STABILITY ANALYSIS OF RIVERBANK SUBJECT TO SEEPAGE

STABILITY ANALYSIS OF RIVERBANK SUBJECT TO SEEPAGEYan LU', Yongjun LU, and Xingnong ZHANGABSTRACTThe stability of riverbanks subject to seepage is studied experimentally and theoretically in this paper.By including seepage in a 3-dimensional theoretical analysis, the study first shows how the critical slopeor angle of repose of a cohesionless material is related to the ratio of the hydraulic gradient of seepage toits critical value under the fluidization condition. The critical stable slope is shown to be related to notonly the hydrauli gradient but also the seepage direction. Measured lboratory data reasonably fit wellwith the theoretical relationship for the case of injection and suction. The data reveal that the slope isreduced with injection and increased with suction, respectively. Additionally, the study identifies theseepage direction which results in a minimum critical stable slope for a certain hydraulic gradient ofseepage.Key Words: Seepage, Riverbank, Stability, Critical stable slopes, Angle of repose, Erosion1 INTRODUCTIONRiver bank collapse is a common phenomenon in alluvial channels, rivers and streams that gives rise todiverse practical problems. Extensive studies on alluvial bank collapse have been undertaken withvarying degrees of success, but their scope has often been restricted to impermeable boundary. Quiteoften, banks may fail after periods of high stage and in locations where deposition would be anticipated.This is because outflow of seepage, if concentrated, tends to remove solid particles in the exfiltration zoneto create tubular underground conduits, i.e, piping, and consequently results in instability of the bank(Hagerty, 1991a; 1991b). Although the magnitude of natural seepage flow is often small compared withthe total flow, the effect produced by it can be significant on the flow characteristics (Cheng and Chiew1998; Dey 2003); incipient motion of sand particles (Cheng and Chiew 1999; Xic and Yu, 2006), withattendant erosion of the river bed and bank; sediment transport (Willets and Drossos, 1975); and thestability of bed and bank (Shen, 1971). Therefore, it is important to study the mechanisms wherebyoutflow of seepage causes bank instability.The stability of the riverbank with respect to mass failure depends on the angle of repose of the sedimentparticles consisting of the bank. Therefore, research work on the angle of repose of sediments deservesto be mentioned here. A large number of experimental data on the angle of repose of submergedcohesionless sediment particles have been reported; but the discrepancies among the results aresignificant which may mainly be explained by the difference in testing methodologies and in sedimentproperties (Chien and Wan, 1998). Eagleson and Dean (1959) conducted a series of experiments inwhich the angle of repose of spherical particles resting on two different types of natural sand, withmedian diamieters = 1.83 mm and 0.79 mm were determined. Their experimental data showed that theangle of repose of the top spheres with various sizes and densities ranged from nearly 50° to less than 20°.Miller and Byme (1966), on the other hand, performed experiments to examine particle-shape effects onthe angle of repose and proposed an empirical equation for predicting the angle of repose asφ=5(k/K)", in which φ is the average angle of repose of a single particle on aroughbed; K/K isthe ratio of the diameter of a single particle to the average diameter of the bed particles; ξ is a| Postdoctoral Fellow, Nanjing Hydraulic Research Instute, State Key Laboratory of Hydrology Water Resourcesand Hydraulic Engineering Science, Nanjing, China 210029.中国煤化工Prof., Nanjing Hydraulic Research Institute, Nanjing. China 23 Prof.. Nanjing Hydraulic Research Institute, Nanjing, China 2YHCNMHGNote: The original manuscript of this paper was received in Feb. 2007. Ihe revised version was received in Sept.2007. Discussion open until Dec.2008.-282.International Journal of Sediment Research, Vol. 22, No. 4, 2007, pp. 282- 291parameter incorporating the effects of shape and roundness in both particle and bed; and η is aparameter incorporating the effects of sorting of the bed particles. All these studies were conducted underconditions where seepage is absent. If seepage is present, the sediment particles experience an additionalhydrodynamic force, which renders a more complex boundary condition for slope stability consideration.Very few studies have been done to explore the response of the angle of repose to seepage flow. Amongstthe studies that can be found in the literature is the one by Rhee and Bezuijen (1992), who proposed twostability criteria, namely, the continuum mode and single-particle mode to account for sandy slopestability in the presence of seepage. Based on their experimental results, they concluded that for injection,the continuum mode theory governs the failure process, whereas in the case of suction, the failure modecan be described by the single particle mode theory. More recently, Lu (2006) examined the influence ofseepage force on the critical sandy slope and proposed a general expression for the angle of repose in thecase of injection and suction. In the above staterment, injection and suction refer to the case where seepageflow through the channel bed in an upward and a downward direction, respectively.Seepage through boundaries of alluvial channels, rivers and streams is a common occurrence due to theporosity of the granular material. Water is continuously seeping into or out of the channel banks becauseof the different level between the free water surface in the channel and the adjoining groundwater table. Aseepage flow net within a riverbank slope is schematically shown in Fig. I. The principle objective of thepresent study is to develop certain concepts on how the magnitude and direction of the seepage affects thestability of a riverbank.BA__ 又Fig. 1 Schematic sketch of seepage flow within a riverbank2 THEORETICAL CRITICAL SLOPEFor a sandy slope consisting of non-cohesive uniform spherical sediments in which the particles aresubjected to seepage alone, i.e., without any surface flow; the geometrical layout of and forces acting onthe sediment particles on the top layer is shown in Fig. 2. The figure shows how in a three- dimensionalconfiguration, one sphere (Sphere I) is located on top of three other spheres. At the onset of motion,Sphere 1 experiences seepage force Fs, submerged weight force W, normal force N and tangential force Tfrom either Sphere 3 or Sphere 4 or both (no reaction forces from Sphere 2 are considered because theyare zero when Sphere 1 is just about to move). If Sphere 1 is sliding down, T = μN, where up iscoefficient of friction between the spherical surfaces. Ling et al. (1992) studied a similar problem inwhich the only driving force on the spbere is gravity. They numerically solved the equations of motionand found that when motion just begins, the sphere starts with rolling because T is usually much smallerthan yN. In fact, sliding is limited to consuming up to only about 1% of the kinetic energy of the sphere.In light of the results of their study, rolling is assumed in the present analysis.In the three dimensional configuration, the reaction forces lie on the edges of a tetrahedron formed bythe points of contact of the spheres and the center of gravity' 2h). Points E, F and .G are the contact points of the top particle with the three und中国煤化工03 03 and 04 arethe centers of the four spheres; and Points 0 and 0。arYHCN M H Ges formed by thecenters of the three underlying spheres and that formed by the contact points E, F and G, respectively.International Joumal of Sediment Research, Vol. 22, No. 4, 2007, pp. 282 -291- 283-When Sphere 1 is just about to move, the moment equation about a nominal pivot point P can be writtena;θ,FN42 T4|321Elevation viewPlan viewFig. 2 Forces acting on a single sphereSM=F;e-W.e2=0(1)e =0,P-sin(φ+θ)(2)e, = oP.sin(φ-a)(3)in which e and e, are the moment arms of seepage and submerged weight forces, respectively; a isthe critical angle of the slope with seepage; θ is direction of the seepage flow shown in Fig. 2,measured clockwise from the normal to the slope and ranges from0 to 2π .EE,GPFO2 <0。过AJO3(a)Fig. 3 Geometrical relationships of spheres with different orientationsBy substituting (2) and (3) into (1), one getsF二sin(φ-a)w(4)sin(φ+ 0)The submerged weight of the top sphere is given byW=E D'(p,(5) .中国煤化工where D = diameter of the particle; ρ, = density of spher.RYHCN M H Gad g= gaitionalacceleration.284 -Intemational Joumal of Sediment Research, Vol. 22, No 4, 2007, pp. 282- 291If the hydraulic gradient is taken as positive for seepage directed out of slope, the force exerted byseepage, S, on the sediment particles on the top layer of the slope (Fig. 4) can be derived by using Darcy'slaw, noting the layer width is D:入、SDSeepageFig. 4 Analysis of seepage force on a sliceS = ipgD'l,(6)where i = hydraulic gradient of seepage. Since the volume occupied by the sediment particles over anarbitrary length, Is is D x D x l, the number of particles in this layer isn=61,(1-8)(7)where E = porosity of the sediment. Thus, the seepage force acting on a particle can be computed bydividing (6) with (7)。 ipgD'Fg=6(1-ε)(8)In the case of coarse particles, there is no or negligible pressure gradient on the upper half of theparticles at the top layer of the slope (Rhee and Bezuijen, 1992); therefore, the bydraulic gradient ofseepage flow has effect only on the lower half of the top particles. To account for this, a coefficient β isadded to (8) resulting inp = ipguD'β9)6(1-E)where β is dependent on particle size, and ranges between 0.5 for coarse granular materials and 1 forfine particles. Substituting (5) and (9) into (4) leads toi=(-e)e-P sin(0-a)(10)ρ βsin(φ+θ)Equation (10) implies that φ is the maximal slope angle, ie, the angle of repose of the uniformspherical particle for the case without seepage.By dividing (10) with equation (1 l) that describes the critical condition of fluidization on a horizontalbed (Cheng and Chiew, 1999)i=(-e)B-卫(11)one gets中国煤化工(12)i. βsin(φMYHCNMHGIt is necessary to note that in the case of φ+θ=π，or 2π，a=φ, defining a transition point ofIntemational Jourmal of Sediment Research, Vol. 22, No.4, 2007, pp. 282 -291-285-positive or negative effects on particle stability due to seepage direction. When 2π-φ>θ>π-φ,seepage increases the slope stability; when π-φ>θ>0 and 2π>θ> 2π-φ, seepage reduces thestability of the riverbank slope. For the case of injection or suction that has been investigated moreextensively in the literature, i.e, θ=0 or π, Eq. (12) reduces to (13), noting i takes negative in thecase of suction.sin(φ-a)(13)βsin($)2.1 Hydraulic GradientThe hydraulic gradient that sustains the movement of water through the porous medium is related to theseepage velocity. In most practical cases in soil mechanics the assumption of a linear relationshipbetween velocity and hydraulic gradient based on Darcy' s law is valid because laminar flow condition ispresent. However, if the flow velocity through coarse granular materials is comparatively large,nonlinear relationships could be present, as is proposed by many researchers to account for thetransitional and turbulent seepage. Kovacs (1981) grouped these nonlinear relationships into two maincategories:(a) Binomial formi=av, + bv,(14)(b) Power formi=cv"(15)in which v, = seepage velocity; a, b, C = empirical coefficients; and m = exponent which ranges between1 and 2.In this paper, Equation (15) is used to delineate the nonlinear relationship between the seepage velocityand the hydraulic gradient and the empirical coefficients, c and m, is determined experimentally for eachsediment sample.3 EXPERIMENTAL METHODOLOGYTwo series of experiments (Series-A, and Series-B) are conducted, each with its own distinctiveobjectives: Series-A aims to determine the coefficients of Eq. (15) for each sediment sample; Series-B isto verify the theoretical model, i.e., Eq. (13). Fig. 5 shows the experimental setup. In Series-A, a clearPerspex seepage conduit is used to perform permeability tests for granular materials. The rectangularconduit has a cross section of 14 cm by 21 cm, and a height of 80 cm (Fig 5a). Water is pumped fromthe bottom of the conduit through a special filter to ensure uniform seepage through the granular materials.Four manometers connected to the conduit are used for piezometric head measurements.In Series B, a glass-sided horizontal flume that is 30 m long, 0.7 m wide and 0.6 m deep (Fig. 5b) is .used to examine the effect of seepage on the critical stable slope. For simplicity, only injection andsuction have been investigated experimentally in the study. Water is re-circulated through a submersiblepump installed in the laboratory reservoir. The flow rate, which is monitored using an electromagneticflow meter, is controlled using a speed inverter and a valve. At the entrance to the flume, pipestraighteners are set to achieve uniform flow and to minimize large scale turbulence and circulations.Located at the middle of the flume is the test section in the form of a recess that is 2 m long and 0.2 mdeep, which spans the flume width. The setup of the seepage recess is schematically shown in Fig. 5(c).The permeable bed in the seepage zone is leveled to the elevation of the adjoining bed. Sand is placedon top of a filter net, which in turms, overlays a perforated metal plate. Water is allowed to seep throughthe perforated plate, filter net and sand layer to ensure uniform seepage flow within the granular materials.In the case of suction twelve identical pipes, each with a valve and a flow meter to respectively controland monitor the suction discharge, are fixed onto the bottom of the recess to drain water out uniformly.On the other hand a separate submersible pump is usflow discharge is中国煤化工monitored using another flow meter.TYHCNMHG-286-International Journal of Sediment Research, Vol. 22, No. 4, 2007, pp. 282- 291电To sump88FilterFlow meter一Ir二e-From submersible pump(a) Seepage conduitPipe straightenerTest section.号Headbox一12m8m110mReservoirPump(b) Open-channel flume200 cm|h- 00000020cm1Impermeable bedFilter'Perforated platePerforated pipes弃菩荆要宇垂Flow meter wih valvehor = From submersible numn(C) Seepage reessFig.5 Schematic sketch of experimental setupThree types of sediments are used in Series-A. Table 1 shows the properties of the sediments, in whichthe gradation coffcients is defined by δ=(D:/D% +.中国煤化工。their grain sizeCNMHG,distributions. The sediments, which are placed in the seepagctwluurl, u itst surred to release all airbubbles. Water is then slowly pumped into the conduit. Upon flow stabilization, the piezometricInternational Journal of Sediment Research, Vol. 22, No. 4, 2007, pp.282-291.287-heads are recorded for subsequent computation of the hydraulic gradient. Observation is made on themobility of the sand particles as the seepage velocity is increased gradually until the occurrence of thefluidization condition or the atanment of the maximum velocity. In this way a series of hydraulicgradients are obtained for the sediments.Table 1 Properties of sediments used in studyNo.Median grain diameter (mm)Gradation coefficientPorositySpecific gravity0.91.270.472.621.411.170.442.551.641.230.46100.00-。Sediment I80.00- 0 Sediment 260.00- - Sediment 340.0020.000.000.101.0010.00Sieve size (mm)Fig. 6 Grain size distributions of sediment samplesIn Series- B, the same three types of sediments are used; the relevant information was detailed previouslyby Lu and Chiew (2007).4 EXPERIMENTAL RESULTS AND DISCUSSIONThe value of c and m in (15) is first determined by plotting the experimental data collected fromSeries-A in Fig. 7. The data show that the hydraulic gradient and seepage velocity fit well with theexponential function.m。SedimentI 125.35 1.00450.8| 0 Sediment2 98.21 1.0374| o Sediment3 91.80 1.2129 |0.6 t号0.4-0.2 t0.005 .0.010.015Seepage velocity (m/s)Fig. 7 Nonlinear relationship between hydraulic gradient and seepage velocityIn Fig. 8, the results of the critical stable slope measured from tests conducted in Series-B are plottedagainst the seepage hydraulic gradients for the three sediments used._ The data show that the presence ofseepage causes a significant influence on the critical slo中国煤化工critical slope whilesuction has the opposite effect. The theoretical equatYHCNMHG; of β=0.75 andφ=34° is also superimposed in Fig. 8, and it compares well wIn Ine expermental data. It is-288-Interational Journal of Sediment Research, Vol. 22, No. 4, 2007. pp. 282-291worthwhile noting that the choice of β= 0.75 and φ= 34° in the computation for each particle used is .quite arbitrary, since the parameters vary with the particle size and shape.0r0t。e中口o。Seidment 1a 20口Sediment 20 Sediment 30F. Equation(13)-0.6-0.4-0.200.20.40.6 .SuctionInjectionFig. 8 Critical slope with seepage normal to slopeNow, consider a seepage flow often occurred within a riverbank as shown in Fig. 1. On the slopesurface, the direction of seepage ranges from θ=0° at A to θ=90° at B. A plot of (12) withβ=0.75 and φ=34° is shown in Fig. 9 for the valid range of the direction of seepage flow on theslope surface. A family of curves is generated, each of which corresponds to various values of ilic .Fig. 9 depicts the trend of the curve for a as a function of日, resulting in a minimum value of criticalslope angle with seepage at θ=90°-φ. It is also shown in this figure that a larger seepage flowinduces a smaller critical stable slope, verifying the negative effect on the stability due to seepage withappropriate directions. Series of laboratory tests were conducted by Zhang et al., (2005) to investigatethe stability of riverbank as the extermnal water level was lowered at various rates. Their observationsgave that the increase in slope erosion is directly related to the seepage hydraulic gradient arising fromwithdrawal rate of the water level, which is in good agreement with that described in Eq. (12) or Fig.9.85 rilic = 0.4525 t=0.30重20i/ic =0.455t10203(405060708()0θ (degrees)Fig. 9 Relation of critical slope and seepage direct中国煤化工entsIn Fig. 1, the slope, originally at the angle of repose of theMYHc N M H Gwill cllapse onceseepage flow is formed due to the lower of the water level in the channel. The slope in the presence ofIntemational Journal of Sediment Research, Vol. 22, No.4, 2007, pp. 282- 291-289-seepage will not be stabilized until it reaches the minimum value of critical stable slope close related tothe seepage direction. For example, if the seepage hydraulic gradient is ili, =0.30，the slope willcollapse from a=φ=34° to x=17° , which is calculated from Eq. (12) and clearly shown in Fig. 9.Since the directions of the seepage flow within the riverbank are not necessary to be θ= 90" -φinducing the minimum critical value, the effects of seepage on the slope stability are admittedlyoverestimated. However, they at least afford a good approximation of the critical stable slope ofriverbank subjected to seepage.5 CONCLUSIONSDetailed investigations on the influence of seepage on the stability of riverbank are conducted. Thestudy first presents a theoretical relationship between the hydraulic gradient of seepage flow and thecritical slope angle by including seepage in a 3-dimensional force analysis. Seepage may induce theinstability of riverbank if the directions of seepage are in the ranges of π-φ>θ>0 and2π>θ> 2π-φ; increase the slope stability with their directions ranging from π-φ to 2π-φ.Laboratory data reasonably support the analytical solution on the critical slope in the case of bothinjection and suction. It is worthwhile to note that the seepage hydraulic applied herein is much smallerthan critical value of ig, which may induce the fluidization on a flat bed. If the seepage force is largeenough, say larger than the critical value, injection in any direction may induce the instability of ariverbank. The magnitude of such effects is shown to be relative to not only the hydraulic gradient butalso the seepage direction. The minimum critical stable slope with a certain hydraulic gradient isachieved when seepage flow directedat θ=90°-φ.ACKNOWLEDGEMENTSThe project has been supported by the National Basic Research Program (973) of China (GrantNo.2003CB415206), the Scientific Research Foundation for the Returned Overseas Chinese Scholars ofMinistry of Personnel, the Postdoctoral Foundation of Jiangsu Province, and the Science Foundation ofNanjing Hydraulic Research Institute.REFERENCESCheng, N. S. and Chiew, Y. M.1998, Turbulent open -channel flow with upward seepage. Journal of HydraulicResearch, Vol. 36, No. 3, pp. 415- 431.Cheng, N. S. and Chiew, Y. M.1999, Incipient sediment motion with upward seepage. Journal of Hydraulic Research,Vol.37, No.5, pp. 665-681.Chien, N. and Wan, z. H.1998, Mechanics of sediment transport. ASCE Press, Virginia.Dey, S. 2003, Nonuniform open channel flow with upward seepage through loose beds. 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H.2005, River collapse monitoring and protectionwith newly-technique. Research Report of Nanjing Hydraulic Research Institute (in Chinese).中国煤化工MYHCNMHGInternational Journal of Sediment Research, Vol. 22, No. 4, 2007, pp. 282-291-291-