Gradations of test soils: (a) Natural field soils; (b) Laboratory mixed soils.

Richards and Reddy


The natural soils were also tested for dispersivity usingthe pin hole (ASTM, 2006) and double hydrometertest (ASTM, 2005), and they were confirmed to benon-dispersive.

Test parameters, such as confining stress, porepressure, and seepage angle, were varied to assesshow these parameters may affect the critical velocity toinitiate backward erosion. Confining stresses and porepressures were selected to represent normal stressstates at depths of 3, 6, and 9 m (10, 20, and 30 feet)below an embankment crest. Test results are presentedand discussed in detail by Richards and Reddy (2012).

KINETIC ENERGY CALCULATION

Based on the laboratory experiments, seepagevelocity (vs) and the kinetic energy (Ek) of the seepageare calculated as follows:

vs~q

Av

~vA

Av

, ð10Þ

where vs 5 pore seepage velocity (m/s); q 5 discharge(flow rate; m3/s); v 5 computed Darcy flow velocity(5 q/A; m/s); A 5 total cross section of the flow area(5 outlet tube inside area; m2); and Av 5 voidseffective cross sectional area (5 nA, where n isporosity; m2). Eq. 10 can be simplified as follows(substituting Av for nA):

vs~vA

Av

� �~v

A

nA

� �~

v

n: ð11Þ

For a unit volume of soil (1 m3), the mass ofpercolating water is equal to the porosity times thedensity of water (998 kg/m3 at room temperature).Taking this into consideration and substituting theseepage velocity of Eq. 11 into Eq. 9, we obtain

Ek~1

2

� �Mf v2

s~1

2

� �998nð Þ v2

n2

� �

~1

2

� �998ð Þ v2

n

� �~499

v2

n

� �,

ð12Þ

where v 5 the measured Darcy velocity from the TTPTAtest (m/s); n 5 the porosity of the soil (non-dimensional);Mf 5 n 3 1 m3 3 998 kg/m3 (kg); and Ek 5 the kineticenergy of seepage per unit volume of soil mass (J).

CRITICAL KINETIC ENERGY TO INITIATEBACKWARD EROSION

Eq. 12 was used to compute the kinetic energy ofseepage at the point at which backward erosion wasinitiated in each laboratory experiment. This kineticenergy is defined as the critical kinetic energy (Ekcrit) atwhich seepage overcomes the interparticle resisting

Figure 5. Relationship between critical kinetic energy (Ekcrit) andstress conditions, as found in the laboratory soil QS when otherparameters were kept equal: (a) critical kinetic energy as a functionof average effective confining stress (p9), (b) critical kinetic energyas a function of normalized deviatoric stress, and (c) criticalkinetic energy as a function of the effective maximum principalstress (all tests done in non-buoyant states).



forces and the soil particles are mobilized. The criticalkinetic energy required to initiate internal erosion wascalculated for each soil type, including those soils thatare less prone to backward erosion (e.g., uniform sandsoil mixed with 20 percent montmorillonite). It wasfound that the kinetic energy–based method was notappropriate for the soils with plastic fines because theTTPTA measures the velocity of seepage, and theseepage velocity is very low as a result of the lowpermeability of the soil due to plastic fines. In themontmorillonite-added soil, the failure criterion forinternal erosion was high pore pressure–induced crack-ing rather than kinetic energy of interparticle seepage,and the mode of internal erosion is scour. Hence, theresults with montmorillonite-added soils were not usedto assess critical kinetic energy for backward erosion.

The critical kinetic energy calculated based on thelaboratory tests using the uniform, medium-grainedlaboratory soil QS are plotted with respect to effectiveconfining stress, p9 (where, p9 5 [s1 + s3]/2),normalized deviatoric stress, and the effective majorprincipal stress as shown in Figure 5a, b, and c,respectively. Though a strong correlation is not seen,a general trend of increased critical kinetic energywith increase in the normalized deviator stress as wellas the major principal stress is clearly evident.

Figures 6 and 7 show the relationship between thecritical kinetic energy and the initial void ratio of thesoil and the seepage direction, respectively. Thecritical kinetic energy decreased with an increase inthe initial void ratio of the soil (Figure 6). Figure 7shows that the critical kinetic energy increased withan increase in seepage angle with respect to thehorizontal. When the seepage angle was 10u abovehorizontal, the critical kinetic energy was 50 percentgreater than that required when the seepage washorizontal. Seepage at an angle of 10u below horizontalrequired about 50 percent less kinetic energy than didhorizontal seepage (Figure 7).

In order to assess the effects of buoyancy or presenceof pore pressure, the results of tests conducted withcontrolled pore pressure conditions are plotted as theeffective minor principal stress versus the criticalkinetic energy, as shown in Figure 8. Where porepressure was less than the minimum principal stress,tests required significantly higher critical kinetic energyto induce backward erosion. These results also showthat the pore pressure did not play a significant role inlowering the critical kinetic energy for backwarderosion until the soil was in a partial buoyant state(when pore pressure exceed the minimum principlestress), at which point the critical kinetic energy couldbe substantially lower.

The soil type, particularly the amount and types offines, are shown to greatly influence the initiation ofbackward erosion. Figure 9 compares the averagecritical kinetic energy for all soil types tests in thelaboratory—natural field soils (1C, PD, and BC) andnatural soil (1C) mixed with 10 percent and 20percent kaolin (10K, 20K). The critical kinetic energy

Figure 6. Relationship between critical kinetic energy (Ekcrit) andthe initial void ratio found for the laboratory soil QS when otherparameters were kept equal.

Figure 7. Relationship between critical kinetic energy (Ekcrit) andthe seepage angle found for the laboratory soil QS when otherparameters were kept equal.

Figure 8. Relationship between critical kinetic energy (Ekcrit) andthe effective minimum principal stress (difference between porepressure and the minimum principle stress, s39 (buoyant state ats39 5 0).

Richards and Reddy


of field soils varied significantly. The field soil 1Cadmixed with 10 percent to 20 percent kaolin had acritical kinetic energy value that was an order ofmagnitude lower than that of the field soil 1C itself. Adetailed discussion of the effect of soil type oninitiation of internal erosion is given by Richardsand Reddy (2012).

KINETIC ENERGY–BASED FACTOR OFSAFETY METHDOLOGY

Based on the dependence of initiation of backwarderosion on the seepage velocity in laboratory exper-iments (Richards, 2008; Richards and Reddy, 2012),backward erosion in non-cohesive soils was con-firmed to be primarily influenced by the kineticenergy of seepage, as was also demonstrated theoret-ically. Therefore, it may be possible to use the criticalkinetic energy determined from laboratory experi-ments as a basis in the assessment of the initiation ofbackward erosion and to determine a factor of safetyagainst backward erosion initiation.

The critical kinetic energy can be determined fromthe laboratory experiments using apparatus such asthe TTPTA on soil samples representative of aproject site under the expected void ratio, seepageangle, and stress-state conditions. Eq. 12 can be usedto compute the critical kinetic energy at the initiationof backward erosion. Next, available kinetic energyfor the specific earthen dam or levee conditionsunder consideration can be computed from flownets, seepage models, or field measurements of theDarcy seepage velocity and porosity/void ratio of thein situ soils (e.g. Cedergren, 1977; Harr, 1990). Fornon-cohesive soils, the factor of safety againstinitiation of backward erosion is expressed as

FSsuffusion or backward erosion~Ekcrit(lab)

Ekavailable(field): ð13Þ

As explained by Richards and Reddy (2012), thereis a marked difference in internal erosion behaviorbetween non-cohesive soils and soils with plastic fines.Backward erosion is not the main mechanism ofinternal erosion in cohesive soils; therefore, a differentapproach is recommended for soils with a PlasticityIndex greater than 7. The factor of safety for internalerosion of cohesive soils can be expressed alternative-ly in terms of hydraulic shear stresses as

FSconcentrated leak~tcrit

tavailable

, ð14Þ

where tcrit 5 the critical hydraulic shear stress (kPa)and tavailable 5 hydraulic shear stress present in aconcentrated leak (scour; kPa).

Laboratory methods have already been developedby Arulanandan et al. (1975) and Wan and Fell(2002) to determine the critical shear stress requiredfor initiating scour along a concentrated leak, whichis applicable for cohesive soils. The shear stressavailable can be expressed as

tavailable~cwhpi, ð15Þ

where tavailable 5 available hydraulic shear stress(kPa); cw 5 unit weight of water (kN/m3); hp 5

pressure head in a concentrated leak (m); and i 5

available hydraulic gradient (non-dimensional).The factors of safety using Eqs. 14 or 15 can be

computed for a variety of hydraulic loading conditionsthat may be encountered using standard seepagemodels and the tests discussed above. It is importantthat Ekcrit or the tcrit be evaluated using site-specificsoils, using either undisturbed or carefully remoldedsamples to simulate conditions in the field, and thatthey be tested under the in situ stress states expected inthe field. As mentioned previously, the amount andplasticity of fines in soils significantly affect the processof internal erosion in soils; hence, it is critical to test thesoils in the laboratory to determine the amount andplasticity of fines. If seepage angle differs fromlaboratory test conditions, correction factors will needto be applied in order to account for gravity effects.

APPLICATIONS OF KINETIC ENERGY–BASEDFACTOR OF SAFETY METHODOLOGY

In order to assess the applicability of the proposedkinetic energy–based factor of safety method, limited

Figure 9. Comparison of average Ekcrit values for each of the soilstested. 1C, PD, and BC are natural soils. 10K and 20K are the 1Cnatural soil (medium-grained, SP soil) with 10 percent and 20percent kaolin added, respectively, QS is laboratory-qualityuniform sand. Note how the addition of kaolin to the naturalsoil lowered the critical kinetic energy required to initiateinternal erosion.



SEEP-W (2000) modeling was performed to deter-mine a peak XY-gradient (and compute Ekavailable) foran assumed section of homogeneous embankmentdam on a homogeneous foundation (Figure 10). Theembankment section was assumed to have 1H:1V sideslopes, measuring 10 m height, with a narrow 2-m–wide crest. This is a non-standard dam section, withnarrow crest and steep slopes, which was intentionallyselected to yield high exit gradients. Soil parameterswere assumed to be equivalent to the soil QS tested inthe laboratory under similar conditions (Richardsand Reddy, 2012). The hydraulic conductivity (k) atthe toe of the dam is assumed equal to k as determined

in the TTPTA tests for uniform medium sand (QS) atthe field void ratio under low confining stress andhorizontal flow conditions, which was 0.026 m/s. Theporosity (n) was determined to be 0.37 at the given voidratio. The SEEP-W modeling of this section resulted inan XY exit gradient of 1.68 at a 45u upward angle atthe dam toe. The Darcy seepage velocity is thereforecalculated as

v~ki~(0:026 m=s)|(1:68)~0:0437 m=s,

and the available kinetic energy can be estimated,based on Eq. 13 as

Eavailable~499v2

n

� �~499((0:0437)2=0:37)~2:58 J:

The critical kinetic energy found based on theTTPTA testing on the soil was 0.065 J for similar stressconditions with the test being done with horizontalflow. As the laboratory test was for horizontal flow, acorrection was applied to account for the upward angleof seepage (see Figure 7). It should be mentioned thatthe laboratory testing program included seepage anglesup to 610u, and the results may be reasonablyextrapolated to seepage angles of about 615u. Thoughseepage angle for the example dam section is muchhigher than 15u, the seepage angle of 15u was used tocalculate critical kinetic energy using the correlationshown in Figure 7:

Ekcrit~0:0072(Seepage Angle)z0:1562~0:26 J:

The available seepage kinetic energy (2.58 J)exceeds the critical kinetic energy to initiate backwarderosion (0.26 J). These values result in a factor ofsafety against backward erosion of 0.26/2.58 5 0.1.This is an unacceptable factor of safety, and remedialmeasures would be required if foundation conditionswere comparable to the soil QS assumed in thisanalysis.

Based on the same assessment methodology,factors of safety for a variety of other seepage casesreported in the literature were computed. Table 1summarizes the selected seepage cases for this study,including the calculated maximum hydraulic gradient,maximum Darcy flow velocity, available kineticenergy, and computed factor of safety (FS). Thecomputed factors of safety determined with thekinetic energy–based factor of safety approach agreedwell with the published cases (Table 1). It should bementioned that the soil conditions in all seepage caseswere assumed to be comparable to the soil QS used inthe TTPTA test program.

Figure 10. SEEP-W model of seepage through a homogeneousembankment with pervious foundation: (a) overall model, (b)close-up of toe area, and (c) tabular results of SEEP-W input/output.

Richards and Reddy


SUMMARY AND CONCLUSIONS

Approximately one-third of all internal erosionfailures of dams are caused by backward erosion orsuffusion. Current methods for evaluating backwarderosion potential do not adequately take intoconsideration all of the parameters that affect piping

potential, such as seepage angle, void ratio, finescontent, and their plasticity. A new apparatus, knownas TTPTA, was developed, and several series of testswere conducted using a wide range of soils to studyinitiation of internal erosion (backward erosion andsuffusion in non-cohesive soils and concentrated leakerosion [scour] in soils with cohesive fines) under

Table 1. Comparison of factors of safety using the kinetic energy–based method and exit gradients for published cases of dams with seepageproblems. F.S. (backward erosion) was computed using Ekcrit for a uniform medium sand (SP) (k 5 0.026 m/sec, Ekcrit 5 0.15 Joules forhorizontal flow at minimum confining stress conditions, n 5 0.37).



various void ratios, stress-state, and seepage angleconditions.

Based on theoretical considerations, the initiation ofbackward erosion in non-cohesive soils was defined bycritical kinetic energy, while concentrated leak erosion(scour) in soils with plastic fines can be defined bycritical shear stress. A new kinetic energy–based factorsof safety methodology is proposed to predict initiationof backward erosion (and suffusion) in non-cohesivesoils. The initiation of concentrated leak erosion incohesive soils can also be presented in factor of safety interms of critical and available shear stresses. Thisapproach can incorporate the complex behavior causedby varying soil compositions and stress states.

In order for the application of kinetic energy–basedfactor of safety method to predict backward erosioninitiation, one must first determine the amount andtypes of fines and confirm that they are prone toundergoing backward erosion or suffusion. Next, thecritical kinetic energy should be determined based onthe laboratory test on the site soil under the conditions(e.g., stress state) representative of the actual earthendam or levee that is being analyzed. Then, the availablekinetic energy should be calculated based on flowanalysis of the actual dam or levee section. Finally, thefactor of safety is calculated as the ratio of availablekinetic energy and the critical kinetic energy. Severalexamples have been presented, and the applicability ofthe assessment method has been successfully demon-strated. It should be noted that the assessment methodcan better predict the internal erosion risk at dams andlevees consisting of non-cohesive soils.

Further work is needed to better define therelationships among soil type and piping modes, thecritical velocity, kinetic energy, and the critical shearstress for various soils. Additional investigations maybe warranted to assess the role that water temperatureand density play in piping. More work should be donewith respect to the coupling between confining stress,void ratio, and pore pressures at the critical state forpiping in granular materials. The equations for thefactor of safety recommended should be furtherevaluated using case histories of failed dams and stabledams. The seepage angle correction method needs tobe expanded and the test apparatus for conductingthese tests improved. Additional case history tests needto be conducted to further assess this promisingmethod.

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