2D Modeling of the consolidation of soft soils. Introduction

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1 D Modeling of the consolidation of soft soils Matthias Haase, WISMUT GmbH, Chemnitz, Germany Mario Exner, WISMUT GmbH, Chemnitz, Germany Uwe Reichel, Technical University Chemnitz, Chemnitz, Germany Abstract: Comlex layered structures of tailings onds and their vertical and horizontal flow conditions for the ore water require to overcome the 1 dimensional modeling of the consolidation rocess. A two-dimensional radial symmetric numerical model (ConsolD) based on a fully three-dimensional aroach is introduced and ractical results are discussed. Secial attention is given to the validation of the numerical model with the hel of analytical solutions for secial cases. Introduction The develoment of a remediation strategy for a tailings ond with a fine slime zone includes the assessment of the state of consolidation of the different zones in the ond. Usually the ond is divided in a fine slime zone, a transition zone and a sandy or beach zone. All these zones often built u a very comlex layered structure with horizontal and vertical flow athways for the ore water during the consolidation rocess. There exist different aroaches for the descrition of the consolidation rocess. The basic hysical relationshis in the consolidation theory were first investigated by K. Terzaghi [5] and later more in general by M. A. Biot [1]. In the one-dimensional Terzaghi theory the following two basic assumtion restrict the use of the aroach: Strains are small and The material roerties remain constant during the consolidation rocess. However, soft tailings are known to exhibit significant changes in void ratios during changes in the stress state in resonse to load surcharge and therefor in the ermeability, too. The theory of non-linear finite strain (NLFS) consolidation overcomes these limitations of the Terzaghi theory. One-dimensional NLFS-models are often used for the modeling of the consolidation rocess in tailings onds with success. But in cases where the horizontal ore water flow has an imortant effect on the consolidation rocess the results of such one-dimensional calculations should be viewed with caution. Furthermore the develoment of remediation strategies requires in some cases an evaluation of the effect of vertical drains. Vertical drains are used for the acceleration of the consolidation rocess of soft tailings in the fine slime zone. Standard one-dimensional consolidation models are unable to determine the effects of such drains correctly because of the two-dimensional (D) radial symmetry of the roblem. Two-dimensional analytical solutions of the basic equations exist, but they do not take into account the change of the ermeability with the void ratio (usually referred to as the k-e-relationshi) and the varying stiffness of the material (void ratio-effective stress-relationshi). The analytical solutions also ignore any layer structure, that means they use only one soil tye for the descrition of the roblem.

2 To overcome these limitations we develoed a numerical model for solving such tyical twodimensional consolidation roblems (ConsolD). The model is based on the work of A. Verruijt [6] with some extensions. Tyical features of the one-dimensional NLFS-models and the descrition of the flow conditions and of the stress-strain-roblem in two dimensions are integrated in this aroach. The original three-dimensional aroach is reduced to D radial symmetry during the numerical imlementation (algorithm). This is sufficient to simulate vertical drains or the consolidation rocess of tailings material in an axi-symmetric ond and saves a lot of comutation time. The correctness of the model has been roved by comarison with analytical solutions and field exeriments. Basic equations Assuming full saturation we consider the orous material as a two-hase system consisting of a liquid hase (ore-water) and a solid hase (articles). Starting from the conservation of mass and assuming the incomressibility of the solid articles we find the storage equation (1) ε t vol = nβ t T ρ + q (1) Together with Darcy's law we get the equation for the excess ore ressure : ε vol k = nβ T () t t γ w with the volume strain ε vol, the time t, the orosity n, the comressibility of the ore-water β, the hydraulic conductivity k, and the volumetric weight of the ore-water γ w. A detailed descrition of the derivation can be found in [6]. For a comlete system of equations we also require an exression for the volume strain ε vol. We obtain this exression by considering the roerties of the solid material, including the equations of equilibrium (3), the rincile of effective stress (4), and the stress-strain-relations (5). We resent these equations for an elastic case: T ρϖ ρ σ f = ρ ρ σ = σ ' I ρ ρ σ ' = λε I µε vol (3) (4) (5) where σ ρ ρ ρ denotes the tensor of total stress, σ ' the tensor of effective stress, f the vector of the body forces, I the unit tensor, the excess ore ressure, ρ ε the strain tensor, and λ and µ the elastic coefficients (Lamé arameters). In soil mechanics the Youngs module E and the Possoin ratio ν are often used instead of the Lame Parameter:

3 λ = ( 1+ ν )( 1 ν ) E µ = 1 Eν ( + ν ), (6) The volume strain ε vol is then defined by the trace of the strain tensor ε ρ : ε = ε + ε + ε vol xx yy zz (7) With the comatibility equations ε ε ε ii ij vol ui =, i = xi 1 u u i j = +, x j xi T = u i, j = 1...3, i j (8) the comonents of the strain tensor are related to the dislacements u. So the final system consists of 4 equations T ρ ( u) t T = nβ t k γ T ρ ρ ρ ρ [ λ( u) I µε () u I] f = T w for the 4 indeendent variables, ux, uy, uz The other quantities are deendent and connected by several non-linear material functions. Material functions and numerical realization Material functions The consolidation behavior of the tailings material is mainly characterized by the stress-strainrelationshi and the ermeability-void ratio-relationshi. The stress-strain-relation is described with the comression curve for the soil. Usually a e-log σ -lot is used for fitting this soil behavior, with the void ratio e (Eq. 1). In our case we always load the soil column with the self-weight during filling or with a surcharge. Therefor we can use the virgin or normal consolidation line. Swelling or recomression is not considered here. e = e, Cc logσ V (1) water e = Vsolid The one-dimensional confined elastic modulus E o may be obtained thus: (9) 3

4 E o = ( 1+ e) σ e The Youngs modulus E in (6) is obtained with: ν E = 1 1 ν E o In analogy to (1) we use also a logarithmic aroach for the descrition of the void ratioermeability-relationshi: e = e c log(k) (11) ok + kl The relationshis (1) and (11) we get in two ways: 1. Analyzing measured values and fit them with an aroriate regression line. Using the measured values as starting values in modeling the filling history of the ond with a 1D-NLFS-model and evaluating the arameters e, C c, e ok and c k by means of such indicators like measured void ratio rofiles, tailings heights or settlement rates. In our ractical work we have found that a combination of both methods rovide defensible arameters for further calculations with our D-model. Numerical realization The numerical realization of the model with the finite element method consists of several stes. The first ste is the choice of aroriate function saces and the conversion into a variational formulation. The next stes are the transformation to cylindrical coordinates and the reduction to D-radial symmetry [4]. The Galerkin method with a finite dimensional trial sace was used for the discretization in sace. Several iecewise linear and quadratic elements were imlemented. For the time discretization a Crank-Nicolson-scheme with a fully imlicit start ste was chosen. We imlemented the following features in the numerical model: The ond or the tailings column can consist of u to 9 different soil tyes. For each soil layer a void ratio - ermeability -relationshi e(k) including the anisotroy of k and a void ratio-effective stress - relationshi e(σ) can be defined. Measured void ratio - rofiles e(z) can be used for defining initial conditions. Boundary conditions for the ore ressure and/or the ore water flow can be defined for each boundary grid element. An additional load can switch on during a run for simulating the covering rocess during the tailings ond remediation. Analytical verification The imlementation of the model has been verified by numerical comutations of roblems with known analytical solutions. The verifications were done by M. Exner as art of [3]. 4

5 Terzaghi theory The first solution of a consolidation roblem was given by Terzaghi in 193. The dissiation of the excess ore ressure in a fully drained soil layer was investigated. An illustration of this roblem is given in Figure 1. The artial differential equation in this case is fully onedimensional. To obtain an analytical solution some simlifications of the material roerties have to be made: hydraulic conductivity k=constant stiffness module E=constant With these assumtions the system can be uncouled. The resulting differential equation is t = c v (1) with the consolidation coefficient c v Ek = γ w For the one-dimensional Terzaghi roblem this imlies = c t v z (13) With the boundary conditions = on z= and z=h and the initial condition = at t= equation (13) can be solved. The solution for the excess ore ressure deending on the time t and the height z is ( 1) n 4 π h z ( ) π = cos n + 1 ex ( n + 1) π n= n + 1 h 4 c vt h (14) For ractical comutations, the infinite sum in (14) is aroximated by the summation of the first 5 terms. In the numerical simulation we use a load to roduce an excess ore ressure of 5 kpa in a soil layer with a constant total stress of 1 kpa. The results are resented in Figure. The oscillations in the analytical solution result from the break of the summation of the infinite sum. In the numerical simulation we get oscillations in the first time stes which are caused by slightly incomatible initial conditions. Desite these initial oscillations the imortant long term corresondence of the analytical and numerical solutions is very good. 5

6 Figure 1: Calculated scenario for the one-dimensional Terzaghi-roblem excess ore ressure (normalized) radius (normalized) 1 days analytical days numerical Figure : Left: Excess ore ressure reduction in Terzaghi-theory, numerical (lines) and analytical (+) results at different oints of time (in days). D radial symmetric solution -- surrounding vertical drain The simlifications made by Terzaghi to solve his roblem can also be made for a twodimensional roblem. Now we consider a cylinder with radius a and assume that the ore water can only disaear over the cylinder shell, for an illustration see Figure 3: Calculated scenario for the radial symmetric roblem. The drain surrounds the cylinder.. After the 6

7 transformation to cylindrical coordinates and with the assumtion of radial symmetry the differential equation (15) reduces to t 1 = c v + t r r (15) The boundary conditions are such that the ressure is zero along the outer boundary at r=a, and that the ore ressure is not singular in the origin. In conjunction with the initial condition = at t=, the solution of (15) is = k = 1 J α ( αkr / a) J ( α ) k 1 k α k ct ex a (16) where J (x) and J 1 (x) are the Bessel functions of first kind and order zero and one. The a k are the zeros of the function J (x). As shown in Figure 4 we achieve a good long term corresondence of the numerical and the analytical solution. We also observe some oscillations at start time, which were already discussed in the one-dimensional case. Figure 3: Calculated scenario for the radial symmetric roblem. The drain surrounds the cylinder. 7

8 radius (norm.) 1 days analytical 1 days 5 days 5 days 1 days days 1 days numerical 1 days 5 days 5 days 1 days days Figure 4: Excess ore ressure reduction in the case of a surrounding drain, numerical (lines) and analytical (+) results. D radial symmetric solution -- inner vertical drain The more interesting two-dimensional case is the inner vertical drain instead of the surrounding one. Here we consider a cylinder with an infinite radius and an inner cylindrical drain of radius a, for an illustration see Figure 5. The numerical model is unable to handle a cylinder with an infinite radius. Hence the simulation makes only sense until the excess ore ressure reduction reaches the outer bound of the simulated cylinder. The boundary condition is = at r=a for all t>, and the initial condition is = at t=. The solution of (17) for this scenario is ( r, t) r' [ J ( λr' ) Y ( λa) Y ( λr' ) J ( λa) ] λ = r' = λ ex ( c λ t) v J ( λa) Y ( λa) [ J ( λr) Y ( λa) Y ( λr) J ( λa) ] d r' dλ = The functions J (x) and Y (x) are the Bessel functions of first and second kind and order zero. Figure 6 shows the numerical and analytical solution for the first days. The solutions corresond very well. (17) 8

9 Figure 5: Calculated scenario for the radial symmetric roblem with an inner vertical drain days numerical 1 days 1 days days 5 days 1 days analytical 1 days 1 days days 5 days radius [m] Figure 6: Excess ore ressure reduction in the case of an inner vertical drain, numerical (lines) and analytical (+) results. Modeling of a field test To investigate the effects of vertical drains on the develoment of soil mechanical arameters the Wismut GmbH started some field exeriments in The main objective was to study the influence in vertical drains on the rocess of consolidation in tailings onds. After installing the drains and covering the field measurements were taken at regular intervals. These measurements enable the comarison with the numerical model. Until now there is no ossibility for long term statements, because of the short run time of the field test, which is only a small fraction of the whole consolidation time. 9

10 The total height of the tailings ond in the test field is about m. The vertical drains were ut in as a attern of equal sided triangles with a side length of 1.5 m. The drain deth is 5 m (see Figure 7: Geometry of the testfield and the of the draindesign). The material arameters for the simulation were determined by samles from some bore holes near the test field (DH1 and DH ) and the void ratios were varied in several runs of the simulation. This variation was necessary because of the missing void ratios for the test field. The values had to be guessed from samles of nearby bore holes. Figure 8 shows the results of three different simulation runs for the first days comared with the measured data. Using the two arameter sets from the bore holes DH 1 and DH we get a good agreement between measured and modeled settlement rates. The modeled values for the settlement without drains are also in good agreement with measured settlement rates at locations near to the test field without drains. From Figure 9 it is seen that the accelerating effect of the ore ressure dissiation with the vertical drain is limited to the uer 5 or 6 meters where the drain is located. For accelerating the ore ressure dissiation in the lower art of the fine tailings the use of dee vertical drains would be required. Figure 7: Geometry of the testfield and the of the draindesign 1

11 .6.5 Modeling of the settlement on the test fields Culmitzsch A DH 1 DH Modelling without drain measured values time[d] Figure 8: Consolidation of the test field, measured data and numerical simulation Figure 9: Dissiation of the excess ore ressure with time in a cross section through the tailings column with the vertical drain. 11

12 Conclusion The imlemented numerical model ConsolD was verified by means of several consolidation scenarios. The numerical results show a good agreement with the analytical solutions for these cases. First calculations of the consolidation state under the influence of vertical drains in a field test give hoe, that with ConsolD a good rognosis of the effect of vertical drains under field conditions is ossible. The imlementation by use of finite elements allows the modeling of the consolidation behavior of comlicated geometries (soil layers) and flow conditions. We use the rogram not as a "stand alone" tool. It is integrated in a technological chain that reaches from field and laboratory measurements over one-dimensional consolidation modeling of the filling history and arameter evaluation to the two-dimensional consolidation modeling of drain design or ond cross sections with ConsolD. The model runs on PC under LINUX. A arallel comuting on several machines for very comlex structures is ossible. The following features are under develoment: Choosing a satial (two dimensional) void ratio distribution as initial conditions instead of the deth rofile Testing the rogram using cartesian coordinates ( true D ) Imlementation of a time deendent loading during the run We intend to start a test rogram with aroriate field measurements of the consolidation behavior of tailings imoundments. 1

13 References 1 M. A. Biot. Gerneral theory of three-dimensional consolidation. J. Al. Phys., 1: , J. H. Bramble, J. E. Pasciak, and J. Xu. Parallel multilevel reconditioners. Math. Com., 55:1-, M. Exner. D Modell zur Setzung von Tailingsschlämmen. Dilomarbeit, Fak. f. Mathematik und Physik, Universität Bayreuth, Feb U. Reichel. Numerische Untersuchungen zum Setzungsverhalten von Tailingsschlämmen. Dilomarbeit, Fak. f. Mathematik, TU Chemnitz, Feb K. Terzaghi. Erdbaumechanik auf bodenhysikalischer Grundlage. Deuticke, Wien, A. Verruijt. Comutational Geomechanics. Theory and Alication of Transort in Porous Media. Kluwer Academic Publishers,