The Evaluation of the Thermal Behaviour of an Underground Repository of the Spent Nuclear Fuel

Transcription

1 The Evaluation of the Thermal Behaviour of an Underground Repository of the Spent Nuclear Fuel Roman Kohut, Jiří Starý, and Alexej Kolcun Institute of Geonics, Academy of Sciences of the Czech Republic, Studentská 1768, Ostrava-Poruba, Czech Republic Abstract. The paper concerns the evaluation of the thermal behaviour of an underground repository of the spent nuclear fuel where the canisters are disposed at a vertical position in the horizontal tunnels. The formulation of thermo-elastic problems should regard the basic steps of the construction of the repository. We tested the influence of the distance between the deposition places on the thermo-elastic response of the rock massif. The problems are solved by the in-house GEM-FEM finite element software. One sided coupling allows a separate solution of the temperature evolution and the computation of elastic responses only in predefined time points as a post-processing to the solution of the heat equations. A parallel solution of the arising linear systems by the conjugate gradient method with a preconditioning based on the additive Schwarz methods is used. 1 Introduction Management of high-level, long-lived radioactive waste is an important issue today for all nuclear-power-generating countries. The deep geological disposal of these wastes is one of the promissing options. The design of a safe underground depository of a spent nuclear fuel (SNF) from nuclear power stations requires careful study of the repository construction, reliability of the protecting barriers between SNF and the environment and study of all kinds of risks related to the behaviour of the whole repository system. For the assessment of the repository performance, it is fundamental to be able to do large-scale computer simulations in various coupled processes as heat transfer, mechanical behaviour, water and gas flow and chemical processes in rocks and water solutions. Generally, we speak about T-H-M-C processes and their modelling. The T-H-M-C processes are generally coupled and a reliable mathematical modelling should respect at least some of the couplings. In this paper, we restrict to the modelling of T-M processes with one-directional T-M coupling via the thermal expansion term in the constitutive relations. Thus the problem can be divided in two parts. Firtsly, the temperature distribution is determined by the solution of the nonstationary heat equation, secondly, at given time points the I. Lirkov, S. Margenov, and J. Waśniewski (Eds.): LSSC 2007, LNCS 4818, pp , c Springer-Verlag Berlin Heidelberg 2008

2 426 R. Kohut, J. Starý, and A. Kolcun linear elasticity problem is solved. The numerical solution of both the problems leads to a repeated solution of large systems of linear equations and our aim is to find efficient and parallelizable iterative solution methods. Mathematically, the thermoelasticity problem is concerned with finding the temperature τ = τ(x, t) and the displacement u = u(x, t), governed by the following equations τ : Ω (0,T) R, u : Ω (0,T) R 3 κρ τ t = k i 2 τ + q(t) in Ω (0,T), (1) x 2 i j σ ij x j = f i (i =1,...,3) in Ω (0,T), (2) σ ij = c ijkl [ε kl (u) α kl (τ τ 0 )] in Ω (0,T), (3) kl ε kl (u) = 1 ( uk + u ) l in Ω (0,T), (4) 2 x l x k together with the corresponding boundary and initial conditions. 2 Numerical Methods The initial-boundary value problem of thermo-elasticity (1) (4) is discretized by finite elements in space and finite differences in time. Using the linear finite elements and the time discretization, it leads to the computation of vectors τ j,u j of nodal temperatures and displacements at the time levels t j,j =1,N, with thetimestepsδt j = t j t j 1. It gives the following time stepping algorithm: find τ 0 : M h τ 0 = τ 0, u 0 : A h u 0 = b 0 = b h (τ 0 ), for j=1,...,n: find τ j : B (j) h τ j =[M h + θδt j K h ] τ j = c j, find u j : A h u j = b j. end for Remark: The system u j : A h u j = b j (5) we solve only in predefined time points. Above, M h is the capacitance matrix, K h is the conductivity matrix, A h is the stiffness matrix, c j =[M h (1 θ)δt j K h ]τ j 1 + Δt j φ j, φ j = θq(t j )+(1 θ)q(t j 1 ), b j = b h (τ j )andτ 0 is determined from the initial condition. Here parameter θ {0, 0.5, 1}. It means that in each time level we have to solve the system of linear equations [M h + θδt j K h ]τ j =[M h (1 θ)δt j K h ]τ j 1 + Δt j φ j. (6)

3 The evaluation of the Thermal Behaviour of an Underground Repository 427 For θ = 0 we obtain the explicit Euler scheme, for θ = 1 we obtain the backward Euler (BE) scheme, θ =0.5 gives the Crank-Nicolson (CN) scheme. In our case we will use the BE scheme. If we substitute τ j = τ j 1 + Δτ j into (6), we obtain the system of equations for the increment of temperature Δτ j, [M h + Δt j K h ]Δτ j = Δt j (q j h K hτ j 1 ), (7) where q j h = q(t j). To ensure accuracy and not waste the computational effort, it is important to adapt the time steps to the behaviour of the solution. We use the procedure based on local comparison of the backward Euler (BE) and the Crank-Nicolson (CN) scheme [1]. We solve the system (6) only using BE scheme. If this solution τ j = τ j 1 + Δτ j is considered as the initial approximation for the solution of system (6) for θ = 0.5 (CN scheme), then the first iteration of Richardson s method presents an approximation of the solution of the system (6) for θ =0.5. Thus τ j CN = τ j r j,where r j =(M h +0.5Δt j K h )τ j (M h 0.5Δt j K h )τ j 1 0.5q j h 0.5qj 1. (8) h The time steps can be controlled with the aid of the ratio η = rj τ j.ifη<ε min then we continue with time step Δt =2 Δt, ifη>ε max then we continue with time step Δt =0.5 Δt, whereε min,ε max are given values. For the solution of the linear system B h Δτ j =(M h + Δ j K h )Δτ j = f j (7) we shall use the preconditioned CG method where the preconditioning is given by the additive overlapping Schwarz method. In this case the domain is divided into m subdomains Ω k. The nonoverlaping subdomains Ω k are then extended to domains Ω k in the way that overlaping between the subdomains are given by two or more layers of elements. If B kk are the FE matrices corresponding to problems on Ω k, I k and R k = (I k )T are the interpolation and restriction matrices, respectively, then introduced matrices B kk = R k BI k allow to define the one-level additive Schwarz preconditioner G, g = Gr = m I kb kk 1 R k r. k=1 Note that for the parabolic problems it is proved in [2] that under the assumption that Δ j /H 2 is reasonably bounded, the algorithms based on one-level additive Schwarz preconditioning remain numerically scalable. Here Δ j is in order of the time stepsize and H is the diameter of the largest subdomain. 3 Model Example The model example comes out from the depository design proposed in [3] (see Figure 1). The whole depository is very large, but using symmetry we can solve the problem only on the part of the domain. The model domain contains three

4 428 R. Kohut, J. Starý, and A. Kolcun Fig. 1. The global design of depository depository drifts (a half of the drift), each with four 1.32 m diameter, 4.77 m deep deposition holes and one access drift. The heating canisters (0.67 m diameter, 3.67 m length) simulating the heating from the radioactive waste are emplaced in the holes. The highest allowable temperature on the surface between the canister and the bentonit is restricted to 100 o C. We solve six variants (A, B, C, D, E, F ) which differ in the distance d h between the holes (from 2.5 mforthe variant A to 15 m for the variant F ). The whole model domain is situated 800m under surface. A constructed 3D T-M model of repository is shown in Figure 2. The computation domain is enlarged with increasing distance between the holes from dimensions m with FE grid nodes ( DOF for the heat problem, DOF for the elasticity problem) for the variant A to dimensions m with FE grid Fig. 2. Finite element mesh for repository model

5 The evaluation of the Thermal Behaviour of an Underground Repository 429 Table 1. Material properties E ν density conductivity capacity expansion (MPa) (kg/m 3 ) (W/m 0 C) (J/kg 0 C) (1/ 0 C) granite concrete bentonite steel SNF nodes ( DOF for the heat problem, DOF for the elasticity problem) for the variant F. The thermal source is given by the radiactive waste. The power of SNF in the canister decays exponentialy in time according to formula determined from given data by MATLAB q(t) = (e (t+tc) e (t+tc) e (t+tc) ) Here t c presents the cooling time depending on the burn-up value of the fuel. In our case we suppose two possibilities for the canister power. In the first case the canister power C p is 1500 W when disposed (this power is reached after t c =50.0 years pre-cooling time), in the second case the canister power C p is 1600 W when disposed (this power is reached after t c =42.16 years pre-cooling time). Canister power is a very important parameter because the canister spacing can be reduced, if the power decreases. The materials are assumed to be isotropic, the mechanical properties do not change with the temperature variations.the thermal conductivity k and thermal expansion α of the rock are also assumed to be isotropic (see Table 1). The boundary conditions for the mechanical parts consist of zero normal displacements and zero stresses on all outer faces except of the upper one. For the thermal part, we assume zero heat flux on all outer faces except ofthe bottom one, where the original rock temperature is given. On the faces of the drifts we suppose the heat transfer with the parameter H =7W/m 2 o C, the temperature of air in the drifts is supposed to be constant in time and is equal to 27 o C.The original temperature of rocks is determined by using geothermal gradient. This temperature also gives the initial condition. The computations are done in four subsequent phases: the phase of virgin rocks the initial stresses are determined from the weights of rocks, the initial temperature is determined using the geothermal gradient the drifts are excavated. The elasticity problem is solved using equivalent forces on the faces of drifts initiated by the excavation. The nonstationary heat problem is solved for period of 10 years with the initial condition determined in the phase 1 and with the heat transfer on the faces of drifts the deposition holes are excavated. The elasticity problem is solved using equivalent forces on the faces of holes

6 430 R. Kohut, J. Starý, and A. Kolcun Fig. 3. The temperature on the line parallel with the drifts crossing the center of the canisters for d h =2.5, 5.0, and 10.0 meters the time is 1.6 year(c p = 1500 W) Fig. 4. The temperature on the line parallel with the drifts crossing the center of the canisters for d h =2.5, 5.0, and 10.0 meters the time is 1.6 year(c p = 1600 W) the thermoelasticity problem is solved for the period of 200 years with the initial condition given by the temperature computed in the phase 2. The highest temperature is encountered after about 1.6 years of deposition for both the cases (C p = 1500 W, 1600 W). The results for the first case for the variants A, B, andd (d h =2.5, 5.0, and 10.0) are shown in Figure 3. The results for the second case for the variants A, B and D are shown in Figure 4. Note that the figures present the behaviour of the temperature on the line parallel with the drifts crossing the center of the canisters. We can see that in the first case (C p = 1500 W) the distance d h =5.0 m is sufficient to fulfil the restriction for the temperature on the surface of canister. In the second case (C p = 1600 W) we can situate the holes in the distance d h =10m. Remark: The distance between drifts is supposed to be 25 metres. The results of our tests showed that the canisters deposition in one drift practically do not influence the temperature in the neighbouring drifts. From the groundwater solute transport modelling point of view the knowledge of the stress field is very important. In Figure 5 we present the behaviour of the shear stress intensity for the first case (C p = 1500 W).

7 The evaluation of the Thermal Behaviour of an Underground Repository 431 Fig. 5. The shear stress intensity on the line parallel with the drifts crossing the center of the canisters for d h =2.5, 5.0, and 10.0 meters the time is 1.6 year(c p = 1500 W) Table 2. The numbers of iterations for the domain division in various directions material homogeneous non-homogeneous direction xyzxyzxyzxyzxyz xyz nbr of subdomains nbr of iterations nbr of subdomains nbr of iterations For the solution of the linear systems (5) and (6) we used the preconditioned CG methods with preconditioning given by the additive overlapping Schwarz method. The linear systems were solved in parallel. The parallel computations were performed on: the IBM xseries 455 computer (symmetric multiprocessor (SMP), 8 processors) with Intel Itanium2 1.3 GHz 64bit Processor, 16 GB shared memory the PC cluster THEA with 8 AMD Athlon 1.4 GHz, 1.5 GB RAM computer nodes. The parallel programming uses: OpenMP and MPI paradigms on SMP computer, MPI paradigm on the PC cluster. The division of the domain to subdomains influences the efficiency of the preconditioning given by the additive overlapping Schwarz method if the materials are strong anisotropic or the material parameters have big jumps or the grid is anisotropic (narrow elements). We tested this efficiency in the variant B. In this case the averaged hexahedral element has dimensions m and the material parameters have big jumps on the canisters (see Table 1). Table 2 presents the numbers of PCG iterations for one timestep (Δt j =10,ε=10 6 ), if the division to three or six subdomains in direction x, y or z is done. On the left part of the table we present results for the homogeneous case (we suppose that all materials have the same properties as granit). We can see that in this case

8 432 R. Kohut, J. Starý, and A. Kolcun the numbers of iterations correspond to the averaged dimensions of hexagonal elements. On the right part of the table we present results for the nonhomogenous case. If we use three subdomains, the boundaries of subdomains are not cutting the canisters and the numbers of iterations correspond to the averaged dimensions of elements. In the case of division to six subdomains the division in the direction x does not cut the canisters and the division in the direction y cuts the canisters directly in the centre. This fact distinctively influences the numbers of iterations. Therefore it s necessary to improve the code to enable the using of irregular division of the domain, which can guarantee that the boundaries of subdomains will not cut the areas with jumps of material parameters. 4 Conclusion In the paper, the model problem of geological depository of the spent nuclear fuel is solved. We compare the results of the solution for various distances of the deposition holes. We tested the efficiency of the DD preconditioner from the point of the dependence on the division of the domain. Acknowledgments The work was supported by the Ministry of Education, Youth and Sports under the project 1M0554 and by the Academy of Sciences of the Czech Republic through the project No. 1ET References 1. Blaheta, R., Byczanski, P., Kohut, R., Starý, J.: Algorithm for parallel FEM modelling of thermo mechanical phenomena arising from the disposal of the spent nuclear fuel. In: Stephansson, O., Hudson, J.B., Jing, L. (eds.) Coupled Thermo-Hydro- Mechanical-Chemical processes in Geo-systems, Elsevier, Amsterdam (2004) 2. Cai, X.-C.: Additive Schwarz algorithms for parabolic convection-diffusion problems. Numer. Math. 60, (1990) 3. Vavřina V.: Reference project of the underground and overground parts of the deep depository (in czech), SURAO /EGPI (1999)