ReliabilityAnalysisforBypassSeepageStabilityofComplex ...downloads.hindawi.com/journals/mpe/2019/8261961.pdf · considers 3D seepage analysis and SRSM with simple polynomials. e objective

Research ArticleReliability Analysis for Bypass Seepage Stability of ComplexReinforced Earth-Rockfill Dam with High-Order PracticalStochastic Response Surface Method

Jinwen He 123 Ping Zhang 2 and Xiaona Li 4

1Hubei Key Laboratory of Disaster Prevention and Mitigation China ree Gorges University Yichang Hubei 443002 China2College of Hydraulic amp Environmental Engineering China ree Gorges University Yichang Hubei 443002 China3Research Center on Water Engineering Safety and Disaster Prevention of the Ministry of Water ResourcesYangtze River Scientific Research Institute Wuhan Hubei 430010 China4School of Water Resources and Hydroelectric Engineering Xirsquoan University of Technology Xirsquoan 710048 China

Correspondence should be addressed to Jinwen He hejinwen2004126com

Received 23 April 2019 Revised 4 August 2019 Accepted 16 September 2019 Published 24 October 2019

Academic Editor Federica Caselli

Copyright copy 2019 Jinwen He et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Practical stochastic response surface method (SRSM) using ordinary high-order polynomials with mixed terms to approximatethe true limit state function (LSF) is proposed to analyze the reliability of bypass seepage stability of earth-rockfill dam Firstly theorders of random variable are determined with a univariate fitting Secondly nonessential variables are excluded to identifypossible mixed terms irdly orthogonal table is used to arrange additional samples and stepwise regression is conducted toachieve a specific high-order response surface polynomial (RSP) Fourthly Monte Carlo simulation (MCS) is used to calculate thefailure probability and RSP is updated by arranging several additional samplings around the design point At last the Bantoucomplex reinforced earth-rockfill dam was taken as an example ere are 6 random variables that is the upper water level and 5hydraulic conductivities (HCs) e result shows that a third-order RSP can ensure good precision and the failure probability ofbypass seepage stability is 3680 times 10minus 5 within an acceptable risk range e HC of concrete cut-off wall and the HC of rockfill areunimportant random variables Maximum failure probability at the bank slope has positive correlation with the HC of curtain andthe upper water level negative correlation with the HC of alluvial deposits and less significance with the HC of filled soil With theincrease of coefficient of variation (Cov) of the HC of curtain the bypass seepage failure probability increases dramaticallyPractical SRSM adopts a nonintrusive form e reliability analysis and the bypass seepage analysis were conducted separatelytherefore it has a high computational efficiency Compared with the existing SRSM the RSP of practical SRSM is simpler and theprocedure of the reliability analysis is easier is paper provides a further evidence for readily application of the high-orderpractical SRSM to engineering

1 Introduction

With a long-term operation of earth-rockfill dams struc-tural failure caused by seepage may make a dam destroy [1]About 25 of the damage of earth-rockfill dams is caused byseepage failure worldwide [2] while with a rate of 30 inChina [3] In recent years a large number of earth-rockfilldams have been reinforced in China by adding a newconcrete cut-off wall or an impervious curtain [4] Althoughthe waterhead in the downstream is dropped the bypass

seepage gradient still plays a controlling role [5 6] so thatthe bypass seepage analysis should be considered as a 3Dproblem At present a seepage gradient less than an al-lowable value is adopted to represent the instability criteriaof seepage failure but the deterministic analysis cannotreflect the parameter uncertainty of bypass seepageeffectively

Many investigators have contributed to the reliabilityanalysis of seepage stability Wei and Shen [7] combined thehydraulic method and Rackwitz-Fiessler method (R-F or JC

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 8261961 15 pageshttpsdoiorg10115520198261961

method) to analyze the seepage stability of a homogeneousearth dam Li et al [8] analyzed the failure probability ofseepage in a homogeneous dike and discussed the influenceof the Cov of HC and the changes of flood level with astochastic FEM Hu et al [9] have combined FEM and SRSMwith Hermite polynomials to analyze the reliability ofseepage stability of the clay core wall of Shuangjiangkourockfill dam and studied the influence of the Cov ofvariables

Great progress has been made in this field from thehydraulic method to FEM in deterministic analysismethod from a low homogeneous dam to a high complexdam in analysis object and from explicit formula to usingFEM to construct SRSM in LSF but reliability calculationis still in 2D stage e difficulties in analyzing the bypassseepage stability of a complex reinforced earth-rockfilldam are (1) the LSF of seepage failure is highly nonlinearand cannot be expressed explicitly (2) bypass seepage ofcomplex reinforced earth-rockfill dam is a 3D problemand large number of random variables will make re-liability analysis too computationally intensive At pres-ent JC method is only applicable to an explicit LSF andlowly nonlinear e stochastic FEM has a long devel-opment cycle and it is difficult to develop a uniformstochastic FEM program to suit for different problemse actual LSF can only be well fitted by an ordinary RSMin the vicinity of design point through adjusting responsesurfaces but it can be well fitted by the SRSM with anoptimal order polynomial chaos expansion in the entirespace [10] e SRSM just needs one response surface andcan solve the problem with implicit LSF by using lesssamples than ordinary RSM but high-order Hermitepolynomials are extremely complicated [11] erefore itis necessary to explore a method for the reliability analysisof 3D bypass seepage stability of earth-rockfill dam whichconsiders 3D seepage analysis and SRSM with simplepolynomials

e objective of this paper is to propose a practicalSRSM with ordinary high-order polynomials as LSF toanalyze the reliability of 3D bypass seepage stability ofearth-rockfill dams To achieve this goal this article isorganized as follows In Section 2 a 3D seepage FEM isintroduced briefly In Section 3 procedure of the high-order practical SRSM is discussed in detail and two nu-merical examples are investigated to demonstrate thevalidity of the proposed method Bantou reinforced dam istaken as an example in Section 4 reliability analysis forbypass seepage stability is calculated based on the upperwater level and 5 HCs of materials as random variables einfluence of the orders and Cov of random variables arealso analyzed

2 Seepage FEM for Earth-Rockfill Dam

e seepage model of an earth dam with free surface andseepage surface is shown in Figure 1 e dam body can bedivided into a saturation zone Ω1 and an unsaturated zoneΩ2 According to Darcyrsquos law the waterhead at any point canbe expressed as follows

z

zxkx

zh

zx1113888 1113889 +

z

zyky

zh

zy1113888 1113889 +

z

zzkz

zh

zz1113888 1113889 0 (1)

where h is waterhead and kx kv and kz are the HCs in x yand z directions e corresponding boundaries of formula(1) are listed as follows

(1) AB and CD are the Dirichlet boundaries of water-head hAB H1 and hC D H2

(2) BC is the Neumann boundary of flow the flowqBC 0

(3) AE is the free surface of seepage the interface of Ω1and Ω2 e water level on the surface is equal to thepiezometric waterhead z If there is no flow exchangebetween the two regions hAE z and vAE 0

(4) DE is the seepage surface Flow outflow from the EDsurface that is hDE z and vDE le 0

e solution of equation (1) is equivalent to the minimumvalue of the following formula

I[h(x y)] CΩ

12

kx

zh

zx1113888 1113889

2

+ ky

zh

zy1113888 1113889

2

+ kz

zh

zz1113888 1113889

2⎡⎣ ⎤⎦dx dy dz

(2)

Equation (2) is derived by the node waterhead in theseepage area [K]e h

e 0 can be obtained Functionaldifferential is conducted on all elements multiply theconduction matrix [K]e and the node waterhead h

emoving the nodes with the given waterhead to the right ofthe equation formula (3) can be obtained

[K] h Q (3)

where [K] is conduction matrix h is waterhead and Q isthe initial flow Since formula (3) was solved by Neumann in1973 there have been several constant mesh methods suchas initial flow method [12] improved method of nodalvirtual flux [13] Signorini variational inequality method[14] and equivalent nodal flow method [15]

If h at any FEM nodes is solved the seepage field will bedetermined e total seepage gradient at any point can bedefined as

J

minuszh

zx1113888 1113889

2

+ minuszh

zy1113888 1113889

2

+ minuszh

zz1113888 1113889

2

11139741113972

(4)

3 High-Order Practical SRSM

e SRSM was first proposed in the study of uncertainproblem in environmental and biological system by Isu-kapalli et al [16] in 1998 e SRSM combining a traditionaldeterministic method and the uncertainty method not onlyovercomes the difficulty that the LSF cannot be expressedexplicitly but also greatly decreases the amount of responsecalculation of samples compared with the traditional directMCS An important procedure with the SRSM is to choosethe RSP e SRSM with Chebyshev polynomials to

2 Mathematical Problems in Engineering

determine variable orders was proposed by Gavin and Yau[17] and the SRSM with Hermite polynomials was proposedby Jiang et al [11] In this paper ordinary high-orderpolynomials withmixed terms in equation (5) are used as theLSF surrogate modele algorithm of the proposed methodis called the high-order practical SRSM

g(x) a + 1113944n

i11113944

ki

p1bipx

pi + 1113944

m

q1cq1113945

n

i1x

piq

i (5)

where a is a constant 1113936ki

p1bipxpi is an independent term and

1113937ni1x

piq

i is the possible mixed term where the total numberof random variables (n) the polynomial order (ki) and thecoefficients (bip) correspond to independent terms involvingonly one variable the coefficient cq corresponds to mixedterms involving the product of two important randomvariables the total number of mixed terms (m) and theorder of a random variable in a mixed term (piq)

Figure 2 shows the flow chart mainly including fourstages First was the order identification (OI) of randomvariables Secondly estimate the importance of the levelof random variables and possible mixed terms ese firsttwo stages result in the basic formulation of the RSP as asurrogate model e specific formulation and corre-sponding coefficients of the RSP are determined in thethird stage by stepwise regression based on OI samplingsadditional samplings and design point or mean valuepoint Fourthly MCS is carried out to determine thefailure probability and estimate the design point If thenumber of variables or the coefficient of variation islarge the RSP needs to be updated by adding severaladditional samples around the design point

31Order Identification If a variable order of polynomials ismuch higher than the real LSF ill-conditioned systems ofequations may be encountered [18] So order identificationis the base of the accuracy of the SRSM If univariate fitting istaken the steps of order identification of the variable arelisted as follows

Step 1OI sample arrangement Variable xi is set to[μi minus fσ i μi + fσ i] where μi is a vector of the meanvalue of xi σi is a vector containing the standarddeviation of xi and interpolation coefficient f is setto the value from 1 to 3 All other variables

x1 ximinus 1xi+1 xn are set to their means(μ1 μiminus 1 μi+1 μn)Step 2 Response calculation of the OI samplesgi g(μ1 μiminus 1 xi μi+1 μn)Step 3 Spline curve fitting taken as the real values basedon variable xi and corresponding giStep 4 Polynomial fitting without mixed terms basedon variable xi and corresponding gi can be approxi-mated in equation (6) and the multiple correlationcoefficient R2

1 can be obtained

gi a0 + a1xi + a2x2i + middot middot middot + anx

ni (6)

Step 5 Calculate the difference Rj between spline curveand polynomials by equation (7) from first order to the

Input statistics parametres of random variablesI = 0

OI samplings arrangement and determination ordersof random variables

Estimate the importance level of random variables and the possible mixed terms

I = I + 1

Additional samplings arrangement nondimensionalization andstepwise regression to determine the specific high-order RSP

Generate random number sequences probabilisticanalyses with MCS and estimate the design point

I lt 2Yes

Result output

No

Figure 2 Flow chart of the high-order practical SRSM

H1

H2

Z

AG F

Ω2

Ω1

Free surfaceE Seepage surface

D

CB0

Figure 1 Seepage model with free surface and seepage surface

Mathematical Problems in Engineering 3

sixth order (xiprime yiprime) and (xPrimei yPrimei ) are the scatter points in

spline curve and polynomials respectively

Rj 1113944100

i1

xiprime minus xPrimei1113872 1113873

2+ yiprime minus yPrimei1113872 1113873

21113970

j 1 2 6

(7)

Step 6 Determine the highest order of xi To avoid theRunge phenomenon the order with the minimum Rj istaken as the highest order of variable xi Meanwhile todecrease the number of samplings the orders 1simn withthe multiple correlation coefficient R2

1 greater than0995 for the first time the order n can also be taken asthe highest order of variable xi for R2

1 in polynomialfitting represents the accuracy of the fitting of thescatter and 0995 is in a high level Choose the smallone as the highest order of variable xi according to thetwo criterionsStep 7Determine the other possible order of xi If n is thehighest order of xi the terms xi x2

i xnminus 1i may not

simultaneously exist in equation (6) Multiple correlationcoefficient R2

2 of the remaining terms can be used toexclude the unimportant order of xi If R2

2R21 gt 09995

the remaining terms are the possible orders

e orders of the variable determined by the seven stepsmay exist in independent terms mixed terms or in bothGenerally RSP includes several independent terms andmixed terms

32 Important Level Estimation and Possible Mixed Terms ofVariables e contribution of mixed terms to the LSF isrelated to the importance level of variable On one handinappropriate mixed terms may lead to incorrect estimationof failure probability On the other hand the more the mixedterms the larger the response calculation erefore onlythe mixed terms of the important variables need to beconsidered e importance of variables is highly correlatedwith the variable contribution to the uncertainty of responsee variance σ2gi

of the response of a single variable can beobtained based on equation (6) e importance level can bedefined as follows

Qi σ2gi

1113936ni1σ2gi

times 100 (8)

If Qi is less than a critical value such as 5 the cor-responding variable xi can be considered as an nonessentialone which can be excluded in the mixed terms to reduce thenumber of the samples required for polynomial fitting

To explore the mixed terms of RSP Gavin and Yau [17]had proposed two criteria (1) the power of a variable in amixed term should not be larger than the estimated order ofthe variable alone (2) the total order of the mixed termshould not be larger than the highest-order term Besidesthe regression additional criterion proposed by Li [19] is justto consider the mixed terms including the important

variables However the GavinndashYau criteria two cannotdetermine the RSP with the only mixed term for exampleg xyz e highest order of variables is 1 but the totalorder of the mixed term is 3 So it can be changed to thepower of a variable in the mixed term which should be lessthan the highest order of this variable except the first orderimproved GavinndashYau criterion two

Once the basic formulation of RSP has been estimatedthe specific form and corresponding coefficients of RSP canbe determined by stepwise regression Polynomial fittingrequires samples uniform and representative in the solutiondomain and the number of samples is best chosen to betwice that of polynomial coefficients [19] is paper usesorthogonal table to arrange additional samples for its bal-anced dispersion and neat comparable characteristics in thesamples arranged

33 Additional SamplesArrangement and Stepwise Regressionof High-Order RSP OI samples are just located in thehorizontal line and vertical line and there are no samples ineach quadrant consisting of horizontal and vertical lines(shown in Figure 3) An orthogonal table will be used toarrange the additional samples to fill the blank Every or-thogonal table has two characteristics different numbersappear equal times in each column and the numbers in anytwo columns are arranged in a balanced dispersion situationTake the L4(23) orthogonal table (Table 1) as an examplethere are four combinations in any two columns (11) (12)(21) and (22) Each combination appears at the samefrequency erefore there is strong representation in thesamples arranged by an orthogonal table

An orthogonal table is selected according to the principlethat the maximum factor is slightly great than or equal to thenumber of variables Take a 2-level orthogonal table as anexample each row of the table can be taken n times inarrangement of additional samples according to the meanpoint (or the design point Xd) and standard deviation σx Ifthere are just two random variables L4(23) orthogonal tablewill be chosen Columns 1 and 2 in Table 1 are used to setvalues for variable one and two respectively When thenumber is 1 the corresponding variable is set as Xd minus fσxwhen the number is 2 the corresponding variable is set asXd + fσx Each row of the table can achieve one sample sothere are 4 samples marked black times in Figure 3(a) If theinterpolation coefficient f ranged from 1 to 3 and the or-thogonal table is used 3 times the corresponding variablecan be set as Xd minus σx Xd minus 2σx andXd minus 3σx when thenumber is 1 and Xd + σx Xd + 2σx andXd + 3σx when thenumber is 2ere are 12 samples marked the sign times (shownin Figure 3(a))

Latin hypercube sampling (LHS) method is a popularsampling method Compared with the LHS method thereare two advantages in the additional samples with the or-thogonal table method (Figure 3(b)) (1) more additionalsamples are located at the edge (2) there is orthogonalitycharacteristic in the additional samples with the orthogonaltable method and the number of additional samples is equalin each quadrant consisting of horizontal and vertical lines


e specific formulation and corresponding coefficientsof RSP can be determined by stepwise regression based onOI samplings additional samples design point and thesecorresponding responses calculated e true LSF g(x) issurrogated with an ordinary polynomial in equation (5) andthe least squares method is used to determine the unknowncoefficients ere are three steps to determine the specificformulation of the RSP

Step 1 e first step is the nondimensionalization ofvariables and responses to avoid the occurrence ofldquolarge numbers to eat decimalsrdquo during the polynomialfitting Assuming that xi and xi

prime are the values beforeand after nondimensionalization a linear dimension-less method is defined as follows

xiprime

xi

maxxi

(9)

Step 2e mix terms are added one by one to conductregression analysis based on all possible independentterms estimated in Section 31 If a possible mix item isadded and the fitting multiple correlation coefficient R2

becomes larger the new mixed term is kept If R2 issmaller or unchanged exclude the new mixed itemStep 3 e possible independent items are subtractedone by one to conduct regression analysis based on allpossible independent items and the determined mix

terms If an independent term is subtracted and R2

becomes larger or almost unchanged the independentterm is excluded If R2 is smaller keep the independentterm

34 MCS and Design Point Estimation e true LSF will bereplaced by the RSP determined When solving the failureprobability with MCS N random sequences will be gener-ated according to the distribution characteristics of vari-ables e sampling method for the random numbersequences of the related multidimensional normal distri-bution has been introduced in many references If randomvariables are nonnormal distribution the correlation co-efficient is not consistent after equivalent normalizationtransformation e correlation coefficient conversion ofnonnormal random variable is commonly transformed withthe Nataf method or the Copula function method To ex-plore the change of correlation coefficient some empiricalformulations with the Nataf method have been established[20 21] A two-step numerical integration method has beenproposed by Wen et al [22] to overcome the non-convergence in the numerical calculation of the Natafmethod

After the random number sequences are generated bysubstituting them into RSP and counting the number ofsamples Nf in the failure region the failure probability isexpressed as

pf Nf

N (10)

As for some events with a small probability in engi-neering the total number of samples N should not be lessthan 100Pf In the reliability analysis the design point hastwo characteristics (1) the design point is on the failure

ndash4 0 2 4ndash2x

ndash4

ndash2

0

2

4

y

OI samplesAdditional samplesMean value point

(a)

y

ndash4

ndash2

0

2

4

ndash2 0 2 4ndash4x


(b)

Figure 3 Diagram of samples (a) Orthogonal table method (b) Latin hypercube sampling method

Table 1 L 4(23) orthogonal table

Test number Column 1 Column 2 Column 31 1 1 12 1 2 23 2 1 24 2 2 1


boundary that is the LSF value of the point is 0 (2) thedesign point is with maximum failure probability on thefailure boundary According to characteristic oneN samplesin theMCS are sorted on the basis of the absolute value of theLSF If N is greater than 10^5 500 small absolute values canbe taken as on the LSF According to the characteristic twothe design point is located in the dense area of the failureboundary e distance between the coordinate origin and500 samples in the standard normal space is calculatedrespectively and the density curve of the distance can beobtained en the group with maximum density is chosenand the averages of this grouprsquos variable are taken as thedesign point

Above all practical SRSM is just one initial responsesurface or one updated response surface which is fittedbased on OI samples additional samples and correspondingresponse calculation Besides practical SRSM adopts anonintrusive form not only the reliability analysis and theresponse calculation can be conducted separately but alsothe response of samples can be calculated on differentcomputers or different cores of a machine

35 Numerical Examples Two numerical examples are usedto illustrate the efficiency and accuracy of the high-orderpractical SRSM

Example 1 (see [23]) e third-order nonlinear functiong(x y) x3 + x2y + y3 minus 18 x simN(10 52) y simN(99 52)and ρxy 01 estimate the failure probability

Interpolation coefficient was assigned from 1 to 3 12 OIsamples and corresponding responses are listed in Table 2Figure 4 shows the relationship between single variable andthese corresponding responses the highest order of vari-able x and y is 3 for both and the multiple correlationcoefficient R2

1 is 100 Table 3 shows the process for de-termining the other possible orders of variables x and yWhen x is subtracted to fit polynomials the multiplecorrelation coefficient R2

2 of the remaining terms x2 andx3

is unchanged If x2 or x and x2 subtracted at the same timeR22R

21 will be less than 09995 so the first order of x is an

unimportant order and can be excluded With the samemethod the term y2 can also be omitted erefore thereare 4 possible independent terms x2 x3 y andy3 and onlyone possible mixed term x2y according to the improvedGavinndashYau criteria two 5 terms and the constant result inbasic formulation of RSP

Use the L4(23) orthogonal table 2 times to arrange 8additional samples and conduct nondimensionalization ofthe 21 samples (Table 2) by equation (9) the stepwiseregression process for determining the specific mixedterms and independent items based on the 21 samples islisted in Table 4 When the 4 independent terms areconsidered as LSF R2 is 09820 If the mixed term x2y isadded R2 is increasing to 100 the mixed term will be keptIf x2 or y is subtracted R2 are all unchanged If x3 or y3 issubtracted R2 are decreased so the independent terms x3

and y3 should be kept and x2 and y can be omittedg(x y) minus 000075 + 065589x3 + 064805y3 + 065326x2y

is obtained and the 3 terms and the constant are perfectlydetermined

Taking the RSP as LSF generate 1 times 107 random numbersequences with correlation coefficient 01 with the Natafmethod Table 5 summarizes the results with the othermethods the failure probability of practical SRSM is8112 times 10minus 3 almost consistent with the reference solution(8110 times 10minus 3) in [24] and the number of calling functions isclose to that of the Hermite SRSMe 500 samples with thesmallest absolute value of the LSF are plotted in Figure 5 andthe estimated design point is (17068 19640) It can be seenthat the 500 samples are all located on the true LSF and theestimated design point is located near the bump eseresults indicate that the high-order practical SRSM evaluatesthe failure probability accuracy

Example 2 (see [24]) g(x) 016(x minus 1)3 minus y + 4 minus 004cos(xy)is a third-order highly nonlinear LSF the ran-dom variables are subject to independent standardnormal distributions and the failure probability iscalculated

f is set from 1 to 2 and 8 OI samples can be arrangede highest order of variable x andy is 3 and 1 with themethod in Section 31 If x x2 and the two terms aresubtracted the R2

2R21 are 09905 08368 and 08273 re-

spectively So the three terms cannot be excludedereforethere are 4 possible independent terms x x2 x3 and y andtwo possible mixed terms xy and x2y according to theimproved GavinndashYau criteria two 6 terms and the constantresult in basic formulation of RSP

Use the L4(23) orthogonal table 2 times to arrange 8additional samples and conduct the nondimensionalizatione stepwise regression for determining the specific mixedterms and independent items is listed in Table 6 When the 4independent terms are considered as LSF R2 is 09999 If themixed term xy or x2y is added one by one R2 is still un-changed So the two mixed terms can be omitted If thepossible independent items x x2 x3 yR2 substracted one byone R2 is decreased to 09863 08041 09456 and 07374respectively so all the independent terms should be keptg(x y) 056082 + 021176x minus 044117y minus 063195x2

+ 063529x3 can be obtained Taking the polynomials as LSFgenerating 2 times 105 random number sequences the failureprobability is 33405 times 10minus 2 with MCS For comparisonTable 7 summarizes the results with the other methods notethat the failure probability of practical SRSM is the closest tothat of Hermite SRSM and the number of calling functions is17 which is slightly larger than that of the Hermite SRSM andsubstantially smaller than that of GavinndashYau SRSM eestimated design point plotted in Figure 6 is (minus 1796804805) ese results indicate that the high-order practicalSRSM evaluates the failure probability efficiency andaccuracy

4 Engineering Application

41 Project Profile e Bantou dam (Figures 7 and 8) is anearth-rockfill dam with a height of 503m e crest


elevation is 31288m and the normal water level is 31058me width of dam crest is 58m and the length is 285mesection of dam body mainly consists of filled soil and

downstream rockfill the total thickness of alluvial deposits atthe riverbed and the left bank is small but is more than 20mat the right bank In 2009 the dam was reinforced and the

Table 4 Stepwise regression for determining the mixed terms and independent terms of example one

Items Operation R2 KeepdiscardBasis terms x2 x3 y y3 09820Possible mixed terms x2y Add 10000 Keep

Possible independent terms

x2 Subtract 10000 Discardx3 Subtract 09047 Keepy Subtract 10000 Discardy3 Subtract 08145 Keep

Table 2 Samples of example one

Sample name Serial number x y g(x y) Serial number x y g(x y)

OI samples

1 minus 5 99 10748 7 10 minus 51 33932 0 99 9523 8 10 minus 01 97203 5 99 13248 9 10 40 144604 15 99 65548 10 10 149 577995 20 99 129123 11 10 199 1085266 25 99 227648 12 10 249 189102

Additional samples

1 5 49 3471 5 15 50 460702 0 minus 01 minus 180 6 20 00 798203 5 149 37874 7 15 149 1001744 0 199 78626 8 20 199 238226

Mean value point 1 10 99 29423

30200 10ndash10x

g = x3 + 99x2 + 3 times 10ndash13x + 9523R2 = 1

0

5000

10000

15000

20000

25000

g

(a)

30200 10ndash10y

g = y3 ndash 3 times 10ndash14y2 + 100y + 982R2 = 1

0

5000

10000

15000

20000

g

(b)

Figure 4 Order identification of random variables

Table 3 Determining the possible orders of variables

Variable Reserved items R22R

21 Keepdiscard

x

x x2 x3 10000x2 x3 10000 Discard x

x x3 09991 Keep x2

x3 09980 Cannot discard x and x2 at the same time

y

y y2 y3 10000y2 y3 09976 Keep y

y y3 10000 Discard y2

y3 09947 Cannot discard y and y2 at the same time


reinforcement measures include the following (1) a new80 cm thick low elastic modulus concrete cut-off wall is builton the upstream side and the cut-off wall is directly insertedinto the bedrock at the left bank and only to the bottom ofthe dam at the right bank (2) curtain grouting is constructedunder the dam foundation at the right bank

e FEM model for seepage analysis (Figure 9) consists of50748 nodes and 49143 elementse statistical parameters ofrandom variables are shown in Table 8 e average of the

upper water level is 30958m ranging from 304m to 312msubject to the extremal type I distributione HC of rockfill ismuch larger than filled soil e HC of curtain and cut-off wallis much smaller than that of filled soil e HC of alluvialdeposits and the HC of filled soil are in the same powerAll of the HC are log-normal distribution

Select the check flood condition (the upper anddownstream water level are 31199m and 2649m re-spectively) for 3D seepage analysis e results are shownin Figures 10 to 12 Note that the waterhead in thedownstream dam body is significantly reduced (Figure 10)but the bypass seepage is still serious (Figure 11) after thereinforcement Figure 12 shows the seepage gradient of thedownstream dam foundation Due to the bypass seepagethe maximum seepage gradient appears at the right steepbank slope not at the foot of the main riverbed

42 Reliability Analysis of Bypass Seepage Stability econcrete cut-off wall and curtain do the main seepageprevention of this project Although the seepage gradient of

Table 5 Comparison of calculation result of example one

Method Failure probability (times10minus 3) Number of calling functionsHermite SRSM (m 3) 8067 19High-order practical SRSM 8112 21NESSUSmdashMCS 8110 107

Table 6 Stepwise regression for determining the mixed terms and independent items of example two

Items Operation R2 KeepdiscardBasis terms x x2 x3 y 09999

Possible mixed terms xy Add 09999 Dx2y Add 09999 D


x Subtract 09863 Kx2 Subtract 08041 Kx3 Subtract 09456 Ky Subtract 07374 K

ndash5

ndash2

1

4

7

y

ndash5 0 5 10ndash10x

Approximated failureboundaryDesign point

Mean value point

True LSF

Figure 5 LSF of example one

Table 7 Comparison of calculation result of example two

Method Failure probability(times10minus 2)

Number of callingfunctions

Hermite SRSM(m 5) 33661 13

High-order practicalSRSM 33405 17

GavinndashYau SRSM 32466 43NESSUSmdashMCS 32833 106

ndash2 0 2 4ndash4x

ndash3

ndash1

1

3

5

y


Mean value pointTrue LSF

Figure 6 LSF of example two


cut-off wall and curtain is large it is much smaller than theallowable value e seepage gradient at the bank will play adecisive role in the overall seepage stability of the damerefore the maximum seepage gradient at the right bankis used to judge the overall seepage failure of the dam eLSF can be written as

g(X) Jcr minus J X1 X2 Xn( 1113857 (11)

where J is the maximum seepage gradient at the bankcalculated by 3D seepage FEM Jcr is the critical seepagegradient equal to 045 X1 X2 Xn are random variablesTake the upper water level the HC of concrete cut-off wall

Normal water level 31058m31408m 31288m

Filled soil Filled soil

Concrete cut-off wall Rockfill

26258mAlluvial depositsAlluvial deposits

Original ground line26658m

Figure 7 Typical section

Alluvial deposits

Alluvial depositsCurtain grouting

Concrete cut-off wall

Original ground line

Figure 8 Cut-off wall and curtain grouting

ZY

X

Figure 9 FEM model

Table 8 Statistical parameters of random variables

Variable e upper water levelmHC(cms)

Cut-off wall Filled soil Rockfill Alluvial deposits CurtainMean 30958 1times 10minus 7 2times10minus 4 2times10minus 2 5times10minus 4 5times10minus 6

Cov 006 006 012 016 013 009


filled soil rockfill alluvial deposits and curtain as randomvariables

e importance level of random variables is listed inTable 9 it can be seen that the HC of cut-off wall and rockfillare nonessential variables since the total importance level ofthe two variables is less than 5 and the nonessentialvariables will be taken as constant to decrease the intensivecomputation e OI of random variables is shown inFigure 13 Note that the highest order of HC of filled soil is 3the HC of curtain is 2 and the others are all 1 e orders

lower than the highest order of each variable cannot beexcluded by Step 7 in Section 31ere are 17 possible termsin the polynomials including 7 independent terms 9 mixedterms based on the three criteria and 1 constant Choosingthe L8(26) orthogonal table to arrange samples 3 timesadding the mean point and 24 OI samplings there are 49samples in total 3D seepage program is used to calculate theseepage gradient of additional samples on different com-puter cores Nondimensionalization is conducted on thesamples and corresponding responses e initial response

7000613152634394352526561788919050

Waterhead (m)

Figure 10 Distribution of waterhead at the largest section

040

036

031

027

023

018

014

009

005

Hydro-gradient

Z

Y

X

Figure 11 Distribution of seepage gradient in filled soil

040

035

030

026

021

016

012

007

002

Hydro-gradient

Z

Y

X

Figure 12 Distribution of seepage gradient on the downstream dam foundation


surface is stepwise regressed and the initial design point is(311992 1737 times 10minus 4 3905 times 10minus 4 6148 times 10minus 6) eorthogonal table was used to arrange samples one time andthe interpolation coefficient was set to 025 8 additionalsamplings can be obtained adding the design point and theprevious 49 samples the polynomials regressed is listed informula (12) 5 times 107 random number sequences are gen-erated e failure probability is 3680 times 10minus 5 and thecorresponding reliability index is 3964 with MCS e newdesign point is (311996 1799 times 10minus 4 3744 times 10minus 46263 times 10minus 6) Taking it into the 3D seepage program thebypass seepage gradient is 04483 similar to the critical value045 erefore the calculation results of this method arereliable

g(X) 00085 + 09222X1 minus 06400X3 minus 03861X24

+ 00242X32 minus 07990X1X2 + 04168X1X

22

+ 01724X1X3 + 07012X1X4 minus 00856X2X3

+ 02167X22X3 + 08403X2X4 minus 06684X

22X4

+ 01289X3X4

(12)Figure 14 shows the correlation between the 4 random

variables and the maximum seepage gradient at the rightbank It can be seen that (1) the maximum bypass seepagegradient is highly related to the upper water level and the HCof alluvial deposits and curtain and is irrelevant to the HC of

303 305 307 309 311 313Upper water level (m)

043

039

035

031

027

Seep

age g

radi

ent

y = 00113x ndash 31313R2 = 09965

(a)

038

037

036

035

034Se

epag

e gra

dien

t

R2 = 09955

10 14 18 22 26 30HC of filled soil (10ndash4cms)

y = 00102x3 ndash 00649x2 + 01172x + 03018

(b)

Seep

age g

radi

ent

R2 = 09987

20 32 44 56 68 80HC of alluvial deposits (10ndash4cms)

042

039

036

033

030

y = ndash00209x + 04632

(c)

Seep

age g

radi

ent

R2 = 09971

30 38 46 54 62 70HC of curtain (10ndash6cms)

041

038

035

032

029

y = ndash0003x2 + 00605x + 01268

(d)


Table 9 Importance level of random variables

Variables e upper water levelX1

HC of cut-offwall

HC of filled soilX2

HC ofrockfill

HC of alluvial depositsX3

HC of curtainX4

Importancelevel 5662 005 593 181 2228 1636


filled soil (2) the maximum bypass seepage gradient ispositively correlated with the upper water level and the HCof curtain and negatively correlated with the HC of alluvialdeposits e waterhead difference increases with the in-crease of the upper water level e greater the HC of curtainis the more the water appears at the bank slope and leads alarger seepage gradient e larger the HC of alluvial de-posits the higher the waterhead in the downstream filledsoil and a smaller seepage gradient will be caused

For comparison the ordinary response surface method(RSM) is used for the reliability analysis of this project Takingthe complete cube polynomials without mixed terms as re-sponse surface the reliability index is 376 6 response surfacesand 144 samples are required which is 248 times of the samplesof high-order practical SRSM ese results indicate that thehigh-order practical SRSM is a powerful analysis method

43 Influence of the Coefficient of Variation of RandomVariables Figure 15 shows the relationship between thefailure probability of bypass seepage gradient and the upperwater level and the HC of alluvial deposits and the HC ofcurtain Note that the maximum bypass seepage failureprobability increases more and more slowly with the in-crease of the Cov of the upper water level As the Cov of HC

of alluvial deposits and curtain increases significant in-creases of seepage failure probability can be seen Mean-while the growth rate of bypass seepage failure probabilitywith the Cov of HC of curtain is larger than that of alluvialdeposits Also the HC of curtain has a significant impact onthe seepage gradient of the dam

44 Influence of Variable Order If the HC of filled soil is setfrom 2nd to 4th order the multiple correlation coefficient R2

of the OI curve line is 0984 0996 and 0998 respectivelyand the reliability results are listed in Table 10 e cu-mulative distribution function (CDF) of maximum bypassseepage gradient at the bank slope is shown in Figure 16Note that the reliability index is getting smaller and smallerwith the increasing of the order and is nearly the same whenthe HC of filled soil is from 3rd and 4th order If the intervalof bypass seepage gradient is greater than 034 the threeCDF curve lines are almost in a line For the domain ofbypass seepage gradient less than 034 there is a large dif-ference between the curve lines of 2nd and other orders butthe 3rd and 4th order curve lines are very close

If the HC of filled soil is set from 2nd to 3rd order themultiple correlation coefficient R2 of the order identificationcurve line is 0992 and 0998 and the corresponding


044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0788

(a)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0085


(b)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = ndash0434


(c)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0403


(d)

Figure 14 Scatter plots between the maximum seepage gradient at the right bank and random variables


reliability results are 4105 and 3964 respectively Our re-sults demonstrated that variable order plays a role in theaccuracy of the reliability index e larger the importancelevel the larger the multiple correlation coefficient requiredand the more accurate the reliability index obtained

5 Conclusions

(1) A high-order polynomial with mixed terms is used asRSP e order identification of random variables

the determination of mixed terms and the stepwiseregression of high-order polynomials are simpleehigh-order practical SRSM provides a newmethod toestimate the failure probability of engineeringproblems with a certain high nonlinearity andcannot be explicitly expressed

(2) e high-order practical SRSM adopts a non-intrusive form e order identification samples andthe additional samples arranged the response cal-culation of samples can be performed on different

025 030 035 040 045

2nd SRSP3rd SRSP4th SRSP

Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF

Figure 16 Comparison of CDF for bypass seepage gradient with maximum failure probability

002 005 008 011 014 017Coefficient of variation

14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)

Upper water levelHC of alluvial depositsHC of curtain

Figure 15 Relationship between the failure probability and the Cov of variables

Table 10 Comparison table for the results of different orders of HC of filled soil with SRSM

Order 2nd 3rd 4thR2 0992 0998 0998Reliability index 4085 3964 3933


computers or different cores of a machine greatlyimproving the calculation efficiency erefore therealization of large engineering reliability analysiscan be achieved

(3) e HC of concrete cut-off wall and the HC ofrockfill are unimportant variables e reliabilityindex for bypass seepage stability around the Bantoudam is about 3964 and the failure probability is3680 times 10minus 5 e seepage failure at the bank slope iswithin an acceptable risk range e maximum by-pass seepage gradient has a great correlation with theHC of curtain alluvial deposits and the upper waterlevel and has little correlation with the HC filledsoil e maximum bypass seepage gradient ispositively correlated with the upper water level andthe HC of curtain and negatively correlated with theHC of alluvial deposits As the Cov of the HC ofcurtain increases the bypass seepage failure prob-ability increases dramatically

(4) e RSP of bypass seepage can be considered as the3rd order As long as the reliability calculationperformed using a third order or a larger order RSPstable results can be obtained How to achieve thesensitivity coefficient of the random variables to theseepage failure based on the high-order practicalSRSM needs further research

Data Availability

All data used to support the findings of this study areavailable from the corresponding author by request

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

is work was supported by the Hubei Key Laboratory ofDisaster Prevention and Mitigation (China ree GorgesUniversity) Open Research Program (no 2016KJZ07) theCRSRI Open Research Program (no CKWV2019755KY)and the Chinese National Natural Science Foundation (nos51909137 and 51809212)

References

[1] S A Shivvakumar Shivamanth C H Solanki andG R Dodagoudar ldquoSeepage and Stability analyses of earthdam using finite element methodrdquo Aquatic Procediavol 4 pp 876ndash883 2015

[2] T A Middlebrooks ldquoEarth-dam practice in United StatesrdquoTransactions of American Society Civil Engineering vol 118pp 697ndash722 1953

[3] E Loukola P Reiter C Shen et al ldquoEmbankment dams andtheir foundation evaluation of erosionrdquo in Proceedings of

the International Workshop on Dam Safety Evaluationpp 171ndash188 Grindewald Switzerland 1993

[4] X Q Niu ldquoCharacteristics of reservoir defects and re-habilitation technology in Chinardquo Chinese Journal of Geo-technical Engineering vol 32 no 1 pp 153ndash157 2010

[5] Y L Li S Y Li X F Zhang and R Fan ldquoResearch on bypassseepage of dam abutment deep-thickness sand layer andanti-seepage schemerdquo in Proceedings of the IEEE 2010 Asia-Pacific Power and Energy Engineering Conference pp 1ndash4Chengdu China March 2010

[6] T Jiang J ZhangWWan S Cui and D Deng ldquo3D transientnumerical flow simulation of groundwater bypass seepage atthe dam site of Dongzhuang hydro-junctionrdquo EngineeringGeology vol 231 pp 176ndash189 2017

[7] H Wei and Z Z Shen ldquoReliability analysis on seepage sta-bility of earth dams and its applicationrdquo Chinese Journal ofGeotechnical Engineering vol 30 pp 1404ndash1409 2008

[8] J H Li Y Wang and Q Hu ldquoProbability analysis ofseepage failure of embankments based on stochastic finiteelement methodrdquo Rock and Soil Mechanics vol 27 no 10pp 1847ndash1850 2006

[9] R Hu Y F Chen D Q Li et al ldquoReliability analysis ofseepage stability of core-wall rockfill dam based on stochasticresponse surface methodrdquo Rock and Soil Mechanics vol 33pp 1051ndash1060 2012

[10] S H Jiang D Q Li C B Zhou and L M Zhang ldquoCa-pabilities of stochastic response surface method and re-sponse surface method in reliability analysisrdquo StructuralEngineering and Mechanics vol 49 no 1 pp 111ndash1282014

[11] S H Jiang D Q Li L M Zhang and C B Zhou ldquoSlopereliability analysis considering spatially variable shearstrength parameters using a non-intrusive stochastic finiteelement methodrdquo Engineering Geology vol 168 pp 120ndash1282014

[12] S L Pan Q F Wang and J Yu ldquoImprovement of analysis offree surface seepage problem by using initial flow methodrdquoChinese Journal of Geotechnical Engineering vol 34 no 2pp 202ndash209 2012

[13] H D Cui and Y M Zhu ldquoImproved procedure of nodalvirtual flux of global iteration to solve seepage free surfacerdquoJournal of Wuhan University of Technology (TransportationScience amp Engineering) vol 33 no 2 pp 238ndash241 2009

[14] Y Chen R Hu C Zhou D Li and G Rong ldquoA new parabolicvariational inequality formulation of Signorinirsquos condition fornon-steady seepage problems with complex seepage controlsystemsrdquo International Journal for Numerical and AnalyticalMethods in Geomechanics vol 35 no 9 pp 1034ndash1058 2011

[15] J W He G M Liu and W Qiao ldquoEquivalent nodal flowmethod used to determine overflow surface in seepage withfree surfacerdquo Journal of China ree Gorges University(Natural Sciences) vol 35 pp 34ndash37 2013

[16] S S Isulapalli A Roy and P G Georgopoulos ldquoStochasticresponse surface methods for uncertainty propagationapplication to environmental and biological systemsrdquo RiskAnalysis vol 18 no 3 pp 351ndash363 1998

[17] H P Gavin and S C Yau ldquoHigh-order limit state functions inthe response surface method for structural reliability analy-sisrdquo Structural Safety vol 30 no 2 pp 162ndash179 2008

[18] M R Rajashekhar and B R Ellingwood ldquoA new look at theresponse surface approach for reliability analysisrdquo StructuralSafety vol 12 no 3 pp 205ndash220 1993


[19] H S Li Research on Probability Uncertainty Analysis andDesign Optimization Method Northwestern PolytechnicalUniversity Xirsquoan China 2008

[20] Y Noh K K Choi and L Du ldquoReliability-based designoptimization of problems with correlated input variablesusing a Gaussian Copulardquo Structural and MultidisciplinaryOptimization vol 38 no 1 pp 1ndash16 2009

[21] P-L Liu and A Der Kiureghian ldquoMultivariate distributionmodels with prescribed marginals and covariancesrdquo Proba-bilistic Engineering Mechanics vol 1 no 2 pp 105ndash112 1986

[22] D Z Wen R H Zhuo D J Ding H Zheng J Cheng andZ-H Li ldquoGeneration of correlated pseudo random variablesin Monte Carlo simulationrdquo Acta Physica Sinica vol 61no 22 pp 1ndash8 2012

[23] I Kaymaz and C A McMahon ldquoA response surface methodbased on weighted regression for structural reliability analysisrdquoProbabilistic Engineering Mechanics vol 20 no 1 pp 11ndash172004

[24] Z Z Lv S F Song H S Li et al Structural MechanismReliability and Reliability Sensitivity Analysis Science PressBeijing China 2009


Hindawiwwwhindawicom Volume 2018

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Dierential EquationsInternational Journal of

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Decision SciencesAdvances in


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Submit your manuscripts atwwwhindawicom

method) to analyze the seepage stability of a homogeneousearth dam Li et al [8] analyzed the failure probability ofseepage in a homogeneous dike and discussed the influenceof the Cov of HC and the changes of flood level with astochastic FEM Hu et al [9] have combined FEM and SRSMwith Hermite polynomials to analyze the reliability ofseepage stability of the clay core wall of Shuangjiangkourockfill dam and studied the influence of the Cov ofvariables

Great progress has been made in this field from thehydraulic method to FEM in deterministic analysismethod from a low homogeneous dam to a high complexdam in analysis object and from explicit formula to usingFEM to construct SRSM in LSF but reliability calculationis still in 2D stage e difficulties in analyzing the bypassseepage stability of a complex reinforced earth-rockfilldam are (1) the LSF of seepage failure is highly nonlinearand cannot be expressed explicitly (2) bypass seepage ofcomplex reinforced earth-rockfill dam is a 3D problemand large number of random variables will make re-liability analysis too computationally intensive At pres-ent JC method is only applicable to an explicit LSF andlowly nonlinear e stochastic FEM has a long devel-opment cycle and it is difficult to develop a uniformstochastic FEM program to suit for different problemse actual LSF can only be well fitted by an ordinary RSMin the vicinity of design point through adjusting responsesurfaces but it can be well fitted by the SRSM with anoptimal order polynomial chaos expansion in the entirespace [10] e SRSM just needs one response surface andcan solve the problem with implicit LSF by using lesssamples than ordinary RSM but high-order Hermitepolynomials are extremely complicated [11] erefore itis necessary to explore a method for the reliability analysisof 3D bypass seepage stability of earth-rockfill dam whichconsiders 3D seepage analysis and SRSM with simplepolynomials

e objective of this paper is to propose a practicalSRSM with ordinary high-order polynomials as LSF toanalyze the reliability of 3D bypass seepage stability ofearth-rockfill dams To achieve this goal this article isorganized as follows In Section 2 a 3D seepage FEM isintroduced briefly In Section 3 procedure of the high-order practical SRSM is discussed in detail and two nu-merical examples are investigated to demonstrate thevalidity of the proposed method Bantou reinforced dam istaken as an example in Section 4 reliability analysis forbypass seepage stability is calculated based on the upperwater level and 5 HCs of materials as random variables einfluence of the orders and Cov of random variables arealso analyzed

2 Seepage FEM for Earth-Rockfill Dam

e seepage model of an earth dam with free surface andseepage surface is shown in Figure 1 e dam body can bedivided into a saturation zone Ω1 and an unsaturated zoneΩ2 According to Darcyrsquos law the waterhead at any point canbe expressed as follows

z

zxkx

zh

zx1113888 1113889 +

z

zyky

zh

zy1113888 1113889 +

z

zzkz

zh

zz1113888 1113889 0 (1)

where h is waterhead and kx kv and kz are the HCs in x yand z directions e corresponding boundaries of formula(1) are listed as follows

(1) AB and CD are the Dirichlet boundaries of water-head hAB H1 and hC D H2

(2) BC is the Neumann boundary of flow the flowqBC 0

(3) AE is the free surface of seepage the interface of Ω1and Ω2 e water level on the surface is equal to thepiezometric waterhead z If there is no flow exchangebetween the two regions hAE z and vAE 0

(4) DE is the seepage surface Flow outflow from the EDsurface that is hDE z and vDE le 0

e solution of equation (1) is equivalent to the minimumvalue of the following formula

I[h(x y)] CΩ

12

kx

zh

zx1113888 1113889

2

+ ky

zh

zy1113888 1113889

2

+ kz

zh

zz1113888 1113889

2⎡⎣ ⎤⎦dx dy dz

(2)

Equation (2) is derived by the node waterhead in theseepage area [K]e h

e 0 can be obtained Functionaldifferential is conducted on all elements multiply theconduction matrix [K]e and the node waterhead h

emoving the nodes with the given waterhead to the right ofthe equation formula (3) can be obtained

[K] h Q (3)

where [K] is conduction matrix h is waterhead and Q isthe initial flow Since formula (3) was solved by Neumann in1973 there have been several constant mesh methods suchas initial flow method [12] improved method of nodalvirtual flux [13] Signorini variational inequality method[14] and equivalent nodal flow method [15]

If h at any FEM nodes is solved the seepage field will bedetermined e total seepage gradient at any point can bedefined as

J

minuszh

zx1113888 1113889

2

+ minuszh

zy1113888 1113889

2

+ minuszh

zz1113888 1113889

2

11139741113972

(4)

3 High-Order Practical SRSM

e SRSM was first proposed in the study of uncertainproblem in environmental and biological system by Isu-kapalli et al [16] in 1998 e SRSM combining a traditionaldeterministic method and the uncertainty method not onlyovercomes the difficulty that the LSF cannot be expressedexplicitly but also greatly decreases the amount of responsecalculation of samples compared with the traditional directMCS An important procedure with the SRSM is to choosethe RSP e SRSM with Chebyshev polynomials to



g(x) a + 1113944n

i11113944

ki

p1bipx

pi + 1113944

m

q1cq1113945

n

i1x

piq

i (5)



1113937ni1x

piq






1 can be obtained


ni (6)





I = I + 1



I lt 2Yes

Result output

No


H1

H2

Z

AG F

Ω2

Ω1


D

CB0





Rj 1113944100

i1



21113970

j 1 2 6

(7)







2R21 gt 09995





Qi σ2gi

1113936ni1σ2gi

times 100 (8)












xiprime

xi

maxxi

(9)







pf Nf

N (10)


ndash4 0 2 4ndash2x

ndash4

ndash2

0

2

4

y


(a)

y

ndash4

ndash2

0

2

4

ndash2 0 2 4ndash4x


(b)





















2R21 are 09905 08368 and 08273 re-















OI samples


Additional samples

1 5 49 3471 5 15 50 460702 0 minus 01 minus 180 6 20 00 798203 5 149 37874 7 15 149 1001744 0 199 78626 8 20 199 238226


30200 10ndash10x


0

5000

10000

15000

20000

25000

g

(a)

30200 10ndash10y


0

5000

10000

15000

20000

g

(b)




21 Keepdiscard

x

x x2 x3 10000x2 x3 10000 Discard x

x x3 09991 Keep x2


y

y y2 y3 10000y2 y3 09976 Keep y
















ndash5

ndash2

1

4

7

y



Mean value point

True LSF








ndash2 0 2 4ndash4x

ndash3

ndash1

1

3

5

y














Alluvial deposits





ZY

X

Figure 9 FEM model




Cov 006 006 012 016 013 009





7000613152634394352526561788919050

Waterhead (m)


040

036

031

027

023

018

014

009

005

Hydro-gradient

Z

Y

X


040

035

030

026

021

016

012

007

002

Hydro-gradient

Z

Y

X





+ 00242X32 minus 07990X1X2 + 04168X1X

22

+ 01724X1X3 + 07012X1X4 minus 00856X2X3

+ 02167X22X3 + 08403X2X4 minus 06684X

22X4

+ 01289X3X4




043

039

035

031

027

Seep

age g

radi

ent

y = 00113x ndash 31313R2 = 09965

(a)

038

037

036

035

034Se

epag

e gra

dien

t

R2 = 09955


y = 00102x3 ndash 00649x2 + 01172x + 03018

(b)

Seep

age g

radi

ent

R2 = 09987


042

039

036

033

030

y = ndash00209x + 04632

(c)

Seep

age g

radi

ent

R2 = 09971


041

038

035

032

029

y = ndash0003x2 + 00605x + 01268

(d)




HC of cut-offwall

HC of filled soilX2

HC ofrockfill


HC of curtainX4











044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0788

(a)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0085


(b)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = ndash0434


(c)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0403


(d)




5 Conclusions




025 030 035 040 045


Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF



14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)









Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









g(x) a + 1113944n

i11113944

ki

p1bipx

pi + 1113944

m

q1cq1113945

n

i1x

piq

i (5)



1113937ni1x

piq






1 can be obtained


ni (6)





I = I + 1



I lt 2Yes

Result output

No


H1

H2

Z

AG F

Ω2

Ω1


D

CB0





Rj 1113944100

i1



21113970

j 1 2 6

(7)







2R21 gt 09995





Qi σ2gi

1113936ni1σ2gi

times 100 (8)












xiprime

xi

maxxi

(9)







pf Nf

N (10)


ndash4 0 2 4ndash2x

ndash4

ndash2

0

2

4

y


(a)

y

ndash4

ndash2

0

2

4

ndash2 0 2 4ndash4x


(b)





















2R21 are 09905 08368 and 08273 re-















OI samples


Additional samples

1 5 49 3471 5 15 50 460702 0 minus 01 minus 180 6 20 00 798203 5 149 37874 7 15 149 1001744 0 199 78626 8 20 199 238226


30200 10ndash10x


0

5000

10000

15000

20000

25000

g

(a)

30200 10ndash10y


0

5000

10000

15000

20000

g

(b)




21 Keepdiscard

x

x x2 x3 10000x2 x3 10000 Discard x

x x3 09991 Keep x2


y

y y2 y3 10000y2 y3 09976 Keep y
















ndash5

ndash2

1

4

7

y



Mean value point

True LSF








ndash2 0 2 4ndash4x

ndash3

ndash1

1

3

5

y














Alluvial deposits





ZY

X

Figure 9 FEM model




Cov 006 006 012 016 013 009





7000613152634394352526561788919050

Waterhead (m)


040

036

031

027

023

018

014

009

005

Hydro-gradient

Z

Y

X


040

035

030

026

021

016

012

007

002

Hydro-gradient

Z

Y

X





+ 00242X32 minus 07990X1X2 + 04168X1X

22

+ 01724X1X3 + 07012X1X4 minus 00856X2X3

+ 02167X22X3 + 08403X2X4 minus 06684X

22X4

+ 01289X3X4




043

039

035

031

027

Seep

age g

radi

ent

y = 00113x ndash 31313R2 = 09965

(a)

038

037

036

035

034Se

epag

e gra

dien

t

R2 = 09955


y = 00102x3 ndash 00649x2 + 01172x + 03018

(b)

Seep

age g

radi

ent

R2 = 09987


042

039

036

033

030

y = ndash00209x + 04632

(c)

Seep

age g

radi

ent

R2 = 09971


041

038

035

032

029

y = ndash0003x2 + 00605x + 01268

(d)




HC of cut-offwall

HC of filled soilX2

HC ofrockfill


HC of curtainX4











044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0788

(a)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0085


(b)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = ndash0434


(c)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0403


(d)




5 Conclusions




025 030 035 040 045


Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF



14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)









Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018










Rj 1113944100

i1



21113970

j 1 2 6

(7)







2R21 gt 09995





Qi σ2gi

1113936ni1σ2gi

times 100 (8)












xiprime

xi

maxxi

(9)







pf Nf

N (10)


ndash4 0 2 4ndash2x

ndash4

ndash2

0

2

4

y


(a)

y

ndash4

ndash2

0

2

4

ndash2 0 2 4ndash4x


(b)





















2R21 are 09905 08368 and 08273 re-















OI samples


Additional samples

1 5 49 3471 5 15 50 460702 0 minus 01 minus 180 6 20 00 798203 5 149 37874 7 15 149 1001744 0 199 78626 8 20 199 238226


30200 10ndash10x


0

5000

10000

15000

20000

25000

g

(a)

30200 10ndash10y


0

5000

10000

15000

20000

g

(b)




21 Keepdiscard

x

x x2 x3 10000x2 x3 10000 Discard x

x x3 09991 Keep x2


y

y y2 y3 10000y2 y3 09976 Keep y
















ndash5

ndash2

1

4

7

y



Mean value point

True LSF








ndash2 0 2 4ndash4x

ndash3

ndash1

1

3

5

y














Alluvial deposits





ZY

X

Figure 9 FEM model




Cov 006 006 012 016 013 009





7000613152634394352526561788919050

Waterhead (m)


040

036

031

027

023

018

014

009

005

Hydro-gradient

Z

Y

X


040

035

030

026

021

016

012

007

002

Hydro-gradient

Z

Y

X





+ 00242X32 minus 07990X1X2 + 04168X1X

22

+ 01724X1X3 + 07012X1X4 minus 00856X2X3

+ 02167X22X3 + 08403X2X4 minus 06684X

22X4

+ 01289X3X4




043

039

035

031

027

Seep

age g

radi

ent

y = 00113x ndash 31313R2 = 09965

(a)

038

037

036

035

034Se

epag

e gra

dien

t

R2 = 09955


y = 00102x3 ndash 00649x2 + 01172x + 03018

(b)

Seep

age g

radi

ent

R2 = 09987


042

039

036

033

030

y = ndash00209x + 04632

(c)

Seep

age g

radi

ent

R2 = 09971


041

038

035

032

029

y = ndash0003x2 + 00605x + 01268

(d)




HC of cut-offwall

HC of filled soilX2

HC ofrockfill


HC of curtainX4











044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0788

(a)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0085


(b)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = ndash0434


(c)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0403


(d)




5 Conclusions




025 030 035 040 045


Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF



14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)









Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018











xiprime

xi

maxxi

(9)







pf Nf

N (10)


ndash4 0 2 4ndash2x

ndash4

ndash2

0

2

4

y


(a)

y

ndash4

ndash2

0

2

4

ndash2 0 2 4ndash4x


(b)





















2R21 are 09905 08368 and 08273 re-















OI samples


Additional samples

1 5 49 3471 5 15 50 460702 0 minus 01 minus 180 6 20 00 798203 5 149 37874 7 15 149 1001744 0 199 78626 8 20 199 238226


30200 10ndash10x


0

5000

10000

15000

20000

25000

g

(a)

30200 10ndash10y


0

5000

10000

15000

20000

g

(b)




21 Keepdiscard

x

x x2 x3 10000x2 x3 10000 Discard x

x x3 09991 Keep x2


y

y y2 y3 10000y2 y3 09976 Keep y
















ndash5

ndash2

1

4

7

y



Mean value point

True LSF








ndash2 0 2 4ndash4x

ndash3

ndash1

1

3

5

y














Alluvial deposits





ZY

X

Figure 9 FEM model




Cov 006 006 012 016 013 009





7000613152634394352526561788919050

Waterhead (m)


040

036

031

027

023

018

014

009

005

Hydro-gradient

Z

Y

X


040

035

030

026

021

016

012

007

002

Hydro-gradient

Z

Y

X





+ 00242X32 minus 07990X1X2 + 04168X1X

22

+ 01724X1X3 + 07012X1X4 minus 00856X2X3

+ 02167X22X3 + 08403X2X4 minus 06684X

22X4

+ 01289X3X4




043

039

035

031

027

Seep

age g

radi

ent

y = 00113x ndash 31313R2 = 09965

(a)

038

037

036

035

034Se

epag

e gra

dien

t

R2 = 09955


y = 00102x3 ndash 00649x2 + 01172x + 03018

(b)

Seep

age g

radi

ent

R2 = 09987


042

039

036

033

030

y = ndash00209x + 04632

(c)

Seep

age g

radi

ent

R2 = 09971


041

038

035

032

029

y = ndash0003x2 + 00605x + 01268

(d)




HC of cut-offwall

HC of filled soilX2

HC ofrockfill


HC of curtainX4











044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0788

(a)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0085


(b)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = ndash0434


(c)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0403


(d)




5 Conclusions




025 030 035 040 045


Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF



14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)









Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018
























2R21 are 09905 08368 and 08273 re-















OI samples


Additional samples

1 5 49 3471 5 15 50 460702 0 minus 01 minus 180 6 20 00 798203 5 149 37874 7 15 149 1001744 0 199 78626 8 20 199 238226


30200 10ndash10x


0

5000

10000

15000

20000

25000

g

(a)

30200 10ndash10y


0

5000

10000

15000

20000

g

(b)




21 Keepdiscard

x

x x2 x3 10000x2 x3 10000 Discard x

x x3 09991 Keep x2


y

y y2 y3 10000y2 y3 09976 Keep y
















ndash5

ndash2

1

4

7

y



Mean value point

True LSF








ndash2 0 2 4ndash4x

ndash3

ndash1

1

3

5

y














Alluvial deposits





ZY

X

Figure 9 FEM model




Cov 006 006 012 016 013 009





7000613152634394352526561788919050

Waterhead (m)


040

036

031

027

023

018

014

009

005

Hydro-gradient

Z

Y

X


040

035

030

026

021

016

012

007

002

Hydro-gradient

Z

Y

X





+ 00242X32 minus 07990X1X2 + 04168X1X

22

+ 01724X1X3 + 07012X1X4 minus 00856X2X3

+ 02167X22X3 + 08403X2X4 minus 06684X

22X4

+ 01289X3X4




043

039

035

031

027

Seep

age g

radi

ent

y = 00113x ndash 31313R2 = 09965

(a)

038

037

036

035

034Se

epag

e gra

dien

t

R2 = 09955


y = 00102x3 ndash 00649x2 + 01172x + 03018

(b)

Seep

age g

radi

ent

R2 = 09987


042

039

036

033

030

y = ndash00209x + 04632

(c)

Seep

age g

radi

ent

R2 = 09971


041

038

035

032

029

y = ndash0003x2 + 00605x + 01268

(d)




HC of cut-offwall

HC of filled soilX2

HC ofrockfill


HC of curtainX4











044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0788

(a)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0085


(b)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = ndash0434


(c)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0403


(d)




5 Conclusions




025 030 035 040 045


Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF



14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)









Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018
















OI samples


Additional samples

1 5 49 3471 5 15 50 460702 0 minus 01 minus 180 6 20 00 798203 5 149 37874 7 15 149 1001744 0 199 78626 8 20 199 238226


30200 10ndash10x


0

5000

10000

15000

20000

25000

g

(a)

30200 10ndash10y


0

5000

10000

15000

20000

g

(b)




21 Keepdiscard

x

x x2 x3 10000x2 x3 10000 Discard x

x x3 09991 Keep x2


y

y y2 y3 10000y2 y3 09976 Keep y
















ndash5

ndash2

1

4

7

y



Mean value point

True LSF








ndash2 0 2 4ndash4x

ndash3

ndash1

1

3

5

y














Alluvial deposits





ZY

X

Figure 9 FEM model




Cov 006 006 012 016 013 009





7000613152634394352526561788919050

Waterhead (m)


040

036

031

027

023

018

014

009

005

Hydro-gradient

Z

Y

X


040

035

030

026

021

016

012

007

002

Hydro-gradient

Z

Y

X





+ 00242X32 minus 07990X1X2 + 04168X1X

22

+ 01724X1X3 + 07012X1X4 minus 00856X2X3

+ 02167X22X3 + 08403X2X4 minus 06684X

22X4

+ 01289X3X4




043

039

035

031

027

Seep

age g

radi

ent

y = 00113x ndash 31313R2 = 09965

(a)

038

037

036

035

034Se

epag

e gra

dien

t

R2 = 09955


y = 00102x3 ndash 00649x2 + 01172x + 03018

(b)

Seep

age g

radi

ent

R2 = 09987


042

039

036

033

030

y = ndash00209x + 04632

(c)

Seep

age g

radi

ent

R2 = 09971


041

038

035

032

029

y = ndash0003x2 + 00605x + 01268

(d)




HC of cut-offwall

HC of filled soilX2

HC ofrockfill


HC of curtainX4











044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0788

(a)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0085


(b)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = ndash0434


(c)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0403


(d)




5 Conclusions




025 030 035 040 045


Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF



14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)









Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018




















ndash5

ndash2

1

4

7

y



Mean value point

True LSF








ndash2 0 2 4ndash4x

ndash3

ndash1

1

3

5

y














Alluvial deposits





ZY

X

Figure 9 FEM model




Cov 006 006 012 016 013 009





7000613152634394352526561788919050

Waterhead (m)


040

036

031

027

023

018

014

009

005

Hydro-gradient

Z

Y

X


040

035

030

026

021

016

012

007

002

Hydro-gradient

Z

Y

X





+ 00242X32 minus 07990X1X2 + 04168X1X

22

+ 01724X1X3 + 07012X1X4 minus 00856X2X3

+ 02167X22X3 + 08403X2X4 minus 06684X

22X4

+ 01289X3X4




043

039

035

031

027

Seep

age g

radi

ent

y = 00113x ndash 31313R2 = 09965

(a)

038

037

036

035

034Se

epag

e gra

dien

t

R2 = 09955


y = 00102x3 ndash 00649x2 + 01172x + 03018

(b)

Seep

age g

radi

ent

R2 = 09987


042

039

036

033

030

y = ndash00209x + 04632

(c)

Seep

age g

radi

ent

R2 = 09971


041

038

035

032

029

y = ndash0003x2 + 00605x + 01268

(d)




HC of cut-offwall

HC of filled soilX2

HC ofrockfill


HC of curtainX4











044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0788

(a)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0085


(b)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = ndash0434


(c)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0403


(d)




5 Conclusions




025 030 035 040 045


Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF



14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)









Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018

















Alluvial deposits





ZY

X

Figure 9 FEM model




Cov 006 006 012 016 013 009





7000613152634394352526561788919050

Waterhead (m)


040

036

031

027

023

018

014

009

005

Hydro-gradient

Z

Y

X


040

035

030

026

021

016

012

007

002

Hydro-gradient

Z

Y

X





+ 00242X32 minus 07990X1X2 + 04168X1X

22

+ 01724X1X3 + 07012X1X4 minus 00856X2X3

+ 02167X22X3 + 08403X2X4 minus 06684X

22X4

+ 01289X3X4




043

039

035

031

027

Seep

age g

radi

ent

y = 00113x ndash 31313R2 = 09965

(a)

038

037

036

035

034Se

epag

e gra

dien

t

R2 = 09955


y = 00102x3 ndash 00649x2 + 01172x + 03018

(b)

Seep

age g

radi

ent

R2 = 09987


042

039

036

033

030

y = ndash00209x + 04632

(c)

Seep

age g

radi

ent

R2 = 09971


041

038

035

032

029

y = ndash0003x2 + 00605x + 01268

(d)




HC of cut-offwall

HC of filled soilX2

HC ofrockfill


HC of curtainX4











044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0788

(a)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0085


(b)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = ndash0434


(c)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0403


(d)




5 Conclusions




025 030 035 040 045


Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF



14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)









Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018











7000613152634394352526561788919050

Waterhead (m)


040

036

031

027

023

018

014

009

005

Hydro-gradient

Z

Y

X


040

035

030

026

021

016

012

007

002

Hydro-gradient

Z

Y

X





+ 00242X32 minus 07990X1X2 + 04168X1X

22

+ 01724X1X3 + 07012X1X4 minus 00856X2X3

+ 02167X22X3 + 08403X2X4 minus 06684X

22X4

+ 01289X3X4




043

039

035

031

027

Seep

age g

radi

ent

y = 00113x ndash 31313R2 = 09965

(a)

038

037

036

035

034Se

epag

e gra

dien

t

R2 = 09955


y = 00102x3 ndash 00649x2 + 01172x + 03018

(b)

Seep

age g

radi

ent

R2 = 09987


042

039

036

033

030

y = ndash00209x + 04632

(c)

Seep

age g

radi

ent

R2 = 09971


041

038

035

032

029

y = ndash0003x2 + 00605x + 01268

(d)




HC of cut-offwall

HC of filled soilX2

HC ofrockfill


HC of curtainX4











044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0788

(a)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0085


(b)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = ndash0434


(c)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0403


(d)




5 Conclusions




025 030 035 040 045


Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF



14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)









Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018










+ 00242X32 minus 07990X1X2 + 04168X1X

22

+ 01724X1X3 + 07012X1X4 minus 00856X2X3

+ 02167X22X3 + 08403X2X4 minus 06684X

22X4

+ 01289X3X4




043

039

035

031

027

Seep

age g

radi

ent

y = 00113x ndash 31313R2 = 09965

(a)

038

037

036

035

034Se

epag

e gra

dien

t

R2 = 09955


y = 00102x3 ndash 00649x2 + 01172x + 03018

(b)

Seep

age g

radi

ent

R2 = 09987


042

039

036

033

030

y = ndash00209x + 04632

(c)

Seep

age g

radi

ent

R2 = 09971


041

038

035

032

029

y = ndash0003x2 + 00605x + 01268

(d)




HC of cut-offwall

HC of filled soilX2

HC ofrockfill


HC of curtainX4











044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0788

(a)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0085


(b)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = ndash0434


(c)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0403


(d)




5 Conclusions




025 030 035 040 045


Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF



14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)









Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018
















044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0788

(a)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0085


(b)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = ndash0434


(c)

044

039

034

029

024

Seep

age g

radi

ent

N = 5000R = 0403


(d)




5 Conclusions




025 030 035 040 045


Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF



14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)









Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









5 Conclusions




025 030 035 040 045


Seepage gradient

1E ndash 03

1E ndash 02

1E ndash 01

1E + 00

CDF



14

12

10

8

6

4

2

0Fa

ilure

pro

babi

lity

(10ndash5

)









Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018











Data Availability




Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018






















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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