THM Coupled Modeling in Near Field of an Assumed HLW Deep Geological Disposal Repository

INTRODUCTION

In many countries, disposal in deep geological repositories is one of the most suitable ways under study for the permanent isolation of high-level radioactive waste (HLW).It is of the utmost importance to evaluate the safety of this underground disposal system.An important part of the safety analysis is an assessment of the coupling of thermal, hydraulic and mechanical (THM) processes (Shen et al., 2000b, 1997; Stephansson et al., 1996; Jing et al., 1995) (see Fig. 1).Though the importance of coupled THM processes has been recognized for some years, the current capability of modeling such processes is still uncertain.

Figure  1.  Coupled THM processes in buffer material (Jing et al., 1995).

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In this paper, based on current knowledge of THM coupled processes in near field, a THM coupled model is presented to describe such processes, and a finite element code is developed.The model and code are then used to simulate the temperature, stress and hydraulic head distribution in the near field of an assumed ideal disposal system.

THM COUPLED MODEL

Assumptions Made When Using THM Coupled Model

(1) The medium is isotropic and pore-elastic; (2) Suitable for a saturated or unsaturated porous medium; (3) Darcy's law is valid for the flow of fluid; (4) Fourier's law holds for heat flux; (5) Teraghi's effective stress principle is valid; (6) Fluid density varies depending upon the temperature and pressure of fluid; (7) Heat transfers between solid phase and liquid phase are neglected.

Mechanical Equilibrium Equation

Biot (1941) provided the mechanical stress equilibrium equation for existing porous pressure conditions

where σ_ij is the stress tensor; b_i is the body force vector.

Teraghi's effective stress principle, which is fit to saturated media, can be expanded to unsaturated media, then

where σ'_ij is the effective stress tensor; φ is the piezometric head, φ=h-z, h is the total hydraulic head, z is the location of the hydraulic head; ρ is fluid density; g is gravity acceleration; χ is a function of the degree of saturation, which needs to be obtained from experiments

δ_ij is the Kronecker delta

When the medium is pore-elastic, the linear thermal stress theory can be applied, and the constituent equation is as follows

where D_ijkl is the elastic compliance tensor, which is a fourth-order tensor; ε_ij is the strain tensor; β is the thermal stress coefficient, , αt is the thermal expansion coefficient, E is medium Young's elastic module, υ is Poisson's ratio; T is current temperature; T₀ is initial temperature.

Strain definition is as follows

where ui is the displacement vector.

Fluid Flow Equation

The fluid flow equation is as follows (Bear, 1972)

where θ is water content; v is velocity vector of groundwater flow; ρis fluid density.Considering fluid compression or expansion, fluid density can be shown as

where ρ₀ is the reference density of the fluid in pressure p₀ and temperature T₀; β_T is the thermal expansivity of the fluid, ; β_p is the compressibility of the fluid, , and p is pore pressure.

Heat Transport Equation

If evaporation and condensation are not considered, and the heat transfer between the liquid phase and solid phase is neglected, then the heat transfer equation is (Bear and Corapcioglu, 1981; Fanst and Mercer, 1979)

where n is effective porosity; Sr is degree of saturation; C_v is specific heat; J is heat flux by heat conduction; f stands for liquid phase, s for solid phase; v_f is velocity of liquid phase; ε_s is strain of solid phase; ρ_s is solid phase density.

Initial Conditions and Boundary Conditions

Initial conditions are

Boundary conditions are as follows.

Displacement

Traction

Hydraulic

Flow rate

Temperature

Heat flow

Heat exchange

where T' is the water convection conduction coefficient; α is the convection heat exchange coefficient; T_b is the temperature on the boundary; T_a is the temperature in the surrounding environment.

FINITE ELEMENT METHODS FOR THMCOUPLED MODEL

The Galerkin weighted residual finite element method is employed in this paper.The code is devel - oped under the assumption that it is a plane strain or an axial symmetry problem.A fully implicit mid-interval backward difference algorithm (Thomas and King, 1991) and a predictor-corrector iterative solution procedure (Volckaert et al., 1996) were employed to solve these non-linear ful-l coupled equations.

The validation of the model and code is verified ei - ther with theoretical analysis or with experiments, which were done by the authors or other researchers. Seven examples were used to verify the model in this study, see details in Shen (1998).

THM COUPLED MODELING IN NEAR FIELD

Background

Since the disposal conception of HLW in various countries is different, the following silo-shaped disposal conception was taken in this paper (Shen and Cheng, 2002;Shen et al., 2000a).The disposal unit is an axial symmetry problem with the solidified product in the centre.Around the solidified product is a waste barrel, around the barrel is buffer material, and around the buffer material is rock.The solidified product is 430 mm in diameter and 1335 mm in height, the barrel is 1030 mm in diameter and 1935 mm in height, and the buffer material is 2230 mm in diameter and 4065 mm in height.It is also assumed that the repository is 1000 m under the surface and the distance between two disposal silos is 10 m.

The heat source of an HLW disposal repository is actually from the decay heat of radionuclides in solidi - fied body, and so the distribution of radiation doses in and around the solidified body should be discussed (Shen et al., 2000a, b).In this paper, the dose rate at different times, both inside and outside the solidified product, is calculated by using the radionuclide inventory in the vitrified high-level waste (W1) of Belgium as the initial condition (Marivot, 1991).Then the dose rate is used as the heat source in the THM coupled model.

It must be pointed out that the solidified product will be stored for 50 years on the surface before it is actually disposed of, so we considered this condition in our code (Table 1).

Table  1.  Radionuclides and their activity in the solidified body (Marivot, 1991)

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Grid Determined

The silo-shape disposal system can be treated as an axis symmetry problem, and only 1/4 part of the disposal unit should be taken in the calculation.It is assumed that the HLW will be disposed 1000 m under the ground.The calculation scale: in radial direction, half of the distance between two disposal silos, that is 5 m, is taken; in vertical direction, 20 m is taken.The numerical discretion is done, and the grids are shown in Fig. 2.There are a total of 450 rectangle elements and 496 nodes.

Figure  2.  Finite element grids.In it, from left to right in lower boundary, 1# to 16# are taken as typical points.1# is the center of the solidified product and 16# is far away from the solidified product.1# to 3# are in the solidified product, 3# to 5# are in the package, 5# to 9# are in buffer material, and 9# to 16# are in disposal medium.The typical points showed in the following figures are the same as this.

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Initial Conditions and Boundary Conditions Initial conditions

Temperature field   The disposal repository is 1000 m subsurface, the temperature at the surface is 15℃, the local ground temperature gradient is 2.5 ℃/ 100 m, so the initial temperature at the disposal repository is 40℃.

Hydraulic field  The buffer material is unsaturated high-compacted bentonite, the initial degree of saturation is 80%, and it is assumed that the water head in the bedrock (disposal medium) will be 800 m as soon as the disposal is completed.

Stress field  It is assumed that the bedrock is at stress equilibrium state after the disposal is completed.

Boundary conditions

Temperature field  The left, right and lower boundaries are heat fluxes zero boundaries, and the upper boundary is the varying fluxes boundary, which can be treated as that in Luo and Chen (1988).

Hydraulic field  The upper and right boundaries are the stable water head boundaries; the lower and left boundaries are the zero flux boundaries.

Stress field  The upper and lower boundaries are the zero vertical displacement boundaries, and the left and right boundaries are the zero landscape orientation displacement boundaries.

Parameters

Table 2 shows the parameters that we used in the model.The parameters of the buffer material are from experiments carried out by Shen (1998).Other parameters are from Stephansson et al.(1996), Dong (1994) and Rehbinder (1995).

Table  2.  Parameters of the various mediums

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Result and Discussion Temperature field

Figure 3 shows the temperature vs.time at several typical points (1#, 5#, 9#, 16#).It shows that the temperature in the central point (1#) of the solidified product will reach the maximum at about 10 a after waste disposal is completed (about 410 K) and then it decreases slowly over time, remaining high even after 10000 a (360 K).The varying trends of the other points in the near field are the same, except that the temperature is a little lower than that of the central point.The greater the distance from the central point, the lower the maximum temperature.

Figure  3.  Temperature vs.time at typical points.

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A very interesting phenomenon is found in modeling, just as shown in Fig. 3, the temperature drops at some periods (0.1-1 a) of heating.We wanted to know why this phenomenon appears, and found that it does not appear if the THM coupled processes are not considered.And if the TM coupled processes only are considered, this phenomenon does not exist either.It appears to be the result of heat convection by fluid flowing.This phenomenon has been not reported by any references so far.

But it must be pointed out that this phenomenon only appeared in this example.In this example, water expansion is caused by a rise in temperature, so the water head in the buffer material is very high and the hydraulic gradient is very large, which means the velocity of fluid is also very high.In these modeled conditions the phenomenon has appeared.In actual condi - tions the groundwater around the disposal repository will be drained before disposal.When the disposal is completed the repository will be closed; the recovery of the water level is gradual, and may take as long as 2 -5 a, so this phenomenon is unlikely to appear.

Figure 4 shows temperature vs.radial distance at different times at the central plane.It shows that the temperature gradient in the solidified product and buffer material is much bigger, and in the package and disposal medium is much smaller.As time goes by, the temperature gradient in the near field gets smaller and smaller.

Figure  4.  Temperature vs.radial distance at the central plane.

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Figure 5 shows temperature distribution in the buffer material in 1 d and 30 a after disposal is completed.It shows that the temperature in the buffer material is 314 K to 319 K in 1 d after disposal is completed, and 360 K to 380 K in 30 a after disposal completion.

Figure  5.  Temperature distribution in buffer material after disposal.(a)1 d;(b)30 a.

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Hydraulic field

Figure 6 shows the hydraulic head vs.time in the buffer material.The fluid in the buffer material is unsaturated in its initial condition.Since it is assumed that the hydraulic heads in the bedrock (disposal medium) remain steady, and the hydraulic head in the solidified body and its package is zero, the hydraulic head change in the buffer material is a consolidation problem with the temperature impact.With the rising temperature, the hydraulic head can not be dissipated in time, so the hydraulic head in the buffer material changes very quickly at the beginning, reaching the maximum in about 15 d.Then the hydraulic head dissipates gradually, becoming steady in about 90 d.

Figure  6.  Hydraulic head vs.time at typical points in buffer material (the points 1#-5# in this figure are actual 5#-9# typical points in Fig. 2).

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Stress field

Figure 7 shows the radial stress vs.time in the buffer material at typical point 6#.Since the stress in this example is actually caused by thermal stress (which is caused by temperature change), and fluid pressure (which is caused by hydraulic head change), it is closely related to temperature and hydraulic head. After about 100 a, the hydraulic head is holding at 800 m and the variation in temperature is very small, so the stress hardly changes.

Figure  7.  Radial stress vs.time at typical point 6#.

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CONCLUSION

The study of THM coupled processes is the new problem which is put forward during HLW deep geological disposal.From above, a model is presented to describe ful-l coupled THM behavior for saturated-unsturated porous media.The preliminary simulation in an assumed disposal condition is performed.The spatial distributions of temperature, water head and stress in the near field of HLW repository in different times are given out.It shows that THM coupled processes are very important in the safety analysis of an HLW deep geological disposal repository.Since THM coupled processes are very complicated, much work needs to be done to fully understand THM coupled processes in near field, even in far field with disposal medium.