METHODOLOGY FOR DRENABILITY STUDIES OF STACKED MINING COARSE TAILINGS

Transcription

1 MEHODOLOGY FOR DRENABILIY SUDIES OF SACKED MINING COARSE AILINGS J. C. Machado Júnior Departamento de Engenharia Civil, Universidade Federal de Ouro Preto W. L. Oliveira Filho Departamento de Engenharia Civil, Universidade Federal de Ouro Preto Abstract: his paper presents a methodology for drenability studies of coarse mining tailings. hese studies have a particular impact in the wall stability and construction methods of stacked mining coarse tailings structures (Gomes et al. 1999a). In that context, flow processes on unsaturated soil are the major complicated factor. For that purpose, a short review of this flow problem is presented, including theory and flow material functions needed in the analysis and their determination. Some common scenarios of flow problems related to tailings piling up are discussed and analyzed using a commercial code, SEEP/W, as a numerical tool. his paper also aims to make a contribution to the diffusion of unsaturated soil mechanics out of the academic circles. 1. INRODUCION Nowadays the mining tailings disposal management is often an activity of primary importance for the success of a mining operation. he economics and also the environmental issues associated with that activity have made the mining industries to optimize the disposal areas and to develop new processes to discard tailings. o pile up coarse tailings is one of these new techniques for the coarse fraction of the waste generated by the iron ore processing plants (Gomes et al. 1999a). In this regard, one of the key issues of the stability analysis of these structures is the question of drenability of the tailings. he project will be more economical and safe, depending on how precise a model can be made of the flow processes taking place in the tailings. However, a factor that complicates the flow analysis is that the process of drainage needs to deal with unsaturated porous media, and a complete new field of study for geotechnical engineers has to be faced. In this paper, an effort to popularize unsaturated flow analysis out of the academic circles is made by addressing a methodology to study a real problem on mining tailings disposal. 2. FUNDAMENALS OF UNSAURAED FLOW brief review of unsaturated flow theory is worth to be presented here since there is a lack of general understanding in that subject within our community of geotechnical engineers HE GOVERNING EQUAION he governing equation of the flow processes in soils is well known from textbooks of basic soil mechanics (Craig 1996, Lambe and Whitman 1979, Ortigão 1996). his equation can be expressed in a vectorial form like, dm ( ρ ) dx dydz w wv = (1), dt where, ρ w is the water density, v is the darcian velocity vector, is a differential operator so that, = [ / x / y / z ], M W is the water mass, and t is the time. 233

2 Equation (1) relates the mass balance in a three dimensional set, x, y and z, with the storage term on the right hand side. he latter can be expressed in terms of soil porosity, n, and degree of saturation, S, so that, M w = ρw S n dxdydz (2) and (1) becomes ( ρ Sn W ) ( ρ W v ) = (3). Depending on the way that the properties ρ w, S, and n are considered, different soil flow processes are described by (3). he usual simplification for most of soil engineers is to consider ρ w, S and n as constants, and therefore their derivatives are equal to zero. he right hand side then is equal to zero and this is the case of saturated, steady state flow analysis. ( v ) = 0 (4). In another case, when hypotheses of isothermal conditions and saturated porous media are introduced, the derivatives on ρ w and S are zero, and (3) becomes n ( v ) = (5), which is the general equation for (large strain) consolidation analysis. he less known form of (3), at least for geotechnical engineers, appears when the derivatives of ρ w and n are zero, and (3) becomes ( v ) = n S (6). Considering the Darcy s law, (6) becomes ( K h) = n S (7), where K is the hydraulic conductivity tensor and h is the total head. he governing equation in this form describes the transient analysis of unsaturated flow of incompressible porous media and isothermal conditions. 2.2 Richard's equation Further developments of the governing equation for unsaturated flow are possible. he concept of volumetric water content, θ, is very often needed in such cases. he relation between water volume in the voids, V w, and total soil volume, V, defines this quantity. hus, Vw θ = = V n S (8). Also, the total hydraulic head, h, can be divided in two other terms, the elevation head, z, and pressure head, ψ. hus, h = z + ψ (9). Equation (7) then becomes, θ ( Ke + K ψ) = (10), where e is a vector with the component z unitary, so that e =[ ]. he last development of the flow equation is to take into account that K and θ are functions of pressure head in unsaturated flow. It means that, K = K (ψ) (11) and θ = θ(ψ) (12), which implies, θ = c( ψ) ψ (13), where c(ψ)= θ/ ψ is called specific capacity retention. By considering these relationships, the primary dependent variable in the flow equation becomes ψ and the governing equation assumes the form 234

3 ( K( ψ) e + K( ψ) ψ) = C( ψ) ψ (14). his ψ-based equation of flow for transient flow through an unsaturated media is called Richard's equation (Freeze and Cherry 1979). It s a nonlinear second order partial differential equation. he nonlinearity on (14) is because the derivative coefficients are not constants but dependent on the primary variable, ψ. he solution of (14) must satisfy the problem boundary conditions. hese can be a prescribed pressure head boundary condition (Dirichlet condition) so that, ψ (x,t) = ψ on Γ 1 (15), or a prescribed normal flux boundary condition (Neuman condition) so that, [ K( ψ) e + K( ψ) ψ] v n = on Γ 2 (16), where Γ=Γ 1 +Γ 2, is the boundary of the volume, V; ψ and v are prescribed values, n is the unit normal vector on Γ 2 and x is a position vector. he solution of (14) must also satisfy the initial condition, ψ( x,0) = ψ 0 (17), where ψ 0 is a known value. o solve (14), flow material functions (11) and (12) are required. he determination of these functions are discussed in section 3. Closed form solutions of (14) exist only in very few cases what makes the use a numerical scheme a necessity. 2.3 Applications o illustrate the use of the flow equation, the analyses of two very common field situations involving flow in unsaturated media are presented Infiltration in a soil column o study this situation, an unsaturated soil layer is thought to receive on its top boundary a constant rain precipitation, R, with dimensions [L/]. Because the problem can be considered one-dimensional, the governing equation, (14), can be written as ψ ψ ψ ψ z k ( )( 1 + z c z ) = ( ) (18). From Darcy's law, v=ki, where i = h/ z and h/ z = 1+ ψ/ z, hence at top boundary R = + 1 ψ z k ( ψ) (19) or ψ R = 1 (20). z k( ψ) At the bottom boundary, a Dirichlet prescription is usually assumed, like ψ = 0 (21). (18), (20) and (21) constitute the boundary value problem for infiltration in a soil column Drainage in a soil column his situation can also be considered onedimensional and therefore the governing equation looks the same as in the previous case. ψ ψ ψ ψ z k ( )( 1 + z c z ) = ( ) (22). he top boundary is now impervious which implies R = 0 or ψ/ z = -1 (23). At the bottom, it is usually assumed again, ψ = 0 (24). 235

4 (22), (23) and (24) defines the boundary value problem for drainage in a soil column. 3. DEERMINAION OF HE FLOW MAERIAL FUNCIONS As it was mentioned, the determination of k= k(ψ) and θ = θ(ψ) is required for the solution of the governing equation. hose functions are called characteristic curves. he first is the hydraulic conductivity function and the second is the soil-water characteristic curve (SWCC). hese curves often have sigmoid shapes and present hysteresis for infiltration and drainage cycles. he determination of characteristic curves can be achieved by specific laboratory or field tests or even by correlation (especially the hydraulic conductivity function). Benson and Gribb (1997) present a comprehensive review of the procedures available. In the laboratory the water retention curve can be obtained through the pressure plate test according to the ASM D he setup of the test is schematically shown in Fig. 1. A chamber houses the soil specimen that is placed on a porous disc of high air entry value. he disc is in contact with a water reservoir at atmospheric pressure. he difference between the pore-air, u a, and pore-water, u w, pressures is called matric suction and it s created by pressurizing the chamber with compressed air. he amount of water that leaves the specimen with the increasing pressure is monitored, allowing the determination of a relationship between the degree of saturation and suction. Both drainage and infiltration cycles are run in the test. It is possible to measure the permeability function directly but intrinsic test difficulties and lack of accuracy very often prevent its large use and lead the practitioner to use indirect methods. he usual procedure is to estimate the hydraulic conductivity function from the SWCC. he greater reliability and short period of time required for the SWCC determination favor this approach. he correlation also has its strength in the intimacy of the two relationships. he SWCC describes the amount of water present in a soil under various matric suctions. his curve essentially indicates the space available for the water to flow through the soil at various matric suctions since water can only flow through the waterfilled pores. herefore the shape of the curve can be used to estimate the permeability function. he function decreases monotonically from the saturated permeability, k sat, at zero matric suction and assumes lower values as the matric suction increases. Figure 1 Schematic diagram of a pressure plate extractor (Rahardjo and Leong, 1997) here are many empirical equations that relate the hydraulic conductivity to the volumetric water content or degree of saturation of the soil (Gardner, 1958, Brooks and Corey, 1964; van Genuchten, 1980). he van Genuchten equation is given below: β 1/ 2 β 1 k r = θ e 1 1 θ e β where, k r relative hydraulic conductivity, β - constant, and hydraulic conductivity is given by (21) k = k sat k r (22). he β coefficient of (21) is a fitting parameter obtained from the experimental data of SWCC and the theoretical model given below also due to van Genuchten, θ e = 1 [ 1+ ( h) ] α β β 1 1 (23) 236

5 and θ e θ θr = θ θ s r (24) where, θ - volumetric water content, θ e equivalent volumetric water content, θ r residual volumetric water content, θ s saturated volumetric water content, h = ( u a - u w ) matric suction, α, β - constants. Figure 2 Soil water characteristic curve (SEEP/W user s guide) 4. NUMERIC ANALYSIS In the following subitems a complete example of a numeric simulation is given. he analyses were performed using a commercial computer program called SEEP/W. his is a finite element software used to model seepage problems, developed by GEO-SLOPE International Ltd. 4.1 Flow material functions he characteristic curves used in the analyses were obtained from the piece of literature (SEEP/W user s guide) and correspond to a fine sand, k sat = 4.3 x 10-6, n=0.22, d 10 = mm. Figure 2 and 3 illustrate the flow material functions used in the analyses. 4.2 Scenarios for the numeric analyses Several field situations can be analyzed within a tailings management strategy. In this paper three scenarios were envisioned for the study of tailings drenability. hese case studies are very often found in practice and the flow processes happen to be gravitational and onedimensional (Collins and Znidarcic, 1997). he boundary conditions for these situations are presented in terms of total head, H, total nodal flow, Q, and flow per unit length, q. In the first scenario, a one-meter thick tailings column, initially saturated, is underlain by a drainage layer, and is left to drain freely. his situation is shown in Figure 4(a). Figure 3 Hydraulic condutivity function (SEEP/W user s guide) he second scenario is a follow-up to the previous case with the initial conditions corresponding to the end of that analysis. In the sequence, a constant flow rate is imposed in the upper boundary, simulating rain precipitation over the drained layer. Figure 5 (a) is a schematic drawing of that case study. he last scenario allows the evaluation of the flow pattern considering multiple layers in different drainage stages. In the actual case, the behavior of a newly disposed layer (saturated) and its iteration with the underlain layer at certain drainage stage are analyzed. his scenario is sketched in Figure 6 (a). 4.3 Analysis Results he results of the numeric simulation of a draining soil column are shown in Figure 4 (b) in terms of pressure head x depth x time. he instant t = 0 reflects the initial conditions of the problem (a hydrostatic water column). It can be seen that for t > 0 the imposed boundary conditions are Q = 0 at top boundary and H = 0 237

6 at the bottom boundary. Hydrostatic conditions firstly develop at top portion of the column and progress downwards with time. At the end steady state conditions are reached with a hydrostatic suction profile throughout the entire column. he results of the infiltration case are shown in Figure 5 (b). At t = 0, the initial conditions corresponding to a hydrostatic suction profile. For t > 0, the boundary conditions are q = 4.3 x 10-6 m/s at the top boundary, and H = 0 at bottom boundary. At the beginning of the simulation, at the top portion, it can be seen that high gradients are necessary to overcome the low permeability values associated with high matric suctions and to allow seepage to take place. As the degree of saturation increases, the entrance gradients are smaller, tending towards a gravitational value of 1. Because q = k sat, the steady state solution gives a profile with constant suction from top to bottom. he corresponding degree of saturation profile would be similar with a constant S = 1.0 from top to bottom. he last case results are shown in Figure 6 (b). he initial conditions reflect two different situations. From 0 to 1m a suction profile corresponding to a layer at certain stage in a drainage process is shown. From 1 to 2 m, a hydrostatic positive porewater pressure profile is seen. his one corresponds to a newly disposed saturated layer. In the initial lapses of time, t > 0, a drainage process of the top layer is taking place while the bottom layer is experiencing a saturation phase. At a certain point in those flow processes, a typical steady state infiltration profile can be seen but only instantaneously, because for later lapses of time, a soil column drainage pattern governs. In the last stage, the two layers behave like one, in a very similar way to the results of the first case study. he boundary conditions in this case are Q = 0 at the top boundary and H=0 at the bottom. Figure 4 - (a) Finite element mesh and boundary conditions to scenario #1; (b) SEEP/W simulation results to scenario #1. 238

7 Figure 5 - (a) Finite element mesh and boundary conditions to scenario #2; (b) SEEP/W simulation results to scenario #2. Figure 6 - (a) Finite element mesh and boundary conditions to scenario #3; (b) SEEP/W simulation results to scenario #3. 239

8 4.4 Discussion Questions about flow rate pressure head and time required for water to flow can now be easily and rationally addressed. Some variations of the case studies described before can be envisioned. Even more sophisticated situations can also be considered such as twodimensional flow and different materials, without introducing any major change in the methodology here described. his kind of analysis can be used in the design of stacked mining coarse tailings structures in areas such as wall stability studies, monitoring programs and construction methods (Znidarcic, 1999). hese aspects will be object of future work. 5. CONCLUSION From the simple cases presented in this paper it is apparent that the methodology has great capability to expand the horizons to study flow processes in unsaturated soils. he requirements for this development include: to perform new but not sophisticated tests, to use a competent and reliable computer code such as SEEP/W and others, and most of all to get acquainted with unsaturated flow processes. An effort to bring about this issues has been made in this work. 6. REFERENCES Benson, C. H. and Gribb, M. M. (1997).Measuring Unsaturated Hydraulic Conductivity in Laboratory and Field, Unsaturated Soil Engineering Practice. Geothecnical Special Publication (68) ASCE, Brooks, R. H. and Corey, A.. (1964). Hydraulic Properties of Porous Medium. Hydrology Paper No.3, Civil Engineering dept., Colorado State University, Fort Collings, Colorado. Craig, R. F. (1997) Soil Mechanics, Chapman & Hall, sixth edition. Collins, B. D., and Znidarcic, D. (1997), riggering Mechanics of Rainfall Induced Debris Flows, Proceedings of 2 nd Pan- American Symposium on Landslides and 2 nd Brazilian Conference on Slope Stability, II PSL/COBRE, Vol.1, Freeze, R.A. & Cherry,J.A. (1979). Groundwater, Prentice-Hall, Inc., Englewood Cliffs, New Jersey. Gardner, W. R. (1958). Some Steady State Solutions of the Unsaturated moisture Flow Equation with Aplications to Eva poration from Water able. Soil Science, 85, Geoslope International Ltd., User Guide SEEP/W for finite element seepage analysis. Version 3 Gomes, R. C., Araújo,L. G., Oliveira, W. L.F., Ribeiro, S. G. S., Nogueira, C. L., (1999a). Concepção e Projeto Básico da Disposição em Pilhas de Rejeito de Minério de Ferro em Cava Exaurida de Mineração. VI Congresso Brasileiro de Geotecnia Ambiental, REGEO 99. Lambe,. W., and Whitman, R. V.,(1979), Soil Mechanics, SI Version, John Wiley & Sons Ortigão, J. A. R., (1995), Introdução àmecânica dos Solos dos Estados Críticos, Livros écnicos e Científicos S. A.. Rahardjo, H. and Leong, E. C. (1997).Soilwater Characteristic Curves and Flux Boundary Probems, Unsaturated Soil Engineering Practice. Geothecnical Special Publication (68) ASCE, Van Genuchten, M.. (1980). A Closed Form Equation for Predicting the Hydraulic conductivity of Unsaturated Soils. Soil Science Society of America Journal, 44, Znidarcic, D. (1998). Report on the Review of Germano Exhausted Open Pit Project 240