The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India Prediction of the Geotechnical Effects Induced by Deep Drainage in Urban Area A. Cividini, S. Bonomi, G.C. Vignati, G. Gioda Dept. of Structural Engineering, Technical University (Politecnico) of Milan, Milan, Italy Keywords: seepage flow, granular soil, erosion, laboratory tests, finite elements ABSTRACT: The geotechnical effects are discussed of the erosion of fine particles caused by the seepage flow within a granular deposit. First some laboratory tests on reconstituted soil samples are illustrated that provide the quantity of eroded particles during time depending on the imposed hydraulic gradient. The experimental results permit defining an erosion law that has been implemented in a finite element code for erosion-transport analysis. After a brief description of the solution procedure, the results are presented of an application concerning a deep drainage well located within an urban area. The results of calculation provide the density of eroded particles in the vicinity of the well and a prediction of the possible settlements of the foundations of nearby buildings. 1Introduction The problem at hand originates from the construction of a number of new buildings within a large dismissed area located downtown Milan (Northern Italy). When fully operational the technical plants installed in the building will requireanotablewaterinflowfortheirfunctioning.Sincethephreatictableislocatedafewmetersbelowthe ground surface, a deep drainage system is planned to recover the necessary quantity of water from the saturated granular soil. The geotechnical design of the drainage system involves a series of experimental and numerical steps that can be summarized as follows: -Evaluation of the coefficient of hydraulic conductivity of the deposit through the back analysis of in situ pump-ing tests; -In situ investigation aimed at determining the deformability parameters of the granular soil; -Numerical evaluation of the water table lowering induced by the drainage system; -Estimation of the foundation settlements caused by the change of the effective stresses; -Laboratory tests on reconstituted soil samples to calibrate an empirical law relating the hydraulic gradient to the quantity of fine particles eroded by seepage; -Estimation of the settlement caused by the erosion process induced by the seepage flow. Here the discussion is limited to the erosion problem and to the steps that have been undertaken to evaluate the consequent foundation settlements. First a series of laboratory tests on reconstituted samples of the granular soil is illustrated aimed at defining the rate of fine particle erosion depending on the hydraulic gradient imposed during the tests. An erosion law is then derived from the experimental results suitable for implementation in a finite element code for erosion and trans-port analysis. After illustrating the main aspects of the solution procedure the finite element approach is applied to the evaluation of the quantity of eroded particles in the vicinity of the deep drainage well. These results are finally used as input data for evaluating the settlements that the erosion process could induce in nearby buildings. 2Testing procedure Various experimental studies have been presented in the literature concerning the erosion of the fine fraction of granular soils. Most of them are related to the design of filters for earth dam and for pumping wells. Sherard el al. (1984) and Lone et al. (2005) used samples having an upper part of fine soil and a lower part of fil-ter material. The samples where subjected to downward seepage flow to study the effectiveness of the filter. Simi-lar tests were carried out by Fannin and Moffat (2006) and by Tomlinson and Vaid (2000). They studied the inter-nal stability (i.e. the capacity of the coarse fraction of the soil to prevent the migration of its fine fraction under given hydraulic gradient) of sandy gravel and of artificial soils consisting of small spheres of glass. Skempton and Brogan (1994) carried out tests on samples consisting of sandy gravel subjecting them to upward seepageflow.Thispermitteddetectingtheonsetofcriticalgradientconditions(involvingvanishingeffective stresses) and to compare them with the corresponding theoretical predictions. Upward seepage flow was adopted also by Sterpi (2003) for tests on reconstituted samples of natural granular soil. They aimed at determining an The 12th International Conference of Geomechanics (IACMAG) 1-6 October, 2008 Goa, India experimental relationship betw ithout reaching critical condi-tions, and the quantity as necessary to this purpose; it permitted in fact recovering the eroded materialhe sample top measuring its increase during time. It also avoided possible clogging of the filter cles, condition that was observed un-der downward flow. The experimental setup adopted in t gure 1 and has been developed following the scheme presented in (Sterpi, 2003). lic head also this connection has a diameter of 8 of ASTM along the sample. An additional piezometer provides the hydraulic head at the base of the sample. The outflow of water from the sample top is conveyed to a lower container. irst the sample is saturated by slowly increasing the level of the upper reser-saturated, and hydrostatic conditions are reached, the reservoir is brought to the iameter) was eliminated before preparing the re-onstituted specimens to keep a ratio of about 1/10 between the diameter of the largest grains and that of the sample. As previously mentioned the soil was compacted in layers of small thickness within the permeameter using the International Association for Computer Methods and Advances in een the hydraulic gradient imposed during the tests, w of eroded fine particles. The application of an upward flow wfrom t used to recover the eroded fine partihe present study is shown in Fi Figure 1. Experimental setup for the erosion tests. The equipment consists of four cylindrical permeameters (only one of them is shown in Figure 1) with internal di-ameter of 8 cm. The permeameter bottom is connected to an upper water reservoir the level of which is kept con-stant during the test. To reduce to a minimum the loss of hydraucm. The granular soil sample is reconstituted within a cylindrical mould (7.4 cm in diameter and 20 cm in height) through the moist tamping technique (Ladd, 1978). The mould bottom consists of a steel wire mesh and No. 200 sieve to prevent the loss of soil. After sample preparation the mould is sealed within the permeameter so thatnowaterleakagecanoccurbetweenthem.Fourpiezometersplacedalongthepermeameterprovidethe pore pressure distributionThe test involves two main steps. Fvoir. When the sample is fully constant level chosen for the hydraulic loading phase of the test. The lower container receiving the water from the sample top is replaced at selected time intervals. The volume of water within it is measured, then the container is oven dried and the weight of the eroded fine soil is determined. The test terminates when the rate of erosion practically vanishes for a sufficient number of subsequent time inter-vals. Depending on the applied hydraulic gradient this condition was attained after more than 500 hours. The tests have been performed following a multi stage procedure. After reaching a vanishing rate of erosion for a given value of the hydraulic gradient, the elevation of the upper reservoir is increased and the test is continued under a higher gradient. 3Experimental results Thegranularsoilwasobtainedfromboringsperformedintheconstructionarea.Itconsistsofsiltysandand gravel with a content of fine particles (with diameter less than 0.074 mm) varying between 19.4% and 30.9% in weight. The gravel fraction (with grains exceeding 9.52 mm in dc The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India moist tamping technique. TDr of 70% and 30%. T ile the second one was in-vestigated to get some further insighton process. A total number of 16 tests was performed under hy is here considered to provide an insight into tThe grain Dr of 70%, which corresponds to dryequal to 44.4% (the minimum and maximum dry unit w st under hydraulic gradient i increasing from 0.20 to 0.99 (f0 initial density of the fine fraction; er total density of eroded particles). wo sets of tests were performed on samples with relative densityhe first one represents a reasonable approximation of the actual in situ conditions, whinto the characteristics of the erosidraulic gradient varying between 0.2 and 1.0. Only one of them he experimental results.size distribution of the tested material is shown in Figure 2. The sample has a relative density unit weight d equal to 1.50 g/cm3 and to porosity neights are 1.18 g/cm3 and 1.70 g/cm3). Figure 2. Grain size distribution of the tested sample. Figure 3 shows the results of a multi stage test in which the hydraulic gradient i was increased from 0.20 to 0.99. Heref0istheinitialvalue(atthebeginningoftest)ofthenondimensionaldensityf(t)offineparticles(ex-pressed as the ratio between the initial weight of the fine fraction and the total weight of the sample)er(t) is the current non dimensional eroded density of the fine fraction (depending on the weight of the eroded material). The dots represent the experimental data while the solid lines are their interpretation based on the erosion lawthat will be described subsequently. These results show that for constant hydraulic gradient i the rate of erosion decreases with time and that the long term density of fine particles f (at the end of test) decreases with increasing hydraulic gradient. and Figure 3. Results of a multi stage te00 50 100 150 200 250 300 350 400 450 500Time [h]i=0.20 0.010.030.040.050.060.070.08er/f0i=0.80 i=0.90 0.02i=0.40 i=0.60 0200.001 0.01 0.1 1 10 100Particle size D [mm]406080100Percentage passingi=0.99 The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India The plot of f/f0o multi stage tests having dif-ferent values, 0.2 and 0.6, ofeen f/f0 and i is barely affected by depends on the gradient im-posed duringprovides an adequate a suitable base for developing a law governing the erosion process. ifor different llowing characteristics of an erosion law suitable for implementation in a reaches its long term value f(i), which is a function of the hydraulic gradient i. -The ratio f/f0 is a continuous, monotonous, decaying function of the gradient and t ds to aasymptotic value c. - f versus i is shown in Figure 4. This figure reports the results of twthe starting gradient. It can be observed that the relationship betw the starting value of the hydraulic gradient and that it basically each step of the test. This indicates that the adopted experimental procedure description of the phenomenon at hand and that the described results represent0.960.981.00f/f0 Figure 4. Ratio between final and initial density f/f0 of the fine fraction versus hydraulic gradient starting gradients (i=0.2 dots; i=0.6 crosses). 4Erosion law The experimental results suggest the fofinite element code: -The beginning of the erosion process is defined by the initial density of the fine fraction f0. -When the process terminates the densityennon vanishing When the hydraulic gradient tends to zero no erosion occurs, i.e. f(i0)=f0, and the variation of f van-ishes, i.e. [f(i)/i]i=0=0. -The variation f/i first increases with increasing hydraulic gradient then tends to decrease. -On these bases the following relationship is suggested that provides the long term density of fine particles as a function of the hydraulic gradient i, ( )( ) ( )+ = c i a cib fexp 1 f 0. (1) The non dimensional coefficients a and b, and the asymptotic value c, ar ate through he bacsis of the experimental data (cf. Figure 4). -s creases monotonously during time. This suggest the following expression e evalu d t k analy-The non dimensional density of fine particles f(t,i) varies during time from f0 to f(i); the rate of erosion irepresented by the time derivative of f(t,i). -Starting from given initial conditions the rate of erosion increases with increasing hydraulic gradient and, un-der constant hydraulic gradient, defor the rate of erosion, [ ] ) ( ) , ( i i t i dtf f ) , ( i tf= . (2) The dimensional coefficient d is evaluated through the back analysis of the experimental data (cf. Figure 3). The above erosion law can be numerically integrated assuming a linear variation of f during a time increment t. This leads to the following values of the parameters: a=4.07, b=2.44, c=0.93, d=9.0410-3 h-1, and to the dia-grams (solid lines) in Figures 3 and 4. 0.900.920.940.0 0.2 0.4 0.6 0.8 1.0Hydraulic gradient i [--] The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India ms. They shouldgovern,respectively,thechangeofthecoefficientofhydraulic conductivity of the granular soil and the w that the variation of permeability is marginal, due to the limited amount of eroded edures, modify the geometry of the finite element grid is assumed when the pore pressure be-quent erosion and settlements analyses. allows assuming as constants both the free surface shape and ation provides the variation with time of the density of fine particles present in each element. Only the reTeems unlikely that the effects of erosion could inducexceeding the limit mechanical resistance of soil. totVsolid due to erosion and the variation of voids. The variation of the solid volume is directly related to f through the unit weigh f grain g, and e initiaIt should be observed that two additional laws are required for solving boundary value erosion problepossible deposition of the transported particles. Since the performed tests shomaterial,aconstantcoefficientofhydraulicconductivityisassumedk=210-5m/sinthesubsequentnumerical analyses. As to the possible deposition of fine particles, the density of four sections of the sample was determined at the end of tests obtaining fairly similar values. This eliminates any marked effects of deposition under constant or in-creasing gradient. Consequently the effects of deposition were disregarded in the numerical analyses. 5Solution procedure Theevaluationofthesettlementsinducedbytheerosion-transportoffineparticleswasbasedonthreeinde-pendent finite element analyses. Note that the evaluation of settlements due to the effective stress change is not discussed here. In fact the relatively high elastic modulus of the in situ soil involves minor settlements related to the water table lowering. First a steady state unconfined seepage problem is solved for evaluating the lowering of the water table induced by the deep drainage. Two classes of finite element algorithms are available for this purpose (see e.g. Cividini and Gioda, 2000). The approaches of the first group, known as variable mesh procuntil a part of its contour approximates with a sufficient accuracy the shape of the free surface. Those of the sec-ond group, referred to as fixed mesh procedures, operate on meshes of constant geometry and replace the intrin-sic non-linearity of the problem, due to the unknown position of the free surface, with a non-linear pore pressure-permeability relationship. The real value of the hydraulic conductivity is adopted in an iterative solution process if the pore pressure is above the atmospheric pressure. A vanishing valuecomes equal to the atmospheric value, condition that takes place on the free surface and above it. Here a fixed mesh procedure was used (Desai, 1976; Desai and Li, 1983) since it allows using the same finite element grid also for the subseHaving solved the free surface problem, the finite element approach presented by Cividini and Gioda (2004) is adopted for the erosion-transport analysis of fine particles. This calculation is based on the previously described erosion law. As already mentioned, the experimental results show that the erosion produces a marginal variation of the hydraulic conductivity of the samples. This the related seepage velocity field during the erosion-transport analysis. This calcullong term value of the variation f of the fine particle non dimensional density is used in the subsequent settle-ment evaluation. In the last finite element analysis, leading to the estimation of the erosion induced settlements, each element is subjected to a volume deformation depending on the p viously determined eroded density. his analysis is car-ried out in elastic regime since it s e an effective stress state The evaluation of the volumetric deformation due to erosion requires some assumptions that are worth comment-ing upon. The decrease of the total volume of soil V , which governs the settlements, can be subdivided into the variation of the volumes of solids t o s th l unit weight 0 of soil,f tot solidV V / = /0 g,(3) As to the variation of the volume of voids it could be assume at it does not change during erosion, wit d th h a con-sequent increase of the void ratio. In this case the total volume strain vol coincides with that expressed by Equa-tion (3), g f vol /0 =. Another possible assumption is that the void ratio does not change during erosion. Consequently both solid andvoid volumes decrease and the total volume strain becomes, (4) f vol =.(5) It seems then reasonable to assume the following range of variation for the volume deformation caused by the erosion process,f vol g f /0 . (6) The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India is, the erosion-transport analysis carried out. In this calculation a vanishing coefficient of hydraulic conductivity was adopted, and the coefficient was set to 1, for the elements above below it. This sec-ond analysis provides the time each finite element. For briefness only the contour lines of its long term di th a detail of the finite element mesh in the vicinity of the we volumetric strain pre-sented in Equation (6). The calculations have been repeated considering two possible values of Poisson ratio . It should be observed that, if a homogeneous deposit is assumed, the settlements caused by the imposed volume strains do not depend on the elastic modulus of the medium. Inthepresentcasethepenetrometertestsshowthepresenceofthreehomogeneoussoillayersinthezone where the erosion process develops. The calculations were repeated varying the elastic modulus of these layers within the limits suggested by the in situ tests. It turned out that the consequent change of the computed settle-ments is marginal and substantially lower than that caused by a variation of Poisson ratio. 6Results of calculations The described solution procedure has been applied to the evaluation of settlements in the vicinity of a 2 m diame-ter well that reaches a depth of 31 m below the hydrostatic water table, which in turn is located approximately at the same elevation of the foundations of nearby buildings. The total inflow of water is about0.75 m3/min. The calculations were based on a mesh of 4278 quadrilateral, four node isoparametric elements in axisymmetric conditions. The mesh has an extension of 1000 m in the horizontal direction from the vertical axis of the well. The bottom of the mesh is located at depth of 46 m from the original water table where an impervious layer of cohe-sive soil is located. fter evaluating the water table lowering through the unconfined seepage analys Aisc the free surface so that erosion could take place onlydependent variation of the fine particle density instribution are shown in Figure 6, together will and of the shape of the steady state free surface. Figure 6. Contour lines of the ratio between the long term density of fine particles f and its initial value f0. The last calculation is an elastic analysis in which the elements are subjected to the volume deformation due to erosion. The curves a) and b) presented in Figure 7 derive from the two limit values of the The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India -3.0-2.5-2.0-1.5-1.0-0.50.00 20 40 60 80x [m]y [cm]a b 100 Figure 7. Settlement y at the foundation level due to erosion: a) volume strain expressed by Equation (4); b) vol-ume strain expressed by Equation (5) ( =0.3 solid lines; =0.2 dashed lines). 7Conclusions Anexperimentalandnumericalstudyhasbeenpresentedaimingatevaluatingthesettlementsinducedina granular deposit by the possible erosion of its fine fraction due to a deep drainage well. The calculations are based on the finite element method and require a free surface seepage analysis, for evaluat-ing the water table lowering, an erosion and transport analysis for determining the percentage of eroded fine par-ticles and, finally, a stress analysis that provides the consequent settlements.The results of a series of laboratory erosion tests on reconstituted soil samples led to the formulation of an ero-sion law that was implemented in a finite element code for erosion and transport analysis. The experimental results showed that erosion induces a marginal variation of the coefficient of hydraulic conduc-tivity of the examined soil. In addition, no appreciable deposition of fine particles takes place within the sample, i.e. the particles eroded from one part of it do not tend to migrate to other parts of the sample but are removed from it by the seepage flow. On the basis of these observations a constant coefficient of hydraulic conductivity was used in the analysis of the field problem and the possible deposition of fine particles in steady state seepage conditions was neglected. Since the transient flow leading to the gradual lowering of the phreatic surface was not accounted for, the erosion above the steady state free surface and the consequent settlements were neglected. This indicates that the cal-culations could underestimate the actual settlements. On the other hand, during erosion the void ratio could reach values exceeding the limit expressed by Equation (4). This would imply settlements smaller than the computed ones. It is important to consider that the actual moduli of elasticity of the in situ deposit, subjected to a given volume de-formation, has a minor influence on the computed settlements. This indicates, from the one hand, that it is not compulsorytoevaluatethechangeofthedeformabilityparametersofsoilcausedbyerosion.Fromtheother hand it appears that, whilst the quantity of eroded material can be properly estimated on the basis of the tests re-sults, the main source of uncertainty of the finite element analyses is the actual change of the void ratio and the consequent volume strains that take place in the field. The settlements have been estimated assuming a linear elastic behaviour of soil, considering that the effective stress changes due to erosion can hardly lead to local failure conditions.The results of calculations show that the maximum differential settlement y/x (cf. Figure 7) at the foundation level in the vicinity of the well is about 1/1000. This should not induce severe problems in nearby buildings, but could cause some damages to their non structural parts. It seems therefore advisable a change in design of the drainage system adopting a series of small diameter wells. This would reduce the seepage gradient and, conse-quently, the quantity of eroded particles and the induced differential settlements. The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India 8Acknowledgements The present study contains part of the work carried out by the second and third authors for completing their Lau-rea thesis in Civil Engineering. The financial support of RCT srl is gratefully acknowledged. The authors wish to thank Livio Locatelli of Golder Associates for his technical suggestions. 9References Cividini, A., Gioda, G. 2000. 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