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  1. 1. CIV 2552 – Mét. Num. Prob. de Fluxo e Transporte em Meios Porosos2013_1Trab2Problema do Adensamento – Meio heterogêneo (duas camadas)Fluxo hidráulico unidimensionalMétodo das Diferenças Finitas – Formulação em volume de controle1ª QuestãoConsidere a geometria mostrada abaixo relacionada a um problema de adensamento de um meio heterogê-neo em 1D.Considere uma sobrecarga unitária na superfície. Formule o problema em volumes finitos e implemente em MATLAB. Resolva o problema usando o algoritmo de Crank-Nicholson. Compare os seus resultados com aqueles da Fig. 2 do trabalho de Pyrah (1996).Referência:Pyrah (1996), Geotechnique, vol 46, n 3, pp 555-560.
  2. 2. 2ª Questão: Simulação de problema de fluxo em uma camada drenante de areia que liga um rio auma escavaçãoA geometria na figura abaixo mostra o local de uma escavação próxima a um rio. O perfil do solo contémuma camada areia onde a água deverá passar devido ao rebaixamento de H1 para H2.A carga q representa a contribuição distribuída da água de chuva na camada de areia.Dados:K = 8 x 10–4cm/s = 8 x 10–6m/sSs = 1 x 10–4m–1q = 1 x 10–6m/sH1 = 40 mH2 = 5 mL = 80 m (extensão do domínio 1D)a = 1 m (largura do modelo 1D)b = 5 m (espessura da camada de areia)Condição inicial: h(x) = H1 para t = 0 e x de 0 a L.Condições de contorno: h(0) = H1 e h(L) = H2.Determinar h(t) ao longo da camada.Determinar a vazão de água para dentro da escavação.Resolva o problema usando o algoritmo implícito de Crank-Nicholson.
  3. 3. TECHNICAL NOTEOne-dimensional consolidation of layered soilsI. C. PYRAHÃKEYWORDS: consolidation; fabric/structure of soils;numerical modelling and analysis; pore pressures;settlement.INTRODUCTIONThe pioneering work of Terzaghi demonstratedthat for a homogeneous clay layer, subjected to anincrease in vertical load under one-dimensionalconditions, the pore water pressure is related toposition within the clay layer z, the time afterapplication of the load t and the coef®cient ofconsolidation cv. While it is widely appreciatedthat this is a function of both compressibility mvand permeability k, it is often assumed that if cv isconstant throughout a layer it is the only parameterrequired to predict rates of consolidation even ifthe deposit is layered. This apparently reasonableassumption, however, can lead to some misleadingresults and this note, using four simple soilpro®les, highlights the inadequacies of this ap-proach for situations where the soil is nothomogeneous.PROBLEM DEFINITIONThe four idealized soil pro®les (Fig. 1) allconsist of two soil layers of equal depth. Only twosoil types (A and B) are considered and for ease ofpresentation the coef®cients of compressibility andpermeability for soil A are taken as unity, as isthe unit weight of water ãw. Soil B has valuesof compressibility and permeability an order ofmagnitude greater than soil A. The coef®cient ofconsolidation for both soils is fully de®ned bythese parameters (cv ˆ k/mvãw) and is equal toone in both cases. The height of each double soillayer is H, the top is fully drained and the base isimpermeable, i.e. single drainage. The applied loadPyrah, I. C. (1996). GeÂotechnique 46, No. 3, 555±560555Manuscript received 31 March 1995; revised manuscriptaccepted 23 August 1995.Discussion on this technical note closes 2 December1996; for further details see p. ii.à Napier University, Edinburgh.Soil ASoil BSoil BSoil ASoil BSoil BSoil ASoil AFree-draining Free-drainingFree-draining Free-drainingImpervious base Impervious baseCASE (i) CASE (ii)CASE (iii)Impervious base Impervious baseCASE (iv)Soil A k = 1mv = 1cv = 1γw = 1Pore fluidSoil B k = 10mv = 10cv = 1Fig. 1. Assumed soil pro®les
  4. 4. p is constant and four con®gurations are consid-ered(i) soil A on top of soil B(ii) soil B on top of soil A(iii) soil A on top of another layer of soil A(iv) soil B on top of another layer of soil B.The last two cases are, of course, not truelayered soils and are simply introduced for com-parison with the two more interesting soil pro®les.Soil pro®les (iii) and (iv) represent homogeneousstrata, and Terzaghis theoretical solution for one-dimensional consolidation can be applied directly.Both have the same coef®cient of consolidationand behave identically in that the variations ofpore water pressure with time and depth will bethe same. The rates of consolidation will also beidentical, although the settlements for case (iv)will be ten times those for case (iii), as soil B isten times more compressible than soil A. Threemethods for predicting settlements for soil pro®les(i) and (ii) are consolidated below.PREDICTION PROCEDURESApproach 1Ðbased on standard average degree ofconsolidation curveIf separate oedometer tests were performed ontwo soil samples, one taken from soil A and theother from soil B, each would give the same valuefor the coef®cient of consolidation and an engineermight reasonably consider it appropriate to esti-mate the time-dependent settlement by using thestandard solution for a homogeneous clay layer forall four soil pro®les. Using this approach thepredicted pore water pressure isochrones and therate of consolidation would be the same for allcases, irrespective of whether the soil strata areidentical, or whether soil A overlies soil B or viceversa.To estimate how the settlement varies with time,the conventional method is to ®rst estimate the®nal settlement rF from the compressibility charac-teristics of the soil and then to multiply this by theappropriate average degree of consolidation "U toobtain the settlement rt at a particular time, i.e.rt ˆ rF"U (1)whererF ˆ…H0pmv dz"U ˆ21 À…H0ut dzpH3and the pore water pressure ut is a function of z and twith an initial value equal to the applied pressure p.For both soil pro®les (i) and (ii) the ®nalconsolidation settlement is a combination of thesettlement due to the compression of soil A (1pH/2) plus that due to the compression of soil B(10pH/2), i.e. rF ˆ 5´5 pH. As the average degreeof consolidation, as de®ned above, is independentof whether soil A or soil B is next to thepermeable boundary, so too is the settlement rt.Whether this is reasonable is explored below.Approach 2Ðbased on isochrones from standardsolutionThe above solution assumes that the degree ofsettlement is the same as the average degree ofconsolidation based on the distribution of porepressure with depth which, although true for auniform deposit, is incorrect for one in whichcompressibility varies with depth.For any clay deposit the pore water pressure inthe soil closest to a free-draining surface willdissipate much more quickly than that in the soilfurthest away from a free-draining boundary. Thus,in the early stages of consolidation the surfacesettlement will be controlled mainly by thecompressibility of the soil adjacent to the free-draining boundary, while the compressibility of thesoil away from this boundary will be moresigni®cant during the later stages of consolidation.In case (i) the more compressible soil (soil B) isnext to the impervious boundary and the settlementwill be much slower than in case (ii), where soil Boverlies the stiffer soil A. The effect of thedifferent compressibilities must be taken intoaccount in the solution, and this is not done ifthe standard theoretical average degree of con-solidation/time factor relationship is used.A more consistent approach is to evaluate thesettlement directly from the change in effectivestress at each point in the soil layer. The changein effective stress due to the dissipation of theexcess pore pressure is a function of positionand time, and is the difference between the initialvalue of the pore pressure p and its current valueut, i.e.rt ˆ…H0(p À ut)mv dz (2)The pore water pressures may be obtained fromthe standard isochrones and use of these togetherwith a different value of mv for each soil layer,rather than the average degree of consolidationtogether with an average value for mv as was doneusing approach 1, would seem a more appropriateprocedure. Unfortunately, this approach is stillincorrect for soil pro®les (i) and (ii).556 PYRAH
  5. 5. Correct methodÐbased on mv and k rather thanthe single parameter cvWhile the settlement prediction using approach2 takes account of the different compressibilities ofthe two soils, no consideration has been given tothe difference in their permeabilities or the effectthis has on the dissipation of pore pressure and theresulting time-dependent settlements. Correct solu-tions can be obtained only if solid±¯uid continuityis taken into account throughout the whole soildeposit, including layer boundaries. Continuitybetween the clay layers requires that the porepressures and ¯ow rates in adjacent layers at alayer interface are the same; this requirement isignored in the simple approaches outlined above.Correct solutions may be obtained using avariety of analytical and numerical techniques(Schiffman & Arya, 1977). Numerical techniquesinclude both ®nite difference and ®nite elementmethods based on either a diffusion or a coupled(Biot) approach. For one-dimensional problems bothgive the same solution if formulated correctly,although care must be taken in calculatingsettlements if the diffusion approach is used. Ifthe isochrones are not interpreted correctly, forexample using equation (1) and an average valueof mv rather than equation (2) with different valuesof mv for each soil layer, errors similar to thosediscussed in the previous section will beintroduced.The results reported in this technical note wereobtained using the ®nite element method and adiffusion approach in which the assemblage mat-rices are formulated in terms of k and mv ratherthan the single parameter cv (Desai, 1979). Withthis formulation the nodal pore water pressures arethe only unknowns, and the method is computa-tionally ef®cient. However, because of possibleerrors in the interpretation of the resulting iso-chrones, the results were checked against solu-tions obtained using a fully coupled (Biot) ®niteelement program (Abid & Pyrah, 1988). With thisapproach, where the unknowns include nodaldisplacements as well as pore pressures, the sur-face settlements are given directly rather thanbeing dependent on an interpretation of the porepressure distributions. Both techniques considercontinuity between connecting elements and, pro-vided every layer boundary is also an elementboundary, no special considerations are required atthe layer interfaces.RESULTSThe correct solutions are shown in Figs 2±4 forall four soil pro®les. As the results are plotted non-dimensionally (Tv ˆ cv t/H2), the solutions for soilpro®les (iii) and (iv) are identical and the same asthe standard Terzaghi solution; these results alsorepresent the solutions obtained for all four soilpro®les if approach 1 is used.Figure 2(a) shows the pore pressure distributionsfor soil pro®le (ii) (soil B overlying soil A), Fig.2(b) shows the standard solution for a uniform soil,cases (iii) (A/A) and (iv) (B/B), and Fig. 2(c)shows the isochrones for soil pro®le (i) (A/B). Thesolutions for rate of settlement and rate ofdissipation of pore water pressure at the imper-vious boundary are shown in Figs 3 and 4respectively. The reasons for the signi®cant differ-ences in these curves can be understood byexamining the dissipation of excess pore waterpressure (Fig. 2) for each soil pro®le.For case (i) (Fig. 2(c)), where soil A overliessoil B, it is the permeability of soil A and thecompressibility of soil B that govern the behaviour.Because of its high compressibility, soil B has toexpress a relatively large amount of water from itsvoids during consolidation, but the rate at whichthis can be done is mainly controlled by the lowerpermeability of the overlying soil. Hence most ofthe total settlement, which is due to the compres-sion of soil B, is correspondingly delayed. Con-versely, for case (ii) (Fig. 2(a)), where the morecompressible soil lies next to the free-draining topboundary, most of the settlement occurs relativelyrapidly. As the contribution of soil A to the overallsettlement is small, the rate of settlement is mainlygoverned by the consolidation of soil B. As thissoil layer is adjacent to the free-draining boundaryand has a thickness H/2, the rate of settlement isapproximately four times higher than that of thestandard solution. For case (i), where the morecompressible soil is overlain by the less permeablesoil, the rate of settlement is signi®cantly lower;the results indicate a rate approximately 1/40 ofthat for the standard solution.DISCUSSIONWhile the correct method for analysing the one-dimensional behaviour of layered soils has beenknown for many years, the examples aboveillustrate the signi®cant effect that variations inpermeability and compressibility can have on thebehaviour of a soil deposit. As was intimated byWroth (1989), the consolidation of a two-layer soilmay be likened to the heat-conduction problem ofbaked Alaska. For those not familiar with thisdessert, it is made by covering ice-cream withwhisked egg white and sugar and placing this in avery hot oven for a few minutes. The outercovering bakes quickly to form a crisp shell, whilethe ice cream, protected by the low conductivity ofthe covering, remains cold. The result is adelicious sweet, a combination of cold, soft icecream surrounded by warm, crisp meringue. Thisis analogous to case (i); the analogy for case (ii)ONE-DIMENSIONAL CONSOLIDATION OF LAYERED SOILS 557
  6. 6. Porewaterpressure(u/p)Porewaterpressure(u/p)Porewaterpressure(u/p)0.000.501.[B/A](b)uniformsoil(c)case(i)[A/B]Time(Tv)0.00050.00500.02500.05000.10000.25000.50001.00000.100.200.300.400.500.600.700.800.901.000.600.700.800.901.00Fig.2.Excessporepressuredistributionsfordifferentsoilpro®les558 PYRAH
  7. 7. would be to surround the whisked egg white withice cream and to place this in a very hot oven. It isunlikely that this has been attempted, but theoutcome would be signi®cantly different fromBaked Alaska and clearly illustrates the differencebetween the two cases. (Note: For the analogy tobe strictly correct the values of the thermaldiffusivity k of the meringue and the ice creamshould be identical; k ˆ k/cr where k is thermalconductivity, c is speci®c heat and r is density.Although the thermal diffusivity for the twomaterials may not be the same, and hence theanalogy with two soils having the same value of cvnot exact, the relative values of thermal conduc-tivity and of the product of speci®c heat anddensity are such that the problem is a usefulillustration of the behaviour of layered materials.)To return to the examples involving fourSoil profile [B/A]Uniform soilSoil profile [A/B]10090807060504030201001.00E−05 1.00E−04 1.00E−03 1.00E−02 1.00E−01 1.00E+00 1.00E+01 1.00E+02Elapsed time (TV)Surfacesettlement(%)Fig. 3. Rate of settlement for different soil pro®les10090807060504030201001.00E−05 1.00E−04 1.00E−03 1.00E−02 1.00E−01 1.00E+00 1.00E+01 1.00E+02Elapsed time (Tv)Excessporewaterpressure(%)Soil profile [B/A]Uniform soilSoil profile [A/B]Fig. 4. Dissipation of pore water pressure at impervious boundaryONE-DIMENSIONAL CONSOLIDATION OF LAYERED SOILS 559
  8. 8. different soil pro®les: these illustrate the impor-tance not only of treating a saturated soil as a two-phase material, including solid±¯uid compatibilityand the effects of time, but also of modelling thecomposite nature or fabric of the soil in order tocapture its true behaviour. In geotechnical en-gineering there are many instances where thepresence of different materials, having differentstress±strain and permeability characteristics, has asigni®cant effect on the mass behaviour of the soil.An examination of the effect of the disposition ofsuch non-uniformities is likely to lead to a betterunderstanding of soil behaviour. This applies toboth natural and man-made ground.CONCLUSIONSSimple examples involving the one-dimensionalconsolidation behaviour of layered soils consistingof two layers with the same value of the coef®cientof consolidation, but with different compressibilityand permeability characteristics, have been used toillustrate the importance of adopting correctprocedures in predicting their behaviour. Theexamples also illustrate the signi®cant effect thatthe arrangement of the different constituents canhave on the behaviour of a soil.REFERENCESAbid, M. M. & Pyrah, I. C. (1988). Guidelines for usingthe ®nite element method to predict one-dimensionalconsolidation behaviour. Comput. Geotech. 5, 213±226.Desai, C. S. (1979) Elementary ®nite element method.Englewood Cliffs, NJ: Prentice-Hall.Schiffman, R. L. & Arya, S. K. (1977). One-dimensionalconsolidation. Numerical methods in geotechnicalengineering (edited by C. S. Desai and J. T.Christian), pp. 364±398. London: McGraw-Hill.Wroth, C. P. (1989). Private communication.560 PYRAH