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Coupled Thermal, Hydraulic and Mechanical Simulation with a Theoretical Model for Swelling Characteristics
553 COUPLED THERMAL, HYDRAULIC AND MECHANICAL SIMULATION WITH A THEORETICAL MODEL FOR SWELLING CHARACTERISTICS Hideo Komine\ Hiroshi Kurikami^, Masakazu Chijimatsu^, Akira Kobayashi"^, Yutaka Sugita^ and Yuzo Ohnishi"^ ^) Ibaraki University, Japan ^) Japan Nuclear Cycle Development Institute, Japan ^) Hazama Corporation, Japan "*) Kyoto University, Japan Abstract: Buffer material as a part of an engineered barrier system for high-level waste disposal is expected to seal the initial gaps between the buffer material and the surrounding rock during the resaturation process. Since the self-sealing ability represents the swelling characteristics of the buffer material, it is very important to estimate the swelling behavior during the resaturation process. This paper expands the theoretical model for evaluating the swelling characteristics of saturated buffer material proposed by Komine to deal with unsaturated media, and it is applied to the coupled thermal, hydraulic and mechanical (THM) model used in Japan. The new model is validated by comparison with a series of laboratory experiments quoted from the literature.
1. INTRODUCTION Buffer material for high-level waste disposal is expected as a barrier with groundwater preventive ability, self-sealing ability, adsorption ability, etc. As a buffer material, bentonite has attracted attention because its swelling characteristics are useful for sealing the initial cracks within the buffer material or the initial gaps between the buffer material and the surrounding rock. Komine & Ogawa (1996, 2003) have proposed a theoretical model for evaluating the swelling characteristics of saturated compacted bentonitebased buffer material. This model is based on the Gouy-Chapman diffuse double layer theory and the van der Waals force. Specifying the physical and chemical properties of the buffer material, such as the density of montmorillonite, exchangeable ion capacity, etc, the swelling pressure or swelling strain of the saturated buffer material can be calculated. However, the buffer material is unsaturated during the resaturation process when the swelling takes place. Therefore, evaluation of swelling characteristics in an unsaturated condition is also important. During the resaturation process, coupled THM behavior occurs. Although analytical and experimental studies on the coupled processes have been performed for a few decades, the swelling characteristics have not been strictly modeled. This paper proposes a new equation to predict the swelling pressure of the buffer material during the resaturation process based on Komine's model, and introduces it to the coupled THM model proposed by Chijimatsu et al. (2000) to build a
more reliable model. In this model, the swelling pressure is assumed to be a function of the degree of saturation by combining the swelling potential to the free energy of the unsaturated buffer material. The proposed model is validated by comparison with a series of laboratory experiments conducted by Chijimatsu et al. (1999) and Suzuki et al (1999).
2. SWELLING CHARACTERISTICS 2.1 Evaluation of maximum swelling pressure in saturated media Swelling characteristics of compacted bentonite is dominated by the repulsive and the attractive forces between two parallel clay layers. Komine has introduced new ideas to the previous model to deal with the compositions and conditions of compacted bentonite (Komine & Ogata(1996, 2003)). In the new equation, it is assumed that the swelling clay minerals such as montmorillonite in bentonite dominate the swelling characteristics of the buffer material independently of the nonswelling minerals and that each exchangeable cation (Na"", Ca"*, K"^, Mg^*) contributes the forces. Using the new model, not only the compositions of bentonite but also the conditions of environment such as groundwater composition, temperature, dry density can be taken into account. Table 1 shows the constants and parameters required to the model. If the coupled THM processes are considered, temperature and dry density in the environmental properties are most important because they are variables in the coupled processes. If the effects of
554 Table 1. Constants required Physical constans | Electonic charge 1.602x10'^ C 1 Boltzmann constant 1.38x10''J/K Hamaker constant 2.2x10'^ J Abogadro's number 6.023x10'' Bentonite properties (bentonite Kunigel VI) Density of montmollironite 2.77x10'kg/m' Density of the other minerals 2.81x10'kg/m' Density of sand 2.66x10'kg/m' Specific surface of 810 montmorillonite Specific surface of the other minerals
Table 2. Definition of F against saturation degree Swelling Pressure Function F (A)
1
n -^ dS, ^.vH=^.HmaxV^
0,y,=0,^„^Jr
(B) (C)
F= 2 . _ 5 . | ^
(D)
^.K
^sw
=<^swnuxSr
~
^swimx^r
0
Content of montmorillonite 48% exchange capacity of Na* 0.405 mequiv./g exchange capacity of Ca^^ 0.287 mequiv./g exchange capacity of K* 0.009 mequiv./g exchange capacity of Mg^"^ 0.030 mequiv./g 1 Nonhydrated radius of Na* 1 Nonhydrated radius of Ca^^ 2 1 Nonhydrated radius of K* 1 1 Nonhydrated radius of Mg^"^ 2 1 Thickness of montmorillonite 9.60x10'° m 1 layers 1 Environmental parameters Static permittivity of pore water, Temperature, Ion consentration, Dry density, Content of bentonite
chemical reactions or salinity in groundwater are considered, this model is useful to build coupled thermal, hydraulic, mechanical and chemical (THMC) model.
0.0
0.2 0.4 0.6 0.8 Degree of saturation, 5^
1.0
Figure 1. Swelling pressure with degree of saturation gravity potential is negligible by comparison with the osmotic and the matric potential, where the suction is sum of the osmotic and the matric potential. Therefore, the swelling pressure can be expressed by the suction. Although the unsaturated buffer material has large suction, it is consumed during the resaturation process not only by the swelling but also by the soil structure changes. We define the function F that illustrates the ratio of the suction contributing to the swelling pressure.
2.2 Evaluation of process of swelling pressure in unsaturated media
Ao,
During the resaturation process of the buffer material, the swelling pressure in unsaturated condition is important, though the model explained in the previous section is limited to the saturated condition. In this section, a new equation is proposed to treat swelling pressure in the unsaturated condition. Free energy change, which consists of matric potential, osmotic potential and gravity potential, is equal to potential swelling pressure (Nakano et al.(1984)). In the case of buffer material, it is approximately equal to the suction, because the
where yr is suction and a,y, is swelling pressure. The function F must satisfy the following equation.
-FAi//
0^..max=p'^V^
(2)
(3)
is the maximum swelling pressure where CT^H from Komine's model. The prospective functions for F are shown in Table 2. Where F is assumed to depend only on degree of saturation Sr- Figure 1 shows the relationship between the swelling
555
^
^
+ ^,,+(2 = 0
^
(5)
dt
Initial state
where 0 is volumetric water content, p^ is water density, / is time, <7, is groundwater flux vector and Q is sink/source.
! ^
3.1.3 Law of heat energy conservation S z
0
Law of heat energy described as
Degree of saturation, 5
Figure 2. Swelling pressure against degree of saturation
conservation
can be
(6) dt
pressure and the degree of saturation for each function given in Table 2. The model (A) represents the condition under that swelling pressure rises in the early stages of the resaturation process, while the model (D) represents the condition under that the swelling pressure cannot be expected for the self-sealing ability in the early stages. Figure 2 shows the resaturation process where initial degree of saturation is S^. When the resaturation begins, the swelling pressure rises to the initial value (7^^ immediately and continues to rise with change of degree of saturation gradually.
3. COUPLED T-H-M PROCESSES IN BUFFER To introduce the new model for the swelling characteristics to the fully coupled THM processes in the resaturation stages, the model explained above is applied to THAMES that is the finite element simulator treating the fully coupled THM process which was improved by Chijimatsu et al(2000). This chapter describes the governing equations of the coupled processes.
3.1 Conservation laws 3.1.1 Law of momentum conservation Conservation law of momentum represents the equilibrium equations as (4)
where Acr, is tensor of total stress increment.
3.1.2 Law offluidmass conservation Law of fluid mass conservation can be written as
where {pc)n, is specific heat of mixture of gas, liquid, and solid, T is temperature, q^i is heat flux vector and ^ is heat sink/source.
3.2 Constitutive equations for bentonite This paper explains only constitutive equations in the unsaturated bentonite because the other constrains are almost general. Assuming the elasticity of bentonite, the increment of the total stress in unsaturated media is written as follows. Aa, = iCy^(AW,,, + Aw/.,)-^G,J,J
-^5,^
(7)
where C,ju is elastic modulus tensor, u, is displacement, P is the parameter regarding to thermal stress and S,j is Kronecker's delta. The second term which represents swelling pressure is calculated by the Equation (2). Groundwater flux is expressed by q, •-PADO\OJ-PADT\TJ
(8)
where (D^Xy is water diffusivity and (Dr)y is thermal water diffusivity. Dg is dependent of both degree of saturation and temperature and shows concave up when it is plotted against the volumetric water content. It represents that the moisture movement in the unsaturated media occurs both in liquid and gas phase, where the former is dominant in high saturation, while the latter is dominant in law saturation. Dg of the bentonite in Japan is given by the following experimental equation, which is the combination of the moisture movement in gas and liquid phase. {Dg\=S,jiDa+Dj
(9)
556 Table 3. Experimental conditions Isothermal tests
Case
Anisothermal test (d)
(a), (b)
(c) Bentonite Kunigel VI
Material
Mixture of Kunigel VI and sand
Specimen size [mm]
0 20 X height 20
0 50 X height 100
0 50 X height 100
Dry density [kg/m^]
1.8X10'
1.8X10'
1.6X10'
Initial water content [%]
0.5, 9.0
10.0
7.0
Temperature [^C]
25
25
30 at upper end, 40 at lower end
Infiltration pressure [mH20]
0.0 from lower end
1.0 from upper end
1.0 from upper end
Max swelling pressure [MPa]
3.7
3.7
0.35
Table 4. Material properties Bentonite Kunigel VI
Mixture
Dry density [kg/m^]
1.8X10'
1.6X10'
Young's modulus [MPa]
1140.0-55.64w
58.74-1.87 w
Poisson ratio [-]
0.3
0.3
Intrinsic permeability K [m^]
4.00X10"
4.00 XIO""
0A-]
0.333
0.403
0r[-] cr[l/m]
0.000
0.000
6.0X10-^
8.0X10"
Van Genuchten parameters
2.5
1.6
a\
1.76X10Y-3.04X10"'
2.99X10"°r-3.78X10'
^2
-1.48X10'7+2.98X10°
-1.50XlO°r+1.49X10'
n[-]
Water diffusivity [cm~/s]
"- (e-^X*,-^J " b,ie-b,)
bx
-3.68X10'
>2.49X10'
bj
5.22X10-'7+2.68X10'
5.59X10*7+3.93X10'
Thermal water diffusivity DT [m7s/K] Thermal conductivity A^ [W/m/K]
2.0 XiO'"
7.0X10"
5.58X10"'+6.17X10"'H'
4.44XiO'+1.38X10-'w
-5.28X10V-9.67X10-V
+6.14X10V-1.69X10V
(32.3+4.18w)/(100+w)
(34.1+4.18w)/(100+w)
Specific heat capasity c [kJ/kg/K]
; water content, T: temperature [°C] where subscript / means liquid and v means vapor, or gas phase. And the isotropy is assumed. On the other side, Dj is assumed a constant because of its difficulty to define. Water retention curve is described by van Genuchten model shown as bellow. Sr
_ e-e, =UM"Y
.1-i
(10)
where Or and 6s is residual and maximum volumetric water content, respectively. And a, n and m are van Genuchten parameters. Heat flux, q^, consists of the advective term, q", and the dispersive term, q'', as (11)
where q] --S,qJ
(12) (13)
557 0.5 0.4 0.3
1 • VTC^K^
Experiments • 8hrs O 48hrs A 16hrs A 72hrs • 24hrs D %hrs Simulation
r /
0.2 H
0.1
Experiments Simulations (A) (B)
D
1
\ * \A
O (C) (D)
0.0
0
5
10
15
20
Distance from infiltration end, mm
Figure 3. Distribution of volumetric water content in simulation and experiments of the test (a)
where Arr, is thermal conductivity of the mixture.
4. VALIDATION 4.1 Experimental conditions To examine the applicabiUty of the new model, it was applied to a series of laboratory swelling experiments conducted by Chijimatsu et al.(1999) and Suzuki et al(1999). These experiments were performed to estimate the hydraulic and the mechanical characteristics of the buffer during the resaturation process. Table 3 and Table 4 show the experimental conditions and the material properties, respectively. The parameters shown in Table 1 are used for Komine's model. There were three tests (a), (b) and (c) in which bentonite Kunigel VI was used on isothermal condition and one test (d) in which mixture of Kunigel VI and sand (sand 30wt%) was used under temperature gradient condition. The object of the test (a) was the evaluation of the infiltration behavior. The distribution of the water content at each time was measured by using several specimens under the same condition, where the specimen was divided into every 2mm height and the water content of each piece was measured. On the other side, the test (b) and (c) were performed to examine the swelling pressure. The experimental conditions of the test (b) were the same as those of the test (a), though the initial water contents were different. The size of the specimen of the test (c) is larger than that of the test (a) and (b). The object of the test (d) was to estimate the swelling pressure under temperature gradient condition during the resaturation process. In the test, horizontal pressure at the upper, the middle and the lower parts and the
0
100
200
300
400
500
600
Time, hrs
Figure 4. Time histories of stress in the test (b) vertical pressure at the top surface of the specimen were measured as time. Comparing this series of tests with the numerical results comprehensively, the numerical model can be validated.
4.2 Results The height of each finite element was 1mm, which resulted in 20 elements for the test (a) and (b) and in 100 elements for the test (c) and (d). The boundary conditions are as follows. Slide boundaries were applied to all the boundaries. The constant hydraulic pressure and constant temperature were specified to the upper and/or the lower boundary following the test conditions shown in Table 3, while the other boundaries were impermeable and adiabatic. As the initial conditions, 25 °C and 30 °C were applied to the isothermal and the anisothermal tests, respectively. Initial water content was also shown in Table 3. To examine the effect of the function F proposed in this paper, the models (A) to (D) in Table 2 were applied to the simulation of the test (b). In the other tests, only the model (A) was used. Figure 3 shows the comparison of the volumetric water content distribution of the test (a) between the experiment and the simulation. Good agreement gives the validity of the moisture movement in the unsaturated bentonite. Figure 4 shows the time histories of the vertical stress in the test (b) corresponding to each F with the experimental results. The resaturation time of all the simulations was about 150 hours and was almost same as that of the experiment. The figure shows that the model (A) is the most suitable for F compared with the other models, though the initial stress is slightly large. It suggests that swelling pressure rises mainly in the early stages of the infiltration with respect to the compacted bentonite. That is the reason why the model (A) was applied to the other tests.
558
0.4
6a.
5-
0.3
4(U
n
-
2-
O
00
0.2
Experiments - Simulation
1-
^-^^
ifJsw^
o
^ksS^^' /~-* ' •
0.1 2000
4000
6000
8000
Time, hrs
—-^ =fiHfi=e-i A^-^
^u^T^ y^
n
3- ^sisx-
''\
Simulation Experiment Vertical stress - o Horizontal stress Lower part —O— —A— Middle part - A— Upper part —V—
—»—
• ~^^
no 0
2000
4000
6000
8000
Time, hrs
Figure 5. Time histories of stress in the test (c) Figure 6. Time history of stress in the test (d) Figure 5 shows the time history of the vertical stress in the test (c). As well as the test (b), good agreement between the experiment and the simulation was obtained. It means that the model proposed is useful for every scale. Figure 6 shows the time histories of the vertical and the horizontal stress with the experimental result for the test (d). Simulated vertical stress agrees with the monitored. On the other hand, the horizontal stress was slightly different. One reason is the initial disturbance of the experiment because of the complicatedness of the apparatus. However, the most influential reason is the anisotropy of the swelling pressure. The direction of compaction of the buffer material plays an important role for the swelling behavior. The further study on the anisotropy has to be continued.
5. CONCLUSIONS This paper proposes a new model for evaluating the swelling pressure during the resaturation process, and introduces it to the coupled thermal, hydraulic and mechanical model. Compared with the series of laboratory experiments, the following conclusions were drawn. Assuming that the swelling pressure is a function of the degree of saturation, good agreement between the simulation and the experiment was obtained. This indicates a oneto-one correspondence between the swelling pressure and the degree of saturation. Swelling pressure is likely to occur in the earlier stages of the resaturation process, and the value is strongly related to the initial degree of saturation.
Although the proposed model cannot strictly simulate the anisotropy of swelling pressure, the results are useful. Since the model involves the Komine's theoretical model for evaluating swelling pressure in saturation, it has the potential to include the effects of chemical change in bentonite and pore water. Using this model, fully coupled thermal, hydraulic, mechanical and chemical analysis can be conducted in the future.
REFERENCES Chijimatsu, M. & Taniguchi, W. 1999. Coupled thermal, hydraulic and mechanical analysis in the near field for geological disposal of highlevel radioactive waste (in Japanese). JNC TN8400 99-014. Chijimatsu, M., Fujita, T., Kobayashi, A. & Nakano, M. 2000. Experiment and validatioin of numerical simulation of coupled thermal, hydraulic and mechanical behaviour in the engineered buffer materials. Int. J. Num. Anal. Mech. Geomech. 24: pp.403-424. Komine, H. & Ogata, N. 1996. Prediction for swelling characteristics of compacted bentonite. Can. Geotech. J. 33: pp. 11-22. Komine, H. & Ogata, N. 2003. New equations for swelling characteristics of bentonite-based buffer materials. Can. Geotech. J. 40(2): pp.460-475 Nakano, M., Amemiya, Y., Fujii, K. & Ishida, A. 1984. Water Penetration and Swelling Pressure of Restrained Unsaturated Clay(in Japanese). JSIDRE112:pp.55-66. Suzuki, H., Chijimatsu, M. & Fujita, T. 1999. Measurement of moisture movement and swelling pressure under temperature gradient (in Japanese). JNC TN8400 99-020.
1. INTRODUCTION Buffer material for high-level waste disposal is expected as a barrier with groundwater preventive ability, self-sealing ability, adsorption ability, etc. As a buffer material, bentonite has attracted attention because its swelling characteristics are useful for sealing the initial cracks within the buffer material or the initial gaps between the buffer material and the surrounding rock. Komine & Ogawa (1996, 2003) have proposed a theoretical model for evaluating the swelling characteristics of saturated compacted bentonitebased buffer material. This model is based on the Gouy-Chapman diffuse double layer theory and the van der Waals force. Specifying the physical and chemical properties of the buffer material, such as the density of montmorillonite, exchangeable ion capacity, etc, the swelling pressure or swelling strain of the saturated buffer material can be calculated. However, the buffer material is unsaturated during the resaturation process when the swelling takes place. Therefore, evaluation of swelling characteristics in an unsaturated condition is also important. During the resaturation process, coupled THM behavior occurs. Although analytical and experimental studies on the coupled processes have been performed for a few decades, the swelling characteristics have not been strictly modeled. This paper proposes a new equation to predict the swelling pressure of the buffer material during the resaturation process based on Komine's model, and introduces it to the coupled THM model proposed by Chijimatsu et al. (2000) to build a
more reliable model. In this model, the swelling pressure is assumed to be a function of the degree of saturation by combining the swelling potential to the free energy of the unsaturated buffer material. The proposed model is validated by comparison with a series of laboratory experiments conducted by Chijimatsu et al. (1999) and Suzuki et al (1999).
2. SWELLING CHARACTERISTICS 2.1 Evaluation of maximum swelling pressure in saturated media Swelling characteristics of compacted bentonite is dominated by the repulsive and the attractive forces between two parallel clay layers. Komine has introduced new ideas to the previous model to deal with the compositions and conditions of compacted bentonite (Komine & Ogata(1996, 2003)). In the new equation, it is assumed that the swelling clay minerals such as montmorillonite in bentonite dominate the swelling characteristics of the buffer material independently of the nonswelling minerals and that each exchangeable cation (Na"", Ca"*, K"^, Mg^*) contributes the forces. Using the new model, not only the compositions of bentonite but also the conditions of environment such as groundwater composition, temperature, dry density can be taken into account. Table 1 shows the constants and parameters required to the model. If the coupled THM processes are considered, temperature and dry density in the environmental properties are most important because they are variables in the coupled processes. If the effects of
554 Table 1. Constants required Physical constans | Electonic charge 1.602x10'^ C 1 Boltzmann constant 1.38x10''J/K Hamaker constant 2.2x10'^ J Abogadro's number 6.023x10'' Bentonite properties (bentonite Kunigel VI) Density of montmollironite 2.77x10'kg/m' Density of the other minerals 2.81x10'kg/m' Density of sand 2.66x10'kg/m' Specific surface of 810 montmorillonite Specific surface of the other minerals
Table 2. Definition of F against saturation degree Swelling Pressure Function F (A)
1
n -^ dS, ^.vH=^.HmaxV^
0,y,=0,^„^Jr
(B) (C)
F= 2 . _ 5 . | ^
(D)
^.K
^sw
=<^swnuxSr
~
^swimx^r
0
Content of montmorillonite 48% exchange capacity of Na* 0.405 mequiv./g exchange capacity of Ca^^ 0.287 mequiv./g exchange capacity of K* 0.009 mequiv./g exchange capacity of Mg^"^ 0.030 mequiv./g 1 Nonhydrated radius of Na* 1 Nonhydrated radius of Ca^^ 2 1 Nonhydrated radius of K* 1 1 Nonhydrated radius of Mg^"^ 2 1 Thickness of montmorillonite 9.60x10'° m 1 layers 1 Environmental parameters Static permittivity of pore water, Temperature, Ion consentration, Dry density, Content of bentonite
chemical reactions or salinity in groundwater are considered, this model is useful to build coupled thermal, hydraulic, mechanical and chemical (THMC) model.
0.0
0.2 0.4 0.6 0.8 Degree of saturation, 5^
1.0
Figure 1. Swelling pressure with degree of saturation gravity potential is negligible by comparison with the osmotic and the matric potential, where the suction is sum of the osmotic and the matric potential. Therefore, the swelling pressure can be expressed by the suction. Although the unsaturated buffer material has large suction, it is consumed during the resaturation process not only by the swelling but also by the soil structure changes. We define the function F that illustrates the ratio of the suction contributing to the swelling pressure.
2.2 Evaluation of process of swelling pressure in unsaturated media
Ao,
During the resaturation process of the buffer material, the swelling pressure in unsaturated condition is important, though the model explained in the previous section is limited to the saturated condition. In this section, a new equation is proposed to treat swelling pressure in the unsaturated condition. Free energy change, which consists of matric potential, osmotic potential and gravity potential, is equal to potential swelling pressure (Nakano et al.(1984)). In the case of buffer material, it is approximately equal to the suction, because the
where yr is suction and a,y, is swelling pressure. The function F must satisfy the following equation.
-FAi//
0^..max=p'^V^
(2)
(3)
is the maximum swelling pressure where CT^H from Komine's model. The prospective functions for F are shown in Table 2. Where F is assumed to depend only on degree of saturation Sr- Figure 1 shows the relationship between the swelling
555
^
^
+ ^,,+(2 = 0
^
(5)
dt
Initial state
where 0 is volumetric water content, p^ is water density, / is time, <7, is groundwater flux vector and Q is sink/source.
! ^
3.1.3 Law of heat energy conservation S z
0
Law of heat energy described as
Degree of saturation, 5
Figure 2. Swelling pressure against degree of saturation
conservation
can be
(6) dt
pressure and the degree of saturation for each function given in Table 2. The model (A) represents the condition under that swelling pressure rises in the early stages of the resaturation process, while the model (D) represents the condition under that the swelling pressure cannot be expected for the self-sealing ability in the early stages. Figure 2 shows the resaturation process where initial degree of saturation is S^. When the resaturation begins, the swelling pressure rises to the initial value (7^^ immediately and continues to rise with change of degree of saturation gradually.
3. COUPLED T-H-M PROCESSES IN BUFFER To introduce the new model for the swelling characteristics to the fully coupled THM processes in the resaturation stages, the model explained above is applied to THAMES that is the finite element simulator treating the fully coupled THM process which was improved by Chijimatsu et al(2000). This chapter describes the governing equations of the coupled processes.
3.1 Conservation laws 3.1.1 Law of momentum conservation Conservation law of momentum represents the equilibrium equations as (4)
where Acr, is tensor of total stress increment.
3.1.2 Law offluidmass conservation Law of fluid mass conservation can be written as
where {pc)n, is specific heat of mixture of gas, liquid, and solid, T is temperature, q^i is heat flux vector and ^ is heat sink/source.
3.2 Constitutive equations for bentonite This paper explains only constitutive equations in the unsaturated bentonite because the other constrains are almost general. Assuming the elasticity of bentonite, the increment of the total stress in unsaturated media is written as follows. Aa, = iCy^(AW,,, + Aw/.,)-^G,J,J
-^5,^
(7)
where C,ju is elastic modulus tensor, u, is displacement, P is the parameter regarding to thermal stress and S,j is Kronecker's delta. The second term which represents swelling pressure is calculated by the Equation (2). Groundwater flux is expressed by q, •-PADO\OJ-PADT\TJ
(8)
where (D^Xy is water diffusivity and (Dr)y is thermal water diffusivity. Dg is dependent of both degree of saturation and temperature and shows concave up when it is plotted against the volumetric water content. It represents that the moisture movement in the unsaturated media occurs both in liquid and gas phase, where the former is dominant in high saturation, while the latter is dominant in law saturation. Dg of the bentonite in Japan is given by the following experimental equation, which is the combination of the moisture movement in gas and liquid phase. {Dg\=S,jiDa+Dj
(9)
556 Table 3. Experimental conditions Isothermal tests
Case
Anisothermal test (d)
(a), (b)
(c) Bentonite Kunigel VI
Material
Mixture of Kunigel VI and sand
Specimen size [mm]
0 20 X height 20
0 50 X height 100
0 50 X height 100
Dry density [kg/m^]
1.8X10'
1.8X10'
1.6X10'
Initial water content [%]
0.5, 9.0
10.0
7.0
Temperature [^C]
25
25
30 at upper end, 40 at lower end
Infiltration pressure [mH20]
0.0 from lower end
1.0 from upper end
1.0 from upper end
Max swelling pressure [MPa]
3.7
3.7
0.35
Table 4. Material properties Bentonite Kunigel VI
Mixture
Dry density [kg/m^]
1.8X10'
1.6X10'
Young's modulus [MPa]
1140.0-55.64w
58.74-1.87 w
Poisson ratio [-]
0.3
0.3
Intrinsic permeability K [m^]
4.00X10"
4.00 XIO""
0A-]
0.333
0.403
0r[-] cr[l/m]
0.000
0.000
6.0X10-^
8.0X10"
Van Genuchten parameters
2.5
1.6
a\
1.76X10Y-3.04X10"'
2.99X10"°r-3.78X10'
^2
-1.48X10'7+2.98X10°
-1.50XlO°r+1.49X10'
n[-]
Water diffusivity [cm~/s]
"- (e-^X*,-^J " b,ie-b,)
bx
-3.68X10'
>2.49X10'
bj
5.22X10-'7+2.68X10'
5.59X10*7+3.93X10'
Thermal water diffusivity DT [m7s/K] Thermal conductivity A^ [W/m/K]
2.0 XiO'"
7.0X10"
5.58X10"'+6.17X10"'H'
4.44XiO'+1.38X10-'w
-5.28X10V-9.67X10-V
+6.14X10V-1.69X10V
(32.3+4.18w)/(100+w)
(34.1+4.18w)/(100+w)
Specific heat capasity c [kJ/kg/K]
; water content, T: temperature [°C] where subscript / means liquid and v means vapor, or gas phase. And the isotropy is assumed. On the other side, Dj is assumed a constant because of its difficulty to define. Water retention curve is described by van Genuchten model shown as bellow. Sr
_ e-e, =UM"Y
.1-i
(10)
where Or and 6s is residual and maximum volumetric water content, respectively. And a, n and m are van Genuchten parameters. Heat flux, q^, consists of the advective term, q", and the dispersive term, q'', as (11)
where q] --S,qJ
(12) (13)
557 0.5 0.4 0.3
1 • VTC^K^
Experiments • 8hrs O 48hrs A 16hrs A 72hrs • 24hrs D %hrs Simulation
r /
0.2 H
0.1
Experiments Simulations (A) (B)
D
1
\ * \A
O (C) (D)
0.0
0
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Figure 3. Distribution of volumetric water content in simulation and experiments of the test (a)
where Arr, is thermal conductivity of the mixture.
4. VALIDATION 4.1 Experimental conditions To examine the applicabiUty of the new model, it was applied to a series of laboratory swelling experiments conducted by Chijimatsu et al.(1999) and Suzuki et al(1999). These experiments were performed to estimate the hydraulic and the mechanical characteristics of the buffer during the resaturation process. Table 3 and Table 4 show the experimental conditions and the material properties, respectively. The parameters shown in Table 1 are used for Komine's model. There were three tests (a), (b) and (c) in which bentonite Kunigel VI was used on isothermal condition and one test (d) in which mixture of Kunigel VI and sand (sand 30wt%) was used under temperature gradient condition. The object of the test (a) was the evaluation of the infiltration behavior. The distribution of the water content at each time was measured by using several specimens under the same condition, where the specimen was divided into every 2mm height and the water content of each piece was measured. On the other side, the test (b) and (c) were performed to examine the swelling pressure. The experimental conditions of the test (b) were the same as those of the test (a), though the initial water contents were different. The size of the specimen of the test (c) is larger than that of the test (a) and (b). The object of the test (d) was to estimate the swelling pressure under temperature gradient condition during the resaturation process. In the test, horizontal pressure at the upper, the middle and the lower parts and the
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Figure 4. Time histories of stress in the test (b) vertical pressure at the top surface of the specimen were measured as time. Comparing this series of tests with the numerical results comprehensively, the numerical model can be validated.
4.2 Results The height of each finite element was 1mm, which resulted in 20 elements for the test (a) and (b) and in 100 elements for the test (c) and (d). The boundary conditions are as follows. Slide boundaries were applied to all the boundaries. The constant hydraulic pressure and constant temperature were specified to the upper and/or the lower boundary following the test conditions shown in Table 3, while the other boundaries were impermeable and adiabatic. As the initial conditions, 25 °C and 30 °C were applied to the isothermal and the anisothermal tests, respectively. Initial water content was also shown in Table 3. To examine the effect of the function F proposed in this paper, the models (A) to (D) in Table 2 were applied to the simulation of the test (b). In the other tests, only the model (A) was used. Figure 3 shows the comparison of the volumetric water content distribution of the test (a) between the experiment and the simulation. Good agreement gives the validity of the moisture movement in the unsaturated bentonite. Figure 4 shows the time histories of the vertical stress in the test (b) corresponding to each F with the experimental results. The resaturation time of all the simulations was about 150 hours and was almost same as that of the experiment. The figure shows that the model (A) is the most suitable for F compared with the other models, though the initial stress is slightly large. It suggests that swelling pressure rises mainly in the early stages of the infiltration with respect to the compacted bentonite. That is the reason why the model (A) was applied to the other tests.
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Figure 5. Time histories of stress in the test (c) Figure 6. Time history of stress in the test (d) Figure 5 shows the time history of the vertical stress in the test (c). As well as the test (b), good agreement between the experiment and the simulation was obtained. It means that the model proposed is useful for every scale. Figure 6 shows the time histories of the vertical and the horizontal stress with the experimental result for the test (d). Simulated vertical stress agrees with the monitored. On the other hand, the horizontal stress was slightly different. One reason is the initial disturbance of the experiment because of the complicatedness of the apparatus. However, the most influential reason is the anisotropy of the swelling pressure. The direction of compaction of the buffer material plays an important role for the swelling behavior. The further study on the anisotropy has to be continued.
5. CONCLUSIONS This paper proposes a new model for evaluating the swelling pressure during the resaturation process, and introduces it to the coupled thermal, hydraulic and mechanical model. Compared with the series of laboratory experiments, the following conclusions were drawn. Assuming that the swelling pressure is a function of the degree of saturation, good agreement between the simulation and the experiment was obtained. This indicates a oneto-one correspondence between the swelling pressure and the degree of saturation. Swelling pressure is likely to occur in the earlier stages of the resaturation process, and the value is strongly related to the initial degree of saturation.
Although the proposed model cannot strictly simulate the anisotropy of swelling pressure, the results are useful. Since the model involves the Komine's theoretical model for evaluating swelling pressure in saturation, it has the potential to include the effects of chemical change in bentonite and pore water. Using this model, fully coupled thermal, hydraulic, mechanical and chemical analysis can be conducted in the future.
REFERENCES Chijimatsu, M. & Taniguchi, W. 1999. Coupled thermal, hydraulic and mechanical analysis in the near field for geological disposal of highlevel radioactive waste (in Japanese). JNC TN8400 99-014. Chijimatsu, M., Fujita, T., Kobayashi, A. & Nakano, M. 2000. Experiment and validatioin of numerical simulation of coupled thermal, hydraulic and mechanical behaviour in the engineered buffer materials. Int. J. Num. Anal. Mech. Geomech. 24: pp.403-424. Komine, H. & Ogata, N. 1996. Prediction for swelling characteristics of compacted bentonite. Can. Geotech. J. 33: pp. 11-22. Komine, H. & Ogata, N. 2003. New equations for swelling characteristics of bentonite-based buffer materials. Can. Geotech. J. 40(2): pp.460-475 Nakano, M., Amemiya, Y., Fujii, K. & Ishida, A. 1984. Water Penetration and Swelling Pressure of Restrained Unsaturated Clay(in Japanese). JSIDRE112:pp.55-66. Suzuki, H., Chijimatsu, M. & Fujita, T. 1999. Measurement of moisture movement and swelling pressure under temperature gradient (in Japanese). JNC TN8400 99-020.