- Email: [email protected]
conversion processes applied to smectite-to-illite conversion in HLW buffers
Engineering
Geology
47 (1997)
367
378
A technique for modeling transport/conversion processesapplied to smectite-to-illite conversion in HLW buffers Harald Hiikmark a Cluy
a**, Ola Karnland a, Roland Pus& a,b
Technology> AB, IDEON b Lund University
Received 8 January
Research
Center, S-223 70 Lund, Lund, Sweden
Sweden
of Technolog~~,
1996; accepted 28 November
1996
Abstract This paper describes an application of a technique developed for modeling chemical processes in buffer materials that are controlled by a reaction rate and by the transport of one component, which is essential for the process in question to occur. The application described here is the illitization of smectite by fixation of potassium ions in cation exchange positions. and with diffusion of dissolved potassium being the transport process. The technique is verified by comparison with analytical solutions. An overview, based on small models, is given which outlines under what constellations of assumptions the time scale for conversion of the buffer is controlled by reaction rate parameters and under which conditions transport controls this time scale. Examples are given of calculations performed for deposition holes, with potassium being supplied from the surroundings to the upper parts of the highly compacted bentonite buffer. It is concluded that restrictions in nearfield transport capacity have a very significant effect on the conversion time scale. Towards the end of the heating period about 98% of the smectite is found to remain, even for reaction rates and buffer transport conditions that would have left only 10% of the smectite unconverted without nearfield transport restrictions. It is also concluded that the modeling technique can be applied to other, similar, transport/conversion processes. Keywords:
Bentonite;
Buffer; Diffusion:
Illitization:
Potassium
1. Introduction
Mineralogical alterations of clay buffer components may affect the long-term performance of high level waste repositories. Conversion of smectites to non-expandable minerals, for instance, will result in increased hydraulic conductivity of the * Corresponding author. FAX: +46 46 13 42 30; e-mail: clay&lay-tech.se
buffer materials. One such alteration process, discussed in numerous investigations, is conversion of smectite to illite with fixation of potassium ions in cation exchange positions (Pusch and Karnland, 1988; Pusch, 1993; Karnland et al., 1995). In the investigation reported here, a kinetic model for this process, proposed by Huang et al. ( 1993), is used as a basis for analyzing how the time scale for conversion of the highly compacted bentonite buffer surrounding KBS3 HLW canisters depends
0013-7952,197/517.00 Copyright Q 1997 Elsevier Science B.V. All rights reserved PII
SO01 3-7952(
96)00120-2
368
H. Hiikmuk
rt cd. : Engittrrring
on combined effects of S/I conversion and potassium transport conditions. The geotechnical finite difference code FLAC (Itasca Consulting Group, 1993) has been applied to a large number of rock mechanical problems related to high level waste deposition. The versatility of this code, however, also makes it applicable to problems in which no mechanical aspects are included. The modeling technique described here was developed for simulating smectite-to-illite conversion in the bentonite buffers that are intended for isolation of HLW canisters in the Swedish KBS3 repository concept, but it should be possible to modify or extend it for application to other processes. The basic idea is to use the FLAC system with its model generation routines, screen graphics, plotting facilities and explicit timestepping solution procedure as a platform, while the FLAC built-in programming facilities are used for implementing the reaction mechanism that controls the process under study. Diffusive transport can be simulated using either thermal logic or flow logic. In the present study the FLAC thermal logic was employed for simulating diffusion of potassium ions using the FLAC temperature to represent the porewater potassium concentration.
2. Smectite-to-illite
conversion
Grolog~~ 47 ( 1997) 367- 378
If the porewater potassium concentration is constant over time and the initial smectite fraction is S,, Eq. ( 1) has the solution: SE-
SO 1 +S,,[K+]kt
(3)
Eq. (3) applies if dissolution of potassium in the nearfield and potassium transport in the nearfield and within the buffer keep pace with the potassium consumption associated with the mineral conversion. If no potassium is supplied to the buffer, i.e. if a limited quantity given by the buffer porosity and the initial porewater potassium concentration [Kflo is available. Eq. (1) changes to: [K+],-)“P(S,-S) IL‘m
kS2
where pw is the density of water (g 1~ ‘), p the mass fraction of potassium in completely converted smectite, M’ the water ratio and m the mole mass of potassium (g mol - ‘). 2.2. Potussium avdnhilit,v The initial K+ concentration plete conversion is given by: [K+],--z=O
required for com-
(5)
2.1. Kinetic model Huang et al. ( 1993) give the rate of smectitejillite conversion as: - $
=[K+]kS2
(1)
where S is the smectite fraction, k the rate constant (1 mall’ s-l ) and [KC] the porewater potassium concentration (mol 1- ‘). The rate constant k is given by: (2) where A is the frequency factor (1 mol -i s-l), Ea the activation energy (cal mol -‘), R the gas constant (cal mol - ’ K- ‘) and T the temperature ( K).
The mass fraction parameter p should be about 0.05 (Karnland et al., 1995). For highly compacted and water saturated bentonite, M, is about 0.26. The initial porewater concentration required for complete conversion would then be about 5 mol 1~ ‘, a value that exceeds the concentration in normal rock porewater and in sea water by several orders of magnitude. The quantities of dissolved potassium contained in the rock porewater that are supplied to the buffer during water saturation will thus suffice to convert only insignificant fractions of the smectite in highly compacted 100% bentonite. This means that the availability of potassium from external sources is necessary if smectite-to-illite conversion is to take place on any significant scale in the buffers surrounding KBS3 HLW canisters. Restrictions in
H. Hijkmark
et 01. :’ Enginc~ering
potassium transport capacity, within the buffer as well as between buffer and potassium source, are consequently important for the conversion time scale. In this study, transport is assumed to take place by diffusion only. For bentonite/ballast mixtures in tunnel backfills. ballast particles containing potassium-bearing minerals may serve as internal sources and promote conversion in the bentonite component. Tunnel backfills without bentonite component, or with completely or partly converted bentonite components, may serve as external potassium sources for canister buffers in the deposition holes. The amounts of potassium that can be supplied from the backfill to the highly compacted bentonite buffer in the deposition holes are determined by chemical weathering rates which depend on a number of factors, e.g. mineral composition and specific area of the backfill ballast. The possible production of dissolved potassium in the tunnel backfill has been shown to be considerable (Karnland et al., 1995). In the numerical study reported here, it is assumed that the backfill and the surrounding rock have the potential of supplying the potassium quantities that would be needed for complete conversion of the deposition hole buffers.
3. Modeling method All simulations presented here were performed using version 3.22 of the finite difference code FLAC (Fast Lagrangian Analysis of Continua, Itasca Consulting Group, 1993 ). FLAC is a recognized tool for analyzing rock mechanical and soil mechanical problems, and contains logic for simulation of fluid flow and heat diffusion. The flow logic and the thermal logic can be operated fully coupled to the mechanical model or independently of the mechanical model. In the latter case. mechanical properties, i.e. values for elastic parameters and strength parameters, need not be specified. The problem domain, which can be two-dimensional or axisymmetric, is discretized into a mesh of quadrilateral zones with gridpoints in the zone corners. An explicit timestepping solution pro-
Geology
47 (1997)
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369
cedure is employed that updates all zone variables and all gridpoint variables throughout the model every timestep. The duration of one timestep is automatically set to a value that is small enough to ensure numerical stability, and is determined by zone sizes and by values of parameters that control the velocity at which disturbances are propagated, i.e. elastic moduli (mechanical models), heat diffusivity (thermal models) and conductivity (flow models). FLAC contains a built-in programming language, FISH, that enhances the power and the versatility of FLAC considerably. User-written FISH routines can be called from the FLAC input file, either explicitly at specified instances in the command sequence, or automatically every timestep. FISH routines can be used to define new scalar variables or gridpoint variables and to assign values to them. Also, values of predefined standard FLAC gridpoint variables can be tested and modified. In the present study. smectite-to-illite conversion was implemented by use of this programming language. Two FISH functions were defined: one for initializing the problem and one containing an algorithm for managing the kinetic model. Schematic descriptions are given below. Initidixtion ( 1) Define scalar variables for parameters included in Eqs. (2) and (4), i.e. for E,, T, R, ~1, n?, SO, pW and p. (2) Define a new extra. gridpoint smectite fraction variable, S(ij). (3) Assign constant values to the scalar variables and initial values to S(i,j). Conversion ulgorithnl For each gridpoint ij: ( 1) Test the value of current local K- concentration (i.e. FLAC temperature) and the value of the FLAC timestep, and calculate the reduction in smectite fraction AS during the timestep by direct use of Eq. (1). (2) Calculate the reduction of porewater K+ concentration AC associated with the smectite fraction reduction AS by using the values of p, S,. pW and M.
(3) Update values for gridpoint variables representing porewater K+ concentration and smectite fraction, i.e. FLAC temperature and S(ij). The coupled process of potassium diffusion and simultaneously progressing S/I conversion was simulated by calling the initialization function and then putting the thermal logic in operation, i.e. starting the timestepping solution procedure, with calls to the conversion algorithm function every timestep. Parameter values for the thermal material properties had to be set with regard to the intended effective potassium diffusivity, taking into account the effects of porosity. Since the thermal logic was engaged for simulating potassium diffusion, no heat conduction calculations could be made, meaning that the real temperature of the system had to be represented by an additional, specifically defined variable. In the simulations performed in this study, a global temperature was assumed, but there are no principal difficulties in defining a gridpoint temperature variable and to prescribe any variation in time and space for it. The default timestep, which is automatically determined and set by the FLAC thermal logic, ensures numerical stability and sufficient accuracy in heat conduction calculations. This timestep may, however. be too large for the conversion algorithm to perform properly, and a lower value may have to be prescribed in the FLAC input file. If this is necessary or not depends on the conversion rate, i.e. if AS and AC take on large values. No attempts were made in this investigation to automate the determination of maximum timestep duration or to optimize the calculations otherwise with respect to runtime. Instead, the maximum timestep for different sets of input data was determined empirically by testing cases for which analytical solutions are available.
successively updates values of gridpoint variables representing smectite fraction (FISH variable) and potassium concentration (FLAC temperature). If no transport, i.e. heat conduction, is allowed, then one-zone models are sufficient. The calculated smectite fraction should change with time according to Eq. (4) and verification can be made by direct comparison with solutions to that equation. (2) By verifying the fully coupled system, i.e. the interlacing of conversion algorithm and potassium transport (FLAC heat diffusion). Verification is made by comparison with solutions obtained using closed-form solutions for moving S/I interfaces. 4.2. One-zone models The discrete plot symbols in Fig. 1 were obtained numerically by applying a RungeeKutta method to Eq. (4). The plot lines show corresponding FLAC results obtained in one-zone models using the FISH procedure described above. The following parameter values were used: w=0.26, p,=1ooog1-1, nz=39 gmoll’, [K+],==O.Ol mall-‘. S,= 1, T=423 K, i.e. 150°C. A number of different assumptions are made regarding p, the parameter representing the potassium mass fraction of the solid material that is required for complete conversion. For highly compacted 100% bentonite this parameter should be approximately 0.05 (Karnland et al.: 1995). Much lower values were used in the verification runs. Two different values for the activation energy E, (i.e. different rate constants) were tried. The conversion algorithm appears to perform very accurately: between values calculated by use of the Runge-Kutta method and the FLAC results agree within about 0.1%. 4.3. MO ving S/I in trrfaces
4. Verification of the simulation procedure 4.1. General The procedure described above is verified below in two steps: (1) By verifying the algorithm that tests and
If the potassium diffusion is slow and smectite conversion by potassium fixation is fast, the diffusing ions will be captured almost instantaneously and become irreversibly immobilized as soon as regions of incomplete conversion are encountered. This means that a distinct I/S interface with zero porewater potassium concentration will form, and
H. HcJkmark
et al. / Enginrering
Smectite
content
Geology
47 (1997)
vs time
R~ngeXutta, FLAC,
p=O.OOl,
20
Fig. I, FLAC
compared
that diffusion will take place only in the converted region between the outer boundary and that interface. The interface moves from the concentration boundary at the same rate at which conversion takes place in the interface region. If the amount of potassium per unit volume required for complete conversion is not too small, a steady state approximation of the concentration distribution can be used to calculate the position of the I/S interface as a function of time (Crank, 1975, p. 3 10). Fig. 2 illustrates schematically diffusion and conversion in systems with moving I/S interfaces that separate regions of converted material ( I) from unconverted regions (S). The upper part represents one-dimensional diffusion (linear steady state concentration distribution) and the lower part diffusion in an axisymmetric geometry. In the axisymmetric system the steady state concentration distribution is given by: C(r) =
C Wl4t)l W,lr(~)l
where C(r) is the concentration
p=0.00001,
40 60 Time, years results
(6) at any position
r
with
Ea=23000
p=O.CGUOl , Ea=23000
Rung-s-Kutta,
0
371
367-378
80
solutions
Ea=23000
100
of Eq. (4).
between the outer boundary at R, (concentration C) and the S/I interface at r(t) (concentration 0). Eqs. (7) and (8) below apply for the position of the S/I interface in the one-dimensional and axisymmetric cases, respectively (Hiikmark, 1995 ): I(t)=
J
ZD,mCt ~
(7)
PdP
r(I)2(ln$+~J-$+~=0
(8)
where I(t) is the diffusion length, pd the dry density, D, the effective potassium diffusivity, ~1 the potassium mole mass and 17 the potassium mass fraction in completely converted material. The other symbols are in iaccordance with Fig. 2. Eq. (8) gives the r(t) vs. t relation in implicit form. A numerical equation solving method, e.g. NewtonRaphson, has to be employed to calculate r(t) for a given time t. Fig. 3 shows concentration profiles, calculated using Eqs. (6) and (8) compared with corresponding
372
H. H(ikmark
Pore water concentration
et al. / Enginrrring
potassium
Geology
47 (1997)
c
367-378
Pore water concentration
, /,I
potassium , ,.- _’
,’
c _ ‘.
0
0
[
Pore water concentration
potassium,
Pore water concentration
C ;’ symmetry
j Symmdtyaxis i/
axis
potassium, , , ’’’ , _ ‘.
c
/ 0
r(U)
ro
0
!
Ro
ro
Ro
r(Q)
Fig. 2. Schematic illustration of the one-dimensional system (upper) and axisymmetric system (lower) with quasi-steady-state diffusion between the outer boundary with constant concentration C and the moving, zero concentration, 1;s interface. The positions of the interface at times I, and t, are denoted by the diffusion length I(t) and distance to symmetry axis r(t). The potassium concentration between the outer boundary and the I/S interface is indicated by dashed lines.
1 E-2
9E-3 i 5 8E-3 E g- 7E-3 g
8E-3
8 5 : g! $ I!! g +
5E-3
*
Della10
m2k
4E-3 3E-3 ZE-3 lE-3 OE+O 0.44
Fig. 3. Concentration
profiles
0.49
0.54
0.59 0.84 0.69 Radial Distance, m
for the axisymmetric
FLAC results obtained from the axisymmetric model with reaction rate parameters set to give fast conversion (cf. Fig. 2, lower). Fig. 4 shows corresponding
model.
Lines:
analytical,
0.74
discrete
0.79
symbols:
FLAC
results
smectite fraction profiles. The vertical lines show interface positions calculated using Eq. (8). The FLAC run appears to reproduce the propagation of
H. Hiikmurk
et al. / Enginrering
Geology
47 ( 1997)
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367-378
1.00 0.90
0.80 0' 0.70
: De=%lOm24 [K+]=O.Ol
moWL
j
'c 0.60 E t 0.50 .g y 0.40 E
v, 0.30
0.44
0.49
0.54
0.59 Radial
Fig. 4. Smectite fraction profiles obtained of the S/I interface according to Eq. (8)
in a verification
run
0.64
Distance,
0.69
0.74
0.79
m
for the axisymmetric
model.
Vertical
dotted
lines denote
the position
the S/I interface as well as the shape of the concentration profiles very satisfactorily.
Conditions regarding diffusivity, potassium concentration and rate constant parameters were varied between the different runs.
5. Results
5.2. Smcrll models
Fig. 5 shows a schematic view of a KBS3 deposition hole and a close-up of a section between a waste canister and the borehole wall. Two types of models. both axisymmetric, are analyzed: ( 1) small models (close-up view), and (2) full models of an 8 m deep deposition hole. The small models were used for analyzing the fundamental features of the transport/conversion process in the buffer, without regarding restrictions in transport capacity between potassium source and buffer boundary. The full models include assumptions regarding this capacity. The following data, which correspond to compacted bentonite with a water saturated density of 2000 kg mm3, were assumed for all models: water ratio w = 0.26, porosity II= 0.41, dry density p,=1590kgmP3, and potassium mass fraction of solid, completely converted material, p = 0.05.
The following cases were analyzed: CUSL’1: Constant boundary Kf concentration at the borehole periphery and constant rate constant. i.e. constant temperature. Cuse 2: The K+ concentration at the cylindrical boundary decreases as potassium is transported into the buffer. The temperature is kept constant. Cuse 3: The K’ concentration at the cylindrical boundary is constant. The conversion rate (i.e. temperature) is reduced with time. Fig. 6 shows the results of a large number of runs performed assuming case 1 conditions. The horizontal axis (effective diffusivity times boundary concentration) represents transport, the vertical axis (rate constant times boundary concentration) represents conversion. In the lower left corner both transport and conversion are slow, giving long buffer lifetimes. In the upper right corner both transport and conversion are fast, giving short buffer lifetimes. In the upper left corner conversion
374
H. Hiikmurk
et al. / Engineering
H Potassium
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47 (1997)
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Source I
lo/so SetltlBallast
r,=O.44
m
R"=o.7g ml
8m HCB
A
(-[K+]
q
Diffusion
0.35
m
W
Waste Canister
JI
I Trammissii
Fig. 5. Left: KBS3 deposition hole in the floor showing the geometry of the small model.
Wall Region
of the backfilled
deposition
is fast, but transport is slow, which means that the buffer lifetime is controlled mainly by restrictions in transport capacity. In the lower right corner transport is fast, and the buffer lifetime is controlled by the rate constant, i.e. by temperature. activation energy and frequency factor according to Eq. (3). The vertical dotted straight lines in Fig. 6 represent [K’]D, values that would leave a certain percentage of the 0.35 m buffer unconverted after 5000 years if conversion occurs instantaneously, i.e. independently of temperature, activation energy and frequency factor. These [K-ID, values are calculated using Eq. (8). For example, the interface position corresponding to 50% remaining smectite is r(t)=0.639 m (for the specific radial geometry assumed here). Putting this figure for r(t) and 5000 years (in seconds) for t in Eq. (8) and then solving for CD, gives the value 1.374E-13 (mol l-‘)(m’s-‘). The horizontal dotted straight lines in Fig. 6
tunnel.
Right:
conceptual
Asymmetric
model
with close-up
represent values for [K+]k=[K+]A
exp(-EJRT)
that would leave a certain percentage of the buffer unconverted after 5000 years without transport capacity restrictions. These values are obtained by direct use of Eq. (3). The discrete plot symbols in Fig. 6 represent FLAC results. The labels denote calculated smectite fractions in percent after 5000 years. The solid curves connecting dotted vertical lines to corresponding horizontal lines are percentage isolines based on the FLAC results, i.e. on the discrete plot symbols. The filled plot symbols in Fig. 6 represent runs with a boundary porewater potassium concentration of 0.01 mol 1-l, while unfilled symbols represent concentrations ranging between 0.001 and 0.01 mol 1 - ‘. Note that filled and unfilled symbols fit equally well in the isoline system. Figs. 7 and 8 show smectite fraction profiles
H. Hijkmurk
et al. ; Engineering
[K+]*De,
iE-14
I “““I 75%
1 E-08
Transporl
47 (19973
367-378
lE-11
lE-12
I
lE-10
.’ .’ /
‘1’1’“”
50%
375
(mole/L)‘(m2/s)
IE-13
1 E-07
Geoiog~
25% IQ%
controlled
Smectite fractions after 5000 years
lE-11
lE-12 Fig. 6. Smectite fractions after 5000 years, assuming boundary concentration and rate constant to be fixed. Filled symbols represent simulations performed with the boundary concentration set at 0.01 mol I -‘. Unfilled symbols represent boundary concentrations ranging between 0.001 and 0.01 mol I -‘. The region above the dashed line represents constellations of reaction/transport data that would mean a transport-controlled time scale for conversion of the buffer. Points A and B denote constellations of reaction/transport data selected for additional simulations (see text). lE+O 9E-1 8E-1 -+-
1 x Required amount of K+ availaWe, 500 yeere
-A-
10 x Required amount of K+ available, 500 yeere
--W-
Conetent boundary
7E-1 E 3
6E-1
ii Q)
5E-1
&
1 x Required
4E-1
-A--
10 x Required amount of K+ available, 2000 years
-o-
conetant boundarycollcentratii, 2ooo years
C t;
g
concentration,
500 years
amount of K+ available, 2000 yesre
E
u,
3E-1
1
2E-1
0.54
0.59
0.64
Radial Distance, Fig. 7. Effects of limitations in potassium availability. conditions corresponding to data point A (Fig. 6).
Smectite
fraction
0.69
0.74
0.79
obtained
in the axisymmetric
m
profiles
model
with
initial
H. HGkmark
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lE+O
et al. / Engineering
Geology
47 (1997)
367-378
,
QE-1 8E-1 7E-1 5
‘g 6E-1 E : 5E-1 .E ti Q) 4E-1 5
-k-
10 x Required amount of K+ available, 500 yeam
+
Constant
-u-
1 x Required
-A-
10 x Required amount of K+ available, 2000 years
boundary oonwntrati0n,500 years amount of K+ available, 2000 years
3E-1 2E-1 lE-1 OE+O 0.44
0.54
0.59
Radial Fig. 8. Effects of limitations in potassium availability. conditions corresponding to data point B (Fig. 6).
Smectite
obtained for runs performed assuming case 2 conditions, i.e. with decreasing concentration at the buffer boundary. The effects of a well-stirred tank surrounding the buffer were obtained by appending a small number of zones outside the buffer boundary, and by assigning FLAC thermal properties representing a specified potassium content and a specified initial potassium concentration to them. Two sets of initial conditions were assumed, represented by the data points labeled A and B in the transport/conversion plot in Fig. 6. Both sets correspond to fast conversion and fast diffusion, with point A in the transport controlled part. Two assumptions were made regarding the well-stirred tank: (1) It contains exactly the quantity of potassium that is necessary for complete conversion. (2) It contains 10 times the necessary quantity. Corresponding results obtained assuming constant boundary concentration are shown along with the well-stirred tank results. Fig. 9 shows smectite fraction profiles obtained assuming case 3 conditions. The initial conditions correspond to data point B in the transport/ conversion plot (Fig. 6). The temperature was assumed to be constant at 423 K during the first
0.64
Distance, fraction
profiles
0.69
0.74
0.79
obtained
in the axisymmetric
m model
with
initial
200 years after deposition and then drop linearly 323 K within the next 1800 years. to Corresponding results for the constant temperature system are shown for comparison. 5.3. Full models Results from one simulation are presented below. The initial conditions for the buffer and for the boundary concentration correspond to data point B in the transport/conversion plot (Fig. 6). i.e. A=80 800 1 (mol s))r, E,=24 000 cal mol -l, T=423 K, [K+]=O.Ol mol l-l, D,=2E10 mz s-l and p=O.O5. The temperature was assumed to be constant during the 2000 years that were simulated, i.e. no effects of decreasing conversion rate were accounted for. Instead of having a constant porewater K+ concentration at the radial boundary of the buffer volume (as for the small models in the previous section) the concentration boundary was located at the tunnel floor 1 m above the upper part of the buffer. This means that transport of potassium from the tunnel to the buffer is an essential process in these models. The potassium transport was
377
9E-1
8E-1 g
7E-1
3 6E-1 g a 5E-1 $ 8 4E-1 5f
--u-
Decreashg
temp,
-4- constant temp.ZOCGyears -U-Decreasing temp,500 yealls
3E-I 2E-1 IE-1 OE+O i 0.44
0.64
0.54
Radial Distance,
m
Fjg. 9. Smectite fraction profiles obtained in the axisymmetric model assuming temperature to be constant at 423 K during the first 200 “ears after depositionand then drop to 313 K within the next 1800 years. Filled symbols show corresponding results for constant temperature. The initial conditions correspond to point B in Fig. 6
assumed to take place by diffusion through the 1 m thick lo/90 bentonite/bailast mixture between HCB buffer and tunnel floor, and through an assumed transmissive, 0.05 m wide, annulus around the borehole periphery. The following additional conditions were assumed: ( 1) Potassium fixation in the IO/90 mixture was not accounted for, i.e. potassium was assumed to diffuse without delay through this 1 m barrier. (2) The diffusivity of the lo/90 mixture and the diffusivity of the 0.05 m wide annulus around the set to were both periphery borehole 5E-9 m’s-r. Figs, 10 and 11 show smectite fraction contours and porewater potassium contours, respectively. It is evident that restrictions in nearfield transport capacity have a very significant effect on the buffer lifetime, About 98% of the buffer remains after 2000 years despite conservative assumptions, while for corresponding initial conditions, less than 10% was found to remain in the case of constant concentration on the buffer boundary (Fig. 6).
mawit
/
btessium TUNNEL
M FL-
6. Discussion
It has been demonstrated that the system used in this investigation for simulating the coupled
Fig. 10. Smectite fraction contours after 2000 years
H. Hiikrnark
rr al. ; Enginrrring
Potassium 15E-9
47 ( 1997) 367-378
Predictive and more detailed analyses would require that the change in temperature distribution over time be included. and that an additional gridpoint variable, representing smectite fraction dependent potassium diffusivity, be defined. These modifications could easily be added to the system. In principle, it is also possible to simulate the dissolution of potassium in the backfill, provided that the dissolution rate can be approximated with a function of known quantities (e.g., temperature, specific area of ballast, porewater potassium concentration) for which FLAC variables exist or can be defined by use of FISH functions.
O.O’O 1 :
Geology
diisivity
m2k
0.008 0.004 0.002
Acknowledgment
Fig. I I. Porewater
K+
concentration
contours
after
2000 years.
process of potassium transport and illitization gives results that are logical and consistent with the input assumptions. For cases that could be checked with analytical solutions, the results were found to be correct and accurate. Since no really certain data exist regarding conversion rate parameters and potassium concentrations in the tunnel backfill, the calculations have been performed as a series of unprejudiced numerical experiments with parameter values selected only within ranges in which both transport restrictions and reaction rate control the buffer lifetime. The parameter values proposed by Huang et al. (1993) (I&=28 000 cal mol-‘), for instance, would give a system in which reasonable restrictions in transport capacity would have a very small effect on the time scale for conversion of KBS3 buffers, and have not been included for this purely formal reason. There is, however, no reason to doubt that the simulation system presented here would be capable of describing the conversion process at any level of detail with good accuracy for any set of parameter values.
The authors wish to acknowledge that this paper is a result of work funded by the Swedish Nuclear Fuel and Waste Management Co. (SKB). The views expressed in this paper are those of the authors and do not necessarily coincide with those of the SKB.
References Crank, J., 1975. The Mathematics of Diffusion. Oxford University Press, Oxford. Huang, W.-L.. Longo, J. and Pevear. I)., 1993. An experimentally derived kinetic model for smectite to illile conversion and its use as a geothermometer. Clays Clay Miner.. 41: 162Zl77. Hiikmark, H., 1995. Smectite-to-illite conversion in bentonite butlers. Application of a technique for modeling degradation processes. SKB Progress Report .4R 95-07. SKB, Stockhohn, Sweden. Itasca Consulting Group. 1993. FLAC. Fast Lagrangian Analysis of Continua. Version 3.22. Users Manual. ltasca Consulting Group, Inc., Minneapolis. USA. Karnland, O., Warvfinge, P. and Pus&. R.. 1995. Smectite-toillite conversion models. Factors of importance for KBS3 conditions. SKB Progress Report AR 95-27. SKB, Stockholm. Sweden. Pusch, R. and Karnland, 0.. 1988. Hydrothermal etTects on montmorillonite. SKB Technical Report TR 88-15. SKB, Stockholm, Sweden. Pusch, R.. 1993. Evaluation of models for conversion of smectites to non-expandable minerals. SKB Technical Report TR 93-33. SKB, Stockholm, Sweden.
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47 (1997)
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378
A technique for modeling transport/conversion processesapplied to smectite-to-illite conversion in HLW buffers Harald Hiikmark a Cluy
a**, Ola Karnland a, Roland Pus& a,b
Technology> AB, IDEON b Lund University
Received 8 January
Research
Center, S-223 70 Lund, Lund, Sweden
Sweden
of Technolog~~,
1996; accepted 28 November
1996
Abstract This paper describes an application of a technique developed for modeling chemical processes in buffer materials that are controlled by a reaction rate and by the transport of one component, which is essential for the process in question to occur. The application described here is the illitization of smectite by fixation of potassium ions in cation exchange positions. and with diffusion of dissolved potassium being the transport process. The technique is verified by comparison with analytical solutions. An overview, based on small models, is given which outlines under what constellations of assumptions the time scale for conversion of the buffer is controlled by reaction rate parameters and under which conditions transport controls this time scale. Examples are given of calculations performed for deposition holes, with potassium being supplied from the surroundings to the upper parts of the highly compacted bentonite buffer. It is concluded that restrictions in nearfield transport capacity have a very significant effect on the conversion time scale. Towards the end of the heating period about 98% of the smectite is found to remain, even for reaction rates and buffer transport conditions that would have left only 10% of the smectite unconverted without nearfield transport restrictions. It is also concluded that the modeling technique can be applied to other, similar, transport/conversion processes. Keywords:
Bentonite;
Buffer; Diffusion:
Illitization:
Potassium
1. Introduction
Mineralogical alterations of clay buffer components may affect the long-term performance of high level waste repositories. Conversion of smectites to non-expandable minerals, for instance, will result in increased hydraulic conductivity of the * Corresponding author. FAX: +46 46 13 42 30; e-mail: clay&lay-tech.se
buffer materials. One such alteration process, discussed in numerous investigations, is conversion of smectite to illite with fixation of potassium ions in cation exchange positions (Pusch and Karnland, 1988; Pusch, 1993; Karnland et al., 1995). In the investigation reported here, a kinetic model for this process, proposed by Huang et al. ( 1993), is used as a basis for analyzing how the time scale for conversion of the highly compacted bentonite buffer surrounding KBS3 HLW canisters depends
0013-7952,197/517.00 Copyright Q 1997 Elsevier Science B.V. All rights reserved PII
SO01 3-7952(
96)00120-2
368
H. Hiikmuk
rt cd. : Engittrrring
on combined effects of S/I conversion and potassium transport conditions. The geotechnical finite difference code FLAC (Itasca Consulting Group, 1993) has been applied to a large number of rock mechanical problems related to high level waste deposition. The versatility of this code, however, also makes it applicable to problems in which no mechanical aspects are included. The modeling technique described here was developed for simulating smectite-to-illite conversion in the bentonite buffers that are intended for isolation of HLW canisters in the Swedish KBS3 repository concept, but it should be possible to modify or extend it for application to other processes. The basic idea is to use the FLAC system with its model generation routines, screen graphics, plotting facilities and explicit timestepping solution procedure as a platform, while the FLAC built-in programming facilities are used for implementing the reaction mechanism that controls the process under study. Diffusive transport can be simulated using either thermal logic or flow logic. In the present study the FLAC thermal logic was employed for simulating diffusion of potassium ions using the FLAC temperature to represent the porewater potassium concentration.
2. Smectite-to-illite
conversion
Grolog~~ 47 ( 1997) 367- 378
If the porewater potassium concentration is constant over time and the initial smectite fraction is S,, Eq. ( 1) has the solution: SE-
SO 1 +S,,[K+]kt
(3)
Eq. (3) applies if dissolution of potassium in the nearfield and potassium transport in the nearfield and within the buffer keep pace with the potassium consumption associated with the mineral conversion. If no potassium is supplied to the buffer, i.e. if a limited quantity given by the buffer porosity and the initial porewater potassium concentration [Kflo is available. Eq. (1) changes to: [K+],-)“P(S,-S) IL‘m
kS2
where pw is the density of water (g 1~ ‘), p the mass fraction of potassium in completely converted smectite, M’ the water ratio and m the mole mass of potassium (g mol - ‘). 2.2. Potussium avdnhilit,v The initial K+ concentration plete conversion is given by: [K+],--z=O
required for com-
(5)
2.1. Kinetic model Huang et al. ( 1993) give the rate of smectitejillite conversion as: - $
=[K+]kS2
(1)
where S is the smectite fraction, k the rate constant (1 mall’ s-l ) and [KC] the porewater potassium concentration (mol 1- ‘). The rate constant k is given by: (2) where A is the frequency factor (1 mol -i s-l), Ea the activation energy (cal mol -‘), R the gas constant (cal mol - ’ K- ‘) and T the temperature ( K).
The mass fraction parameter p should be about 0.05 (Karnland et al., 1995). For highly compacted and water saturated bentonite, M, is about 0.26. The initial porewater concentration required for complete conversion would then be about 5 mol 1~ ‘, a value that exceeds the concentration in normal rock porewater and in sea water by several orders of magnitude. The quantities of dissolved potassium contained in the rock porewater that are supplied to the buffer during water saturation will thus suffice to convert only insignificant fractions of the smectite in highly compacted 100% bentonite. This means that the availability of potassium from external sources is necessary if smectite-to-illite conversion is to take place on any significant scale in the buffers surrounding KBS3 HLW canisters. Restrictions in
H. Hijkmark
et 01. :’ Enginc~ering
potassium transport capacity, within the buffer as well as between buffer and potassium source, are consequently important for the conversion time scale. In this study, transport is assumed to take place by diffusion only. For bentonite/ballast mixtures in tunnel backfills. ballast particles containing potassium-bearing minerals may serve as internal sources and promote conversion in the bentonite component. Tunnel backfills without bentonite component, or with completely or partly converted bentonite components, may serve as external potassium sources for canister buffers in the deposition holes. The amounts of potassium that can be supplied from the backfill to the highly compacted bentonite buffer in the deposition holes are determined by chemical weathering rates which depend on a number of factors, e.g. mineral composition and specific area of the backfill ballast. The possible production of dissolved potassium in the tunnel backfill has been shown to be considerable (Karnland et al., 1995). In the numerical study reported here, it is assumed that the backfill and the surrounding rock have the potential of supplying the potassium quantities that would be needed for complete conversion of the deposition hole buffers.
3. Modeling method All simulations presented here were performed using version 3.22 of the finite difference code FLAC (Fast Lagrangian Analysis of Continua, Itasca Consulting Group, 1993 ). FLAC is a recognized tool for analyzing rock mechanical and soil mechanical problems, and contains logic for simulation of fluid flow and heat diffusion. The flow logic and the thermal logic can be operated fully coupled to the mechanical model or independently of the mechanical model. In the latter case. mechanical properties, i.e. values for elastic parameters and strength parameters, need not be specified. The problem domain, which can be two-dimensional or axisymmetric, is discretized into a mesh of quadrilateral zones with gridpoints in the zone corners. An explicit timestepping solution pro-
Geology
47 (1997)
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369
cedure is employed that updates all zone variables and all gridpoint variables throughout the model every timestep. The duration of one timestep is automatically set to a value that is small enough to ensure numerical stability, and is determined by zone sizes and by values of parameters that control the velocity at which disturbances are propagated, i.e. elastic moduli (mechanical models), heat diffusivity (thermal models) and conductivity (flow models). FLAC contains a built-in programming language, FISH, that enhances the power and the versatility of FLAC considerably. User-written FISH routines can be called from the FLAC input file, either explicitly at specified instances in the command sequence, or automatically every timestep. FISH routines can be used to define new scalar variables or gridpoint variables and to assign values to them. Also, values of predefined standard FLAC gridpoint variables can be tested and modified. In the present study. smectite-to-illite conversion was implemented by use of this programming language. Two FISH functions were defined: one for initializing the problem and one containing an algorithm for managing the kinetic model. Schematic descriptions are given below. Initidixtion ( 1) Define scalar variables for parameters included in Eqs. (2) and (4), i.e. for E,, T, R, ~1, n?, SO, pW and p. (2) Define a new extra. gridpoint smectite fraction variable, S(ij). (3) Assign constant values to the scalar variables and initial values to S(i,j). Conversion ulgorithnl For each gridpoint ij: ( 1) Test the value of current local K- concentration (i.e. FLAC temperature) and the value of the FLAC timestep, and calculate the reduction in smectite fraction AS during the timestep by direct use of Eq. (1). (2) Calculate the reduction of porewater K+ concentration AC associated with the smectite fraction reduction AS by using the values of p, S,. pW and M.
(3) Update values for gridpoint variables representing porewater K+ concentration and smectite fraction, i.e. FLAC temperature and S(ij). The coupled process of potassium diffusion and simultaneously progressing S/I conversion was simulated by calling the initialization function and then putting the thermal logic in operation, i.e. starting the timestepping solution procedure, with calls to the conversion algorithm function every timestep. Parameter values for the thermal material properties had to be set with regard to the intended effective potassium diffusivity, taking into account the effects of porosity. Since the thermal logic was engaged for simulating potassium diffusion, no heat conduction calculations could be made, meaning that the real temperature of the system had to be represented by an additional, specifically defined variable. In the simulations performed in this study, a global temperature was assumed, but there are no principal difficulties in defining a gridpoint temperature variable and to prescribe any variation in time and space for it. The default timestep, which is automatically determined and set by the FLAC thermal logic, ensures numerical stability and sufficient accuracy in heat conduction calculations. This timestep may, however. be too large for the conversion algorithm to perform properly, and a lower value may have to be prescribed in the FLAC input file. If this is necessary or not depends on the conversion rate, i.e. if AS and AC take on large values. No attempts were made in this investigation to automate the determination of maximum timestep duration or to optimize the calculations otherwise with respect to runtime. Instead, the maximum timestep for different sets of input data was determined empirically by testing cases for which analytical solutions are available.
successively updates values of gridpoint variables representing smectite fraction (FISH variable) and potassium concentration (FLAC temperature). If no transport, i.e. heat conduction, is allowed, then one-zone models are sufficient. The calculated smectite fraction should change with time according to Eq. (4) and verification can be made by direct comparison with solutions to that equation. (2) By verifying the fully coupled system, i.e. the interlacing of conversion algorithm and potassium transport (FLAC heat diffusion). Verification is made by comparison with solutions obtained using closed-form solutions for moving S/I interfaces. 4.2. One-zone models The discrete plot symbols in Fig. 1 were obtained numerically by applying a RungeeKutta method to Eq. (4). The plot lines show corresponding FLAC results obtained in one-zone models using the FISH procedure described above. The following parameter values were used: w=0.26, p,=1ooog1-1, nz=39 gmoll’, [K+],==O.Ol mall-‘. S,= 1, T=423 K, i.e. 150°C. A number of different assumptions are made regarding p, the parameter representing the potassium mass fraction of the solid material that is required for complete conversion. For highly compacted 100% bentonite this parameter should be approximately 0.05 (Karnland et al.: 1995). Much lower values were used in the verification runs. Two different values for the activation energy E, (i.e. different rate constants) were tried. The conversion algorithm appears to perform very accurately: between values calculated by use of the Runge-Kutta method and the FLAC results agree within about 0.1%. 4.3. MO ving S/I in trrfaces
4. Verification of the simulation procedure 4.1. General The procedure described above is verified below in two steps: (1) By verifying the algorithm that tests and
If the potassium diffusion is slow and smectite conversion by potassium fixation is fast, the diffusing ions will be captured almost instantaneously and become irreversibly immobilized as soon as regions of incomplete conversion are encountered. This means that a distinct I/S interface with zero porewater potassium concentration will form, and
H. HcJkmark
et al. / Enginrering
Smectite
content
Geology
47 (1997)
vs time
R~ngeXutta, FLAC,
p=O.OOl,
20
Fig. I, FLAC
compared
that diffusion will take place only in the converted region between the outer boundary and that interface. The interface moves from the concentration boundary at the same rate at which conversion takes place in the interface region. If the amount of potassium per unit volume required for complete conversion is not too small, a steady state approximation of the concentration distribution can be used to calculate the position of the I/S interface as a function of time (Crank, 1975, p. 3 10). Fig. 2 illustrates schematically diffusion and conversion in systems with moving I/S interfaces that separate regions of converted material ( I) from unconverted regions (S). The upper part represents one-dimensional diffusion (linear steady state concentration distribution) and the lower part diffusion in an axisymmetric geometry. In the axisymmetric system the steady state concentration distribution is given by: C(r) =
C Wl4t)l W,lr(~)l
where C(r) is the concentration
p=0.00001,
40 60 Time, years results
(6) at any position
r
with
Ea=23000
p=O.CGUOl , Ea=23000
Rung-s-Kutta,
0
371
367-378
80
solutions
Ea=23000
100
of Eq. (4).
between the outer boundary at R, (concentration C) and the S/I interface at r(t) (concentration 0). Eqs. (7) and (8) below apply for the position of the S/I interface in the one-dimensional and axisymmetric cases, respectively (Hiikmark, 1995 ): I(t)=
J
ZD,mCt ~
(7)
PdP
r(I)2(ln$+~J-$+~=0
(8)
where I(t) is the diffusion length, pd the dry density, D, the effective potassium diffusivity, ~1 the potassium mole mass and 17 the potassium mass fraction in completely converted material. The other symbols are in iaccordance with Fig. 2. Eq. (8) gives the r(t) vs. t relation in implicit form. A numerical equation solving method, e.g. NewtonRaphson, has to be employed to calculate r(t) for a given time t. Fig. 3 shows concentration profiles, calculated using Eqs. (6) and (8) compared with corresponding
372
H. H(ikmark
Pore water concentration
et al. / Enginrrring
potassium
Geology
47 (1997)
c
367-378
Pore water concentration
, /,I
potassium , ,.- _’
,’
c _ ‘.
0
0
[
Pore water concentration
potassium,
Pore water concentration
C ;’ symmetry
j Symmdtyaxis i/
axis
potassium, , , ’’’ , _ ‘.
c
/ 0
r(U)
ro
0
!
Ro
ro
Ro
r(Q)
Fig. 2. Schematic illustration of the one-dimensional system (upper) and axisymmetric system (lower) with quasi-steady-state diffusion between the outer boundary with constant concentration C and the moving, zero concentration, 1;s interface. The positions of the interface at times I, and t, are denoted by the diffusion length I(t) and distance to symmetry axis r(t). The potassium concentration between the outer boundary and the I/S interface is indicated by dashed lines.
1 E-2
9E-3 i 5 8E-3 E g- 7E-3 g
8E-3
8 5 : g! $ I!! g +
5E-3
*
Della10
m2k
4E-3 3E-3 ZE-3 lE-3 OE+O 0.44
Fig. 3. Concentration
profiles
0.49
0.54
0.59 0.84 0.69 Radial Distance, m
for the axisymmetric
FLAC results obtained from the axisymmetric model with reaction rate parameters set to give fast conversion (cf. Fig. 2, lower). Fig. 4 shows corresponding
model.
Lines:
analytical,
0.74
discrete
0.79
symbols:
FLAC
results
smectite fraction profiles. The vertical lines show interface positions calculated using Eq. (8). The FLAC run appears to reproduce the propagation of
H. Hiikmurk
et al. / Enginrering
Geology
47 ( 1997)
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367-378
1.00 0.90
0.80 0' 0.70
: De=%lOm24 [K+]=O.Ol
moWL
j
'c 0.60 E t 0.50 .g y 0.40 E
v, 0.30
0.44
0.49
0.54
0.59 Radial
Fig. 4. Smectite fraction profiles obtained of the S/I interface according to Eq. (8)
in a verification
run
0.64
Distance,
0.69
0.74
0.79
m
for the axisymmetric
model.
Vertical
dotted
lines denote
the position
the S/I interface as well as the shape of the concentration profiles very satisfactorily.
Conditions regarding diffusivity, potassium concentration and rate constant parameters were varied between the different runs.
5. Results
5.2. Smcrll models
Fig. 5 shows a schematic view of a KBS3 deposition hole and a close-up of a section between a waste canister and the borehole wall. Two types of models. both axisymmetric, are analyzed: ( 1) small models (close-up view), and (2) full models of an 8 m deep deposition hole. The small models were used for analyzing the fundamental features of the transport/conversion process in the buffer, without regarding restrictions in transport capacity between potassium source and buffer boundary. The full models include assumptions regarding this capacity. The following data, which correspond to compacted bentonite with a water saturated density of 2000 kg mm3, were assumed for all models: water ratio w = 0.26, porosity II= 0.41, dry density p,=1590kgmP3, and potassium mass fraction of solid, completely converted material, p = 0.05.
The following cases were analyzed: CUSL’1: Constant boundary Kf concentration at the borehole periphery and constant rate constant. i.e. constant temperature. Cuse 2: The K+ concentration at the cylindrical boundary decreases as potassium is transported into the buffer. The temperature is kept constant. Cuse 3: The K’ concentration at the cylindrical boundary is constant. The conversion rate (i.e. temperature) is reduced with time. Fig. 6 shows the results of a large number of runs performed assuming case 1 conditions. The horizontal axis (effective diffusivity times boundary concentration) represents transport, the vertical axis (rate constant times boundary concentration) represents conversion. In the lower left corner both transport and conversion are slow, giving long buffer lifetimes. In the upper right corner both transport and conversion are fast, giving short buffer lifetimes. In the upper left corner conversion
374
H. Hiikmurk
et al. / Engineering
H Potassium
Geology
47 (1997)
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Source I
lo/so SetltlBallast
r,=O.44
m
R"=o.7g ml
8m HCB
A
(-[K+]
q
Diffusion
0.35
m
W
Waste Canister
JI
I Trammissii
Fig. 5. Left: KBS3 deposition hole in the floor showing the geometry of the small model.
Wall Region
of the backfilled
deposition
is fast, but transport is slow, which means that the buffer lifetime is controlled mainly by restrictions in transport capacity. In the lower right corner transport is fast, and the buffer lifetime is controlled by the rate constant, i.e. by temperature. activation energy and frequency factor according to Eq. (3). The vertical dotted straight lines in Fig. 6 represent [K’]D, values that would leave a certain percentage of the 0.35 m buffer unconverted after 5000 years if conversion occurs instantaneously, i.e. independently of temperature, activation energy and frequency factor. These [K-ID, values are calculated using Eq. (8). For example, the interface position corresponding to 50% remaining smectite is r(t)=0.639 m (for the specific radial geometry assumed here). Putting this figure for r(t) and 5000 years (in seconds) for t in Eq. (8) and then solving for CD, gives the value 1.374E-13 (mol l-‘)(m’s-‘). The horizontal dotted straight lines in Fig. 6
tunnel.
Right:
conceptual
Asymmetric
model
with close-up
represent values for [K+]k=[K+]A
exp(-EJRT)
that would leave a certain percentage of the buffer unconverted after 5000 years without transport capacity restrictions. These values are obtained by direct use of Eq. (3). The discrete plot symbols in Fig. 6 represent FLAC results. The labels denote calculated smectite fractions in percent after 5000 years. The solid curves connecting dotted vertical lines to corresponding horizontal lines are percentage isolines based on the FLAC results, i.e. on the discrete plot symbols. The filled plot symbols in Fig. 6 represent runs with a boundary porewater potassium concentration of 0.01 mol 1-l, while unfilled symbols represent concentrations ranging between 0.001 and 0.01 mol 1 - ‘. Note that filled and unfilled symbols fit equally well in the isoline system. Figs. 7 and 8 show smectite fraction profiles
H. Hijkmurk
et al. ; Engineering
[K+]*De,
iE-14
I “““I 75%
1 E-08
Transporl
47 (19973
367-378
lE-11
lE-12
I
lE-10
.’ .’ /
‘1’1’“”
50%
375
(mole/L)‘(m2/s)
IE-13
1 E-07
Geoiog~
25% IQ%
controlled
Smectite fractions after 5000 years
lE-11
lE-12 Fig. 6. Smectite fractions after 5000 years, assuming boundary concentration and rate constant to be fixed. Filled symbols represent simulations performed with the boundary concentration set at 0.01 mol I -‘. Unfilled symbols represent boundary concentrations ranging between 0.001 and 0.01 mol I -‘. The region above the dashed line represents constellations of reaction/transport data that would mean a transport-controlled time scale for conversion of the buffer. Points A and B denote constellations of reaction/transport data selected for additional simulations (see text). lE+O 9E-1 8E-1 -+-
1 x Required amount of K+ availaWe, 500 yeere
-A-
10 x Required amount of K+ available, 500 yeere
--W-
Conetent boundary
7E-1 E 3
6E-1
ii Q)
5E-1
&
1 x Required
4E-1
-A--
10 x Required amount of K+ available, 2000 years
-o-
conetant boundarycollcentratii, 2ooo years
C t;
g
concentration,
500 years
amount of K+ available, 2000 yesre
E
u,
3E-1
1
2E-1
0.54
0.59
0.64
Radial Distance, Fig. 7. Effects of limitations in potassium availability. conditions corresponding to data point A (Fig. 6).
Smectite
fraction
0.69
0.74
0.79
obtained
in the axisymmetric
m
profiles
model
with
initial
H. HGkmark
376
lE+O
et al. / Engineering
Geology
47 (1997)
367-378
,
QE-1 8E-1 7E-1 5
‘g 6E-1 E : 5E-1 .E ti Q) 4E-1 5
-k-
10 x Required amount of K+ available, 500 yeam
+
Constant
-u-
1 x Required
-A-
10 x Required amount of K+ available, 2000 years
boundary oonwntrati0n,500 years amount of K+ available, 2000 years
3E-1 2E-1 lE-1 OE+O 0.44
0.54
0.59
Radial Fig. 8. Effects of limitations in potassium availability. conditions corresponding to data point B (Fig. 6).
Smectite
obtained for runs performed assuming case 2 conditions, i.e. with decreasing concentration at the buffer boundary. The effects of a well-stirred tank surrounding the buffer were obtained by appending a small number of zones outside the buffer boundary, and by assigning FLAC thermal properties representing a specified potassium content and a specified initial potassium concentration to them. Two sets of initial conditions were assumed, represented by the data points labeled A and B in the transport/conversion plot in Fig. 6. Both sets correspond to fast conversion and fast diffusion, with point A in the transport controlled part. Two assumptions were made regarding the well-stirred tank: (1) It contains exactly the quantity of potassium that is necessary for complete conversion. (2) It contains 10 times the necessary quantity. Corresponding results obtained assuming constant boundary concentration are shown along with the well-stirred tank results. Fig. 9 shows smectite fraction profiles obtained assuming case 3 conditions. The initial conditions correspond to data point B in the transport/ conversion plot (Fig. 6). The temperature was assumed to be constant at 423 K during the first
0.64
Distance, fraction
profiles
0.69
0.74
0.79
obtained
in the axisymmetric
m model
with
initial
200 years after deposition and then drop linearly 323 K within the next 1800 years. to Corresponding results for the constant temperature system are shown for comparison. 5.3. Full models Results from one simulation are presented below. The initial conditions for the buffer and for the boundary concentration correspond to data point B in the transport/conversion plot (Fig. 6). i.e. A=80 800 1 (mol s))r, E,=24 000 cal mol -l, T=423 K, [K+]=O.Ol mol l-l, D,=2E10 mz s-l and p=O.O5. The temperature was assumed to be constant during the 2000 years that were simulated, i.e. no effects of decreasing conversion rate were accounted for. Instead of having a constant porewater K+ concentration at the radial boundary of the buffer volume (as for the small models in the previous section) the concentration boundary was located at the tunnel floor 1 m above the upper part of the buffer. This means that transport of potassium from the tunnel to the buffer is an essential process in these models. The potassium transport was
377
9E-1
8E-1 g
7E-1
3 6E-1 g a 5E-1 $ 8 4E-1 5f
--u-
Decreashg
temp,
-4- constant temp.ZOCGyears -U-Decreasing temp,500 yealls
3E-I 2E-1 IE-1 OE+O i 0.44
0.64
0.54
Radial Distance,
m
Fjg. 9. Smectite fraction profiles obtained in the axisymmetric model assuming temperature to be constant at 423 K during the first 200 “ears after depositionand then drop to 313 K within the next 1800 years. Filled symbols show corresponding results for constant temperature. The initial conditions correspond to point B in Fig. 6
assumed to take place by diffusion through the 1 m thick lo/90 bentonite/bailast mixture between HCB buffer and tunnel floor, and through an assumed transmissive, 0.05 m wide, annulus around the borehole periphery. The following additional conditions were assumed: ( 1) Potassium fixation in the IO/90 mixture was not accounted for, i.e. potassium was assumed to diffuse without delay through this 1 m barrier. (2) The diffusivity of the lo/90 mixture and the diffusivity of the 0.05 m wide annulus around the set to were both periphery borehole 5E-9 m’s-r. Figs, 10 and 11 show smectite fraction contours and porewater potassium contours, respectively. It is evident that restrictions in nearfield transport capacity have a very significant effect on the buffer lifetime, About 98% of the buffer remains after 2000 years despite conservative assumptions, while for corresponding initial conditions, less than 10% was found to remain in the case of constant concentration on the buffer boundary (Fig. 6).
mawit
/
btessium TUNNEL
M FL-
6. Discussion
It has been demonstrated that the system used in this investigation for simulating the coupled
Fig. 10. Smectite fraction contours after 2000 years
H. Hiikrnark
rr al. ; Enginrrring
Potassium 15E-9
47 ( 1997) 367-378
Predictive and more detailed analyses would require that the change in temperature distribution over time be included. and that an additional gridpoint variable, representing smectite fraction dependent potassium diffusivity, be defined. These modifications could easily be added to the system. In principle, it is also possible to simulate the dissolution of potassium in the backfill, provided that the dissolution rate can be approximated with a function of known quantities (e.g., temperature, specific area of ballast, porewater potassium concentration) for which FLAC variables exist or can be defined by use of FISH functions.
O.O’O 1 :
Geology
diisivity
m2k
0.008 0.004 0.002
Acknowledgment
Fig. I I. Porewater
K+
concentration
contours
after
2000 years.
process of potassium transport and illitization gives results that are logical and consistent with the input assumptions. For cases that could be checked with analytical solutions, the results were found to be correct and accurate. Since no really certain data exist regarding conversion rate parameters and potassium concentrations in the tunnel backfill, the calculations have been performed as a series of unprejudiced numerical experiments with parameter values selected only within ranges in which both transport restrictions and reaction rate control the buffer lifetime. The parameter values proposed by Huang et al. (1993) (I&=28 000 cal mol-‘), for instance, would give a system in which reasonable restrictions in transport capacity would have a very small effect on the time scale for conversion of KBS3 buffers, and have not been included for this purely formal reason. There is, however, no reason to doubt that the simulation system presented here would be capable of describing the conversion process at any level of detail with good accuracy for any set of parameter values.
The authors wish to acknowledge that this paper is a result of work funded by the Swedish Nuclear Fuel and Waste Management Co. (SKB). The views expressed in this paper are those of the authors and do not necessarily coincide with those of the SKB.
References Crank, J., 1975. The Mathematics of Diffusion. Oxford University Press, Oxford. Huang, W.-L.. Longo, J. and Pevear. I)., 1993. An experimentally derived kinetic model for smectite to illile conversion and its use as a geothermometer. Clays Clay Miner.. 41: 162Zl77. Hiikmark, H., 1995. Smectite-to-illite conversion in bentonite butlers. Application of a technique for modeling degradation processes. SKB Progress Report .4R 95-07. SKB, Stockhohn, Sweden. Itasca Consulting Group. 1993. FLAC. Fast Lagrangian Analysis of Continua. Version 3.22. Users Manual. ltasca Consulting Group, Inc., Minneapolis. USA. Karnland, O., Warvfinge, P. and Pus&. R.. 1995. Smectite-toillite conversion models. Factors of importance for KBS3 conditions. SKB Progress Report AR 95-27. SKB, Stockholm. Sweden. Pusch, R. and Karnland, 0.. 1988. Hydrothermal etTects on montmorillonite. SKB Technical Report TR 88-15. SKB, Stockholm, Sweden. Pusch, R.. 1993. Evaluation of models for conversion of smectites to non-expandable minerals. SKB Technical Report TR 93-33. SKB, Stockholm, Sweden.