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Migration processes in a model deep repository for the disposal of intermediate-level nuclear waste
Ann. nucLEnergy, Vol. 13, No. 3, pp. 141-158, 1986 Printed in Great Britain
0306-4549/86$3.00+ 0.00 Pergamon Press Ltd
MIGRATION PROCESSES IN A MODEL DEEP REPOSITORY FOR THE DISPOSAL OF INTERMEDIATE-LEVEL N U C L E A R WASTE S. M. SHARLAND a n d P. W. TASKER Theoretical Physics Division, AERE Harwell, Oxon OX11 0RA, England (Received 17 April 1985) Abstract--It is difficult to conceive Of radionuclides escaping from a repository by any means other than migration in groundwater. Simple models of the repository are constructed and various migration processes are identified and assessed, according to the flow speed of water through the repository. Diffusion in static water and advection in fast flows are considered separately initially, but later we examine the effect of slow flows in which both these processes contribute to the removal of radionuclides. Concentration profiles across the repository, fluxes of nuclides and total losses are obtained from the analysis. We investigate the time scales necessary for the steady state to be achieved in the repository and conclude that flow speed is roughly inversely proportional to this time scale, i.e. faster flows establish a steady state sooner than slow ones. We also assess the sensitivity of the results to the physical properties of the components of the repository.
NOMENCLATURE a= Co = C1 = C2 = C8 = D1 = D2 = /5 = F = G= K1= K2= Kd =
Pe = RI = R2 =
Uo = u= v= w= ct = 2= 4, = p = ~b1 = ~'2 =
Radius of wasteform (m) Concentration of nuclides in wasteform (M) Concentration of nuclides in backfill (M) Concentration of nuclides in rock (M) Concentration of nuclides at backfill surface (M) Intrinsic diffusion coefficient in backfill (m 2 s-1) Intrinsic diffusion coefficient in rock (m 2 s- 1) Apparent diffusion coefficient (m 2 s- 1) Groundwater volume flux ( m 3 s - 1) Groundwater volume flux (m 3 s- t) Permeability of backfill and wasteform (m 2) Permeability of rock (m 2) Mass-based equilibrium distribution coefficient (m 3 kg- 1) Peclet number Radius of backfill sphere or cylinder (m) Radius of sphere in rock (m) Darcy velocity of groundwater (m s-1) Darcy speed (m s- t) Darcy speed (m s- 1) Velocity of fluid in pores (m s- 1) Capacity factor Radioactive decay constant Porosity Density of wasteform (kg m - 3) Stream function of flow in backfill/waste (m 3 s- 1) Stream function of flow in rock (m 3 s-1)
1. INTRODUCTION It is envisaged t h a t long-lived intermediate-level radioactive waste (ILW) will be buried in deep u n d e r g r o u n d sites a n d s u r r o u n d e d by a n u m b e r of engineered barriers. T h e repository itself a n d the s u r r o u n d i n g geology are referred to as the near field 141 ANE 13:3-E
while the r e m a i n i n g rock structure c o n t a i n i n g routes to the surface is k n o w n as the far field. The aim of the 'source t e r m ' studies is to develop a realistic model of the repository from which the loss of radionuclides from the near field m a y be calculated, using experimental data. In considering the possible m e c h a n i s m s by which the radionuclides m a y be t r a n s p o r t e d from the source, the model m u s t i n c o r p o r a t e such details as the content of the waste, the form a n d physical properties of its packaging a n d other engineered barriers a n d the type a n d structure of the s u r r o u n d i n g geology. Careful a t t e n t i o n m u s t be paid to the chemical conditions present in the repository since factors such as the a d s o r p t i o n a n d solubility of the various constituents will have a considerable effect o n the overall loss rates. A repository will consist of the waste c o n t a i n e d in a solid matrix, p r o b a b l y concrete. This is c o n t a i n e d in a metal can which in t u r n is s u r r o u n d e d by a backfill material, again p r o b a b l y concrete. B e y o n d t h a t is the s u r r o u n d i n g rock whose mineral c o m p o s i t i o n depends on the particular chosen site. It is difficult to imagine any loss of radioactive material from this site other t h a n by m i g r a t i o n t h r o u g h groundwater. O n e of the barriers to loss of waste material is the choice of a dry site. However, we will assume the p e n e t r a t i o n of the repository by water as a prerequisite to the leaching of the waste. Nevertheless, some corrosion of the canister m a y already have t a k e n place by the action of residual pore water in the backfill. The c o m p o s i t i o n of the g r o u n d w a t e r t h a t will penetrate the site is determined by the n a t u r e of the geology of the region. As it
142
S.M. SHARLANDand P. W. TASKER
permeates the repository it will equilibrate with the backfill. This will lead to important changes in composition, particularly in pH. Ionic diffusion will occur between the backfill and outer minerals, leading to changes in the water chemistry in both regions over a fairly long time scale. No loss of nuclides will occur until the metal canister surrounding the wasteform has been penetrated. One part of the source term analysis will, therefore, be an assessment of the lifetime of the can under the chemical conditions of the repository. Once water has penetrated the concrete wasteform, dissolution of the radioactive elements will occur. These elements are then lost by diffusion and if there is a flow of water by convection. Inclusion of the many complex processes involved in the evolution of the near field of an ILW repository is likely to necessitate a large numerical model. Initially, however, we consider a simple idealized model that is, nevertheless, capable of incorporating all the necessary phenomena as data become available. We construct this model with a view to obtaining analytic solutions. In this way, we can not only determine the magnitude of concentrations, fluxes etc. but can also ascertain the time scales involved in their progressions. In Section 2, we set up our idealized model of the waste repository. In Section 3, we consider diffusion of the nuclides from the wasteform in the stationary groundwater within the repository. Section 4 considers the effect of a fast flow through the model and in Section 5 we consider a situation in which both advection and diffusion of the nuclides take place. 2. THE M O D E L
The idealized model considered initially comprises concentric spheres representing the wasteform, backfill and surrounding rock, as illustrated in Fig. 1. This is not intended to represent the complete repository, but rather the behaviour ofa wasteform near the edge of the site. We have assumed that water has penetrated and that the canister is no longer intact. We also assume that the groundwater has equilibrated with the backfill and wasteform and that the concentration of radionuclides within the wasteform is determined by the equilibrium solubilities of the species and is constant throughout. At time t = 0, we imagine that the concentration outside the wasteform is zero and the concentration within the wasteform is held constant throughout time, i.e. it acts as a permanent source of nuclides. Thus we are ignoring the leaching rate, which is clearly a pessimistic assumption. The source should stop when all the nuclides have been released and this can be incorporated when the exact inventory of the contents is known. Initially we consider only one species of
~
Rock
(FIL
I
\o,
ROCK
Z Fig. 1. Schematicillustration of the simple idealizedmodel of the waste repository.
radionuclide, although the extension to several noninteracting species is trivial. The results presented here have also neglected radioactive decay. This can be readily included but the results for the very long-lived isotopes will not vary significantly. 3. DIFFUSION 3.1. Governing equations We consider loss of nuclides from the source by diffusion only. The equation governing the concentration in the repository is given by
~i O t (r, t) = DiV 2 Ci(r, t ) - 2~iC~(r,t) + Q(r, t), i = 1,2,
(1)
where Cl(r,t) represents the radionuclide concentration in the water contained in the micropores of the backfill and C2(r, t) represents the concentration in the water in the rock. Diffusion in a sorbing, porous medium is described by two parameters, the intrinsic diffusion coefficient, D and the capacity factor, ~. The capacity factor reflects both the physical retention of the nuclide in the porosity (~b)and the bulk chemical retention due to equilibrium sorption. Thus, o~ = c~ + pKd,
where p is the density of the wasteform and K d is the mass-based equilibrium distribution coefficient for the combined solid phases in the wasteform (Lever et al., 1982). The term Q(r,t) represents the phenomenon of leaching from the wasteform. This term may be
Migration processes in a deep nuclear waste repository reasonably ignored if we assume that the water has equilibrated with the wasteform. Clearly we are taking a pessimistic view of the situation in neglecting this. The term 2aiC,(r, t) represents the radioactive decay of the radionuclide under consideration; 2 is the radioactive decay constant and is related to the half-life t~ by 2 = In (2)/t~.
143
where Co is the concentration of nuclides in the wasteform. Conditions (v) and (vi) constrain the concentration and flux of nuclides to be continuous at the backfillrock boundary. We solve equations (2) and (3) with boundary conditions (i)--(vi)using a Laplace transform technique. Details are given in Appendix A. By transforming in the variable t and solving for C~(r,p), C2(r, p) we obtain
CI(r'P) = aC° rp exp [ s l ( r - a ) ] I1 and
C2(r,p) =
DlslCoa sinh [s2(R2-r)] rpz sinh [SE(R2-R1)]
where
and S1 ----(pctl/D1)t-
and
Initially, for simplicity we consider a nuclide with a sufficiently long half-life that its decay will not be significant over the time scales of most interest, and we may neglect this term. However we shall note later on that this phenomenon may be easily included in the analysis. We transform equation (1) to a spherical geometry, coordinates (r, 0, tk) and obtain
~CI (r,t) ~t
Di ~2 = ~ - ~ r 2 [rCt(r,t)]
a <~ r <~ R 1
c3t
De ~2
r cgr2 [rC2(r't)]
R1 <~ r <~ R2, (3)
assuming symmetry in 0 and ~b. The parameter /31.2 is the effective diffusion coefficient of the media and is given b y / ) = D/ot. The boundary conditions of the problem are (i) Cl(r,O) = 0 a <~ r <~ RI, (ii) C2(r,0 ) = 0 R1 ~< r ~< R2, (iii) Cl(a ,t) = C O t >l O, (iv) C2(R2, t) = 0 t/> 0, (v) CI(RI, t ) = C2(RI, t) t >1 0 and (vi) D I ~ ( R I ,
t)= D2~L(RI,
t)
t~>0,
Note that inclusion of the radioactive decay term in equation (1) will merely have the effect of multiplying Cl(r, t) and C2(r, t) by e - l t The flux per unit area of escaping nuclides is given by
~Ci fi(r, t) = -- D i ~-r
i = 1, 2
(2)
and
dCz(r,t)
s2 = (PCtz/Dz) ½.
and an expression for the Laplace transform, f~(r, p) may be readily obtained from the above. Similarly, an expression may be derived for the transform of the total amount of nuclides passing through the sphere after time t, T(r, p), where
T~(r,t) = - |
30
~C, D i-~-r (R,t )A dt'. ?
A is the surface area of the sphere radius R. Details of the derivations and some examples of the transforms are given in Appendix B. Analytic inversion of all these transforms would be extremely difficult, so we have inverted them numerically using an approach used by Talbot (1979) which involves integrating the inversion integrals along the steepest descent contour of the function g(t) = lit.
144
S.M. SHARLANDand P. W. TASKER
3.2. Results Parameters used for initial runs. The parameters chosen for the repository size were a = 0.5 m,
1.8--
?E 1.5-
R~ = 2 m
T , 1.2-
and ~0,9--
%
R2 at infinity. Initially we consider a fairly dry cement as backfill (with a water/cement ratio at mixing of about 0.3), with intrinsic diffusion coefficient Dt = 10 -~2 m 2 s -~ (Atkinson et al., 1983). We take cq = q~l = 10-1 for such a cement (i.e. consider the progress of a nonsorbing ion such as 1291). We assume the rock to be of granitic type and take D 2 = 10 -13 m 2 s -~ and ~t2 = q~2 = 10-2 (Lever and Bradbury, 1983), and we take the source concentration Co = 10- 7 M which is a typical limiting solubility of a species in the wasteform. Clayton and Rees (1983) derive values for the limiting solubilities of the actinides under disposal conditions in the range 10 -22 to 6 x 10 -6 M. However, we note that the analytic expressions obtained for the Laplace transforms of the concentrations etc. depend linearly on Co so an alternative choice of this parameter simply rescales the results. Results from initial runs. Using the data obtained from inverting the Laplace transforms, we have plotted concentration profiles across the repository at various times and obtained graphs showing the variations in flux and total loss with time. Figure 2 shows a profile plot at time 3000 years. The change in slope at r = 2 m reflects the condition that the flux of nuclides across the
WASTEFQRM BACKFILL 1.O D=10-12m 2 s-1
ROCK D=10-13m2 s-1
=o.1
"7 o.6o 0.3-I 500
1000
1 5 O O 2000 Time (yr}
2500
3000
Fig. 3. Flux of radionuclides per unit area through a sphere of radius 3 m against time (loss by diffusion only).
boundary is continuous. For the parameters used we obtain a change of a factor of 10 (increasing on the rock side). Figure 3 shows a plot of flux/unit area with time. The graph will eventually level out to a steady value. The time needed to achieve the steady state is studied in more detail later. Figure 4 shows a plot of total loss through a sphere of radius 3 m against time.
3.3. Sensitivity to parameters By changing the parameters individually or in pairs, one can assess the sensitivity of the results to the conditions present in the repository. (i) Concentration at source, Co. Since C~(r, p) depends linearly on Co then changing Co will only have the effect of rescaling the results. (ii) Diffusion coefficient in rock,/)2- Results for the granitic rock were compared with those obtained from
~ =OOl
0.8
1.25
1.00
x 0.6
~o 0.7.5
~
0.4
?,
o
0.50
0.2
~- 0.25
I 0
0.5
1.0
1.5
2.0
2.5
3.0
I ~z_...J 3.5 " 4.0
Radius (m) Fig. 2. Concentration profile across the repository surrounded by granite after 3000 years (considering loss by diffusion only).
O
30
60
90 120 Time (x 103 yr)
150
Fig. 4. Total loss of nuclides through a sphere at radius 3 m against time (diffusion only).
Migration processes in a deep nuclear waste repository WASTEFORM BACKFILL 1.0 D ,10-12m2 s-1
145
ROCK D : 2 x I O -11 m2 s-1 ¢ =0.146
@=o.1
0.8 _o x
0.6
8 0
20
40
~ 0.4
80 100 120 (x 103 yr)
Time
140
I
160
Fig. 6. Total loss of nuclides through the surface of a sphere in the surrounding sandstone at radius 3 m against time (diffusion only).
o 0.2
0
60
05
1.0
1.5
2.0
Radius
,
2.5 (m)
i
3.0
i
5.5
,,,_2..
5.5
greater than those through the granitic rock. Although Fig. 5 shows a shallower gradient at R = 3 m, D 2 is sufficiently large that the flux through this point and hence the total loss will be greater with a sandstone rock surrounding the backfill. (iii) Diffusion coefficient in backfill, D 1. We consider cements with higher water/cement ratios at mixing, e.g. D1 = 10 T M m 2 s -I and D 1 = 10 -1° m 2 s -1 and compare with the case of D 1 = 10-12 m 2 s - 1, keeping D 2 = 10 -13 m 2 s -1 and ~b2 = 10 -2. Figure 7 shows a plot of the concentration profiles for each of the three cases after 1000 years. As expected, the profile in the backfill is flatter for higher diffusion coefficients. Figure
Fig. 5. Concentration profile across the repository surrounded by sandstone after 3000 years (diffusion only).
using D 2 = 2 x 10 -11 m 2 s -1 and ~b2 = 0.146, which represents a typical sandstone (Lever et al., 1982). Figure 5 shows a concentration profile after 3000 years. The slope at the backfill-rock boundary decreases by a factor of about 10 in this case. Figure 6 shows a plot of total loss of nuclides against time. Comparing with Fig. 4, we see that the total losses are about four times
Z)l = 10-1o m2 s-1 1.00 ~
!1 D2 =I0-13m2 s-i
7,-
0.75
= -
-
x
~_ cJ 0.50
\o,=,o-,2::,
o
0.00
5
1.0
1.5
\
2.0
2.5
3.0
3.5
I
I
4.0
4.5
Radius (m)
Fig. 7. Comparison of the concentration profiles after 1000 years for three different cement mixes as backfill (diffusion only).
146
S.M. SHARLANDand P. W. TASKER 1.8
2
30
1.5
~25
r~O 1.2
620
0.9
~5 ~o
-6 0.6 I~ 0.5
~5 I 20
I 40
I I I 60 80 100 Time (x 103 yr)
I 120
I 140
300
600 Time
900
1200
1500
(xlO 3 yr)
Fig. 8. Total loss through a sphere at radius 3 m (in granite) against time, with a 'wet mix' cement backfill (diffusion only).
Fig. 10. Comparison of total losses from diffusion models with the outer boundary at infinity, 5 m and 3.5 m (calculated at radius 3 m) (diffusion only).
8 shows the total loss at radius 3 m for the case of D~ = 10-lo m 2 s-1. The total is slightly greater than for a 'drier mix' concrete reflecting the larger concentration gradient at r = 3 m (as shown in Fig. 7). (iv) Outer boundary condition distance, R2. So far we have considered the case of zero concentration at infinity. Now we investigate the situation of finite Rv This represents the possibility of having groundwater flow some distance from (but not directly through) the wasteform. All the radionuclides are imagined to be carried away in the stream, hence constraining C2 to be zero at this radius. Taking R2 = 5 m and 3.5 m and examining the short-time concentration profiles, we
find there is little difference between these cases and the 'R2 at infinity' situation. Figure 9 shows the steadystate profiles and Fig. 10 shows the variation in total losses for each of the three cases. Over the time scales of most interest these variations are not significant. F r o m the above analysis, we conclude that the concentrations and fluxes present in the repository are most sensitive to variations in the diffusive properties of the backfill and geology but are not adversely affected by changes in the boundary conditions of the problem. We may also conclude that the analysis is readily adapted to represent a wide range of physical conditions that may be present in the repository.
1.0
BACKFILL
I
ROCK
I
I 0.8
b-
'o ~ 0.6
.~_ E
~
0.4
0.2
o.o 0.5
] .1.0
I 1.5
2.0
I 2.5
I 3.0
5.5
4.0
4.5
I 5.0
Radius of westeform (m)
Fig. 9. Comparison of steady-state concentration profiles for the outer boundary of the diffusion model at infinity, 5 m and 3.5 m (diffusion only).
Migration processes in a deep nuclear waste repository
147
8
3.4. Comparison with the steady state In the steady state we may derive a simple formula describing the flux of nuclides from the repository. The flux at the surface of a sphere radius r is given by
A i
-
"
J = - - 4 n r 2 D i ~r • For the steady state J is constant over r, therefore integrating over r we obtain J
1 dr
D(r) 47zr2
dC
2 0
I
300
600
I
I
900
1200
I
1500
Time(xlO3yr)
_j[-1 (1 LDl
Rl--al #
~O2
= 4r Co
Fig. l 1. Flux of nuclides through a sphere at radius 3 m (in granite) against time, to show how it settles to the steady-state value represented by the dashed line (diffusion only).
and
4nCoDtD2Rla J = [D2(R1-a)+Dla]"
(4)
The fluxes obtained from the Laplace transform method agree well with this expression at long times. For the initial set of parameters considered it was found that the steady state was achieved after approx, l0 s years. Although periods of time of this order are not directly relevant, it is perhaps interesting to note how fluxes of nuclides from the wasteform settle to the steady-state value (Fig. 11). The sharp peak represents the passing of the concentration front and the associated steep concentration gradient at that time. After about 7000 years the steady-state approximation gives an optimistic value for the flux of nuclides. However, after such a long time many of the assumptions made during the mathematical analysis may no longer be valid. For example, the integrity of the concrete backfill can no longer be relied on. Over shorter time scales the steady-state approximation is a pessimistic view of the situation and since the results only differ by about 10-20%, it is reasonable to use this formula to estimate losses from the repository. Nevertheless, a full solution of the equations is needed to obtain the concentration profiles and their evolutions which are necessary if the detailed chemistry cf the problem is to be included.
3.5. Summary 1. Solutions to the governing equation are obtained using a Laplace transform method and numerical inversion, ignoring leaching and radioactive decay. Concentrations, fluxes and total losses from the repository are obtained. 2. Initial calculations using parameters for a typical dry cement embedded in a granitic geology,
indicate losses of about 104 x Co mol/150,000 years (assuming a permanent source of concentration C OM), with time scales to reach a steady state of the order of 10 a years. 3. The results are most sensitive to variations in the diffusive properties of the backfill and rock, but are not as dependent on the boundary conditions of the problem, i.e. the repository dimensions etc. 4. The analysis may be easily adapted to incorporate a wide range of physical parameters that might describe the conditions in the repository. 5. The fluxes (and total losses) from the repository may be approximated using a simple formula derived from considering the steady-state situation, despite the extremely long times necessary for this to be achieved. However, a full solution to the governing equations is necessary to obtain the concentration profiles and their progression with time. 4. ADVECTION
4.1. Governin# equations We now set up a simple model to investigate the effect of water flow through the repository. As before, we regard the waste, backfill and geology as concentric spheres, but now we assume there is a flow through the system. Flow through a porous medium is governed by Darcy's equation : Vp(r)=---
/re(r) and K
V'u(r)=0,
(5)
where p(r) represents the pressure field, and u(r) represents the Darcy velocity field;/~ is the viscosity of
148
S.M. SHARLANDand P. W. TASKER
the fluid and K is the permeability of the medium. We assume that the permeability of the wasteform and backfill are equal. We solve V2p = 0 in a spherical geometry with the boundary conditions (i) u is continuous at r = R 1, (ii) p is continuous at r = R1 and (iii) u ~ U o as r--* oo, where Uo is the velocity of the fluid far from the wasteform, and obtain the pressure field and hence velocity field (details are given in Appendix C) : 3UoK1 ul = ( 2 K ~ l j ( - c o s
0, sin 0,0)
and
o2
. f =
~1-
2R 3 ( K 2 - K 0 q
(6)
ok-cos V-- 7-
s m o / 1-4 -- - - / , L r (2K2 + K 1 ) J
o/ /
where Ul and u2 are the velocity fields in the backfill/wasteform and rock, respectively, and K t and K 2 are the permeabilities of the two regions. From these velocity fields the stream functions ~bl(r, 0) and ~//2(r,/9) may be derived : 3 U ° K 1 sin2 r2
/
2(2K2 + K1 )
~bl(r,O ) -
/
and
] [- (7)
~b2(r, 0) =
U° sin2 Or2 2
1
2R 3 (K2--K1)
1 w(r)=~u(r),
r 3 (2K2 + K 1 ) J ' J
The stream function, ~b, is derived from the mass conservation equation for an incompressible fluid which in the spherical geometry under consideration reduces to
1 0
flow (i.e. lines of constant stream function) for the cases : (i) K I = 10 -19 m 2, K 2 = 10 -18 m 2, i.e. backfill less permeable than rock; and (ii) K 1 = 10 -14 m 2, K 2 = 10-16 m 2, i.e. rock less permeable than backfill. These parameters were chosen as typical for the type of cements and geologies that may be chosen as repository components (Atkinson, 1983). We now consider transport of radionuclides due to fluid flow only, which will be a reasonable approximation for fast flows. We neglect loss due to diffusion completely. Since we are considering a steady flow, the path of a material element of fluid coincides with a streamline. The escaping nuclides will only follow the paths of those elements that have actually passed through the wasteform itself, since we are ignoring the possibility of any diffusion into the stream. Thus the region of contamination in the repository is bounded by the streamlines which coincide with the edge of the wasteform. This is schematically illustrated in Fig. 14. The concentration behind the concentration front will be constant in this region and at worst equal to the limiting solubility of the source Co. If the flow is too fast for chemical equilibrium to be reached between the groundwater and wasteform then the concentration will clearly be less than Co. The progression of the concentration front through the shaded region of Fig. 14 can be easily measured since the fluid velocity is known everywhere. For example, the time, T, taken for the front to reach a point at radius R may be evaluated by integrating 1/[w(r)lalong the streamline through that point, where w(r) is the velocity of the fluid in the pores ; w(r) is related to the Darcy velocity a(r) by
where q~ is the porosity of the medium. In general T approximates to T-
Re Uo'
1
c3 r 2 0~ (r2Ur) + (sin 0 uo) = O. r sin 0 O0
since the streamlines are reasonably parallel for most choices of K 1 and K 2.
We obtain ~k from
1 Ur
0$
r 2 sin 0 00
1 and
uo
0~b
r sin 0 Or'
where ur and u 0 are the radial and tangential components of u, respectively. Physically the stream function may be interpreted as follows : the flux of fluid volume across a curve joining two points is given by the difference in the stream functions evaluated at each point. Figures 12 and 13 show the streamlines of the
4.2. Sensitivity to parameters
To investigate the effect of the relative permeabilities of the backfill/wasteform and the rock we first calculate the flux density across the backfill, using the stream functions. The volume flux between two points is equal to the difference in stream function evaluated at the points. Consider two points A, B at the edge of the backfill, such that the line joining them is perpendicular to the flow. For K 2 < K 1,i-e.the rock less permeable than
Migration processes in a deep nuclear waste repository
/
149
ROCK K2=1~18m 2
c o ,m
-2 ROCK
-4
I
-8
-6
I
--4
]~
I
-2
Distance
F i g . 12.
~
0
I
2
4
J
6
(m)
Streamlines through the repository with the backfill and waste less permeable than the surrounding rock.
the backfill, the flux across the backfill, G is given by G = ffl(A)-~kl(B)
3UoK1R 2 (2K2+K0"
(8)
For K1 < K2, from equation (7), the flux across the backfill, F is given by
3UoK2R 2 F = ~2(A)-~,2(B) = (2K2+K~)'
(9)
In the case of a highly permeable backfill G approximates to G = 3Uo R2 and in the limit of an impermeable backfill, F = 3 UoR2/2. Atkinson (1983) uses these extremes to make estimates of the loss rates. In each case, he also includes an estimate for the diffusive loss into the stream and concludes that limits of extremes of permeabilities only result in about a 20-fold change in the release rate of nuclides. He also shows that the release rate is
4--
" ~
BACKFILL
8c o
A,'2=10-16 m2 0
121
-2
-4
I
-
-6
; ~ -4
I -2
~
I Distonce
F i g . 13.
~
0
I 2
~
I 4
6
(m)
Streamlines through the repository with the backfill and waste more permeable than the rock.
150
S.M. SHARLANDand P. W. TASKER
"-'--"-
ROCK
q
J J
Z///////////~
,I
o CI
_ _
i
ROCK
-4 -8
I
I
-6
--4
1__
I
-2
~
0
2
I
I
4
6
"
Distance (m)
Fig. 14. Schematic illustration of the model indicating the region of contamination by radionuclides for fast flows (loss of nuclides by advection only). independent of the permeability of the repository when K1/K 2 <~ 10 -2 and concludes that under typical conditions the permeability of the repository is not a major factor in determining the rate of loss of nuclides but there is some advantage in having K 1 approximately one or two orders of magnitude less than K2.
4.3. Summary 1. Solutions to Darcy's equation describing flow through a porous medium are obtained in the spherical geometry of our model, assuming the wasteform and backfill are of equal permeabilities. 2. Assuming a fast enough flow that diffusion may be neglected, we may identify a distinct region of the repository into which nuclides will penetrate. The concentration in the region behind the concentration front is independent of time assuming the source concentration remains constant. 3. Estimates of the losses due to advection with some diffusion are fairly insensitive to the relative permeabilities of the backfill and rock but it is preferable to choose the permeability of the backfill to be one or two orders of magnitude less than that of the rock. 5. A D V E C T I O N
AND
DIFFUSION
5.1. Governing equations The equation governing the concentration of radionuclides in the respository in the presence of a
slow water flow is now
cq~-[-(r,t)+u,~r,t)'VCi(r,t) = DiV2Ci(r,t)
i = 1,2. (10)
An analytic solution of this equation in a spherical geometry is difficult since the concentration now has angular dependence. However, one can obtain an analytic approximation in a cylindrical geometry following Chambr6 (1982). He assumes an impermeable infinite cylindrical wasteform embedded in a porous medium. We shall assume an infinite impermeable cylindrical backfill. Although, it is unlikely that a chosen backfill material would have zero permeability, this approximation can still provide a reasonable solution to the problem as shown in Section 4.2. Initially we consider flow normal to the axis of the cylinder but we shall later compare this with flow parallel to the axis. By solving Darcy's equation in a cylindrical geometry and setting K1 = 0, we obtain the flow field u(r, 0) given by u(r, 0) = U 0 [ - ( 1 -R~'~r2 ] cos 0 , ( 1 + r~-)sin 0, 0 ] = (u, v, 0).
(11)
In cylindrical coordinates equation (10) transforms to dC
8C
OC
~ +u(r,O)T/+v(r,O)So = D['O2C
1 8C
1 c~2C'~
( x ~ - - ~ - + r N + r - ~ 8 0 2 ],
(12)
Migration processes in a deep nuclear waste repository where we now drop the subscript 2 from the concentration, diffusion coefficient etc. denoting the properties in the rock. The boundary conditions are (i) C(RI,0, t) = C a 0~<0~<2r~ t~>0, (ii) C(@,O,t)=O 0~<0~<2r~ t~>0 and (iii) C(r,O,O)=O
Rl <~r<~ oo O<~O<<.2n,
where Ca denotes the concentration at the backfill surface which we assume to be constant. We assume the backfill cylinder has radius R1. We introduce the non-dimensional variables z=
Uot ~tR,
and
UoR1
Pe -
D
,
Both the concentration and flux now depend on angular position as well as radial position and time. The concentration is smallest at 0 = 0, where the flux is largest. Figures 15 and 16 show the concentration contours in the rock at 500 and 10,000 years, using parameters for a granitic rock with the concentration at the backfill surface, CB, equal to 8 x 10 -8 M and Uo = 1 x 10 -12 m s -1. The flow direction is from the right in each case. Figure 17 shows the progression of the flux with time at 0 = 0 and 0 = ~. The flux is greatest in the direction of the approaching stream since the concentration gradient is largest here. Chambr6 estimates the time necessary to establish the surface mass flux to 9 9 ~ of its steady value as T, given by
Uo T ctR 1
and equation (12) then transforms to c3z
rE COS
Or -- l + r 2 ]
1 f02C
r
1~C
1 02C~
002 } (13)
forR 1 ~< r ~< oo, 0 ~< 0 ~< 2 h a n d z >1 0. Peis called the Peclet number for the flow and provides a measure of the relative importance of the advection and diffusion terms in the governing equations. For a large Peclet number, advection is d o m i n a n t and for a small Peclet n u m b e r diffusion is more important. The form of equation (13) suggests an asymptotic solution for a large Peclet number. To obtain this we transform equation (13) with i+
to obtain t3C
t~C dC 2R cos 0 ~-~ + 2 sin 0 c30
- 1.2.
For a flow with Uo = 1 x 10-12 m s - 1, T -'- 750 years which is considerably less than the time necessary to establish the diffusive steady state studied earlier.
c~0
=Pe\~r 2 + ~ + 7
r=R1
151
t32C i_0(Pc_ 1/2)' dR 2
5.2. Comparison with flow parallel to the axis of the cylinder The angular dependence of the surface mass flux may be removed by averaging f(R1, 0, z) over the perimeter of the cylinder and an expression may be obtained for the mass transfer from the surface, ~/ . . . . . for a cylinder of length L (in the steady state) : ....
:
4.513 CBL(UoRID) 1/2.
We now make an approximation for the mass loss in the case of flow parallel to the cylinder axis. Following Chambr6's approach, we approximate the cylinder as a flat plate of length L and width 2nR r The flow is in the direction of the length of the plate. The flux, #/~on~,from the surface may be estimated as 5)/lo,g
2xRILDCa
(14) with boundary conditions (i)-(iii). Chambr6 obtains the following solution of this : 2
where 3 is the boundary layer width, and may be taken as
. (15)
The surface mass flux/unit area, f(R1, O,z) may be derived from this expression :
\Uo/ Hence, h)/iong "~ 21tRI(DLUo)I/2.
f(R,, O,z) = - D C3~r
Consider the ratio of these two estimates :
r=Rl
_ DCB y2Pe (coth R1 / ~
2~+cos
0)]'/2.
/~/ ....
~/long
4.513 ( L y / 2
2re \R1,]
;
152
S.M. SHARLANDand P. W. TASKER Concentration contours 7.7800E-08
I -x--
- - - -
5.5800E-08
--~--
------
5.2100E-
08
. . . . .
1.43,00E-08
- - - - -
7.9100E1.0000E-10
09
. . . .
1.0000E08 1.0000E-11
s: lil)!! %\\\ £;///
Q
I',,. -2
~
- ~
Z_Z~ ~SA~ /
-4
-6 -- l
I 1 4
1
I
12
0
I
I
2
4
Distance (m) Fig. 15. Plot of lines of constant concentration of nuclides in rock, assuming an impermeable cylindrical backfillwith flow normal to axis(from the fight).This plot representsthe situationafter500 years (adv¢ction and diffusion).
~/ . . . . > M~o,, for cylinders with L > 1.938 R1, that is the mass loss from a cylinder in a flow normal to its axis will be greater than that from the cylinder with flow parallel to the axis, if its length is approximately greater than twice its radius.
perpendicular to u, assuming K l = 0).
5.3. Estimate offlow regimesfor dominant diffusion and advection
where the volume flux is calculated from the stream functions (7) as before. We equate this quantity with the diffusive flux of nuclides from the surface of the backfill, Fo, and in doing so make the assumption that all the nuclides diffusing into the region between A' and B' are swept away by the stream :
The above analysis is valid for flows with a Peclet number greater than about 10, that is with upstream speeds greater than about 5 x 1 0 - l a m s -a, for the granitic type rocks and repository dimensions considered earlier. We can now consider the flow regimes in which each type of transport process is dominant. Returning to the spherical geometry used earlier we assume a flow with speed U o in the far field and estimate the width of region surrounding the backfill sphere, 6, in which the concentration of radionuclides is non-zero for the steady state. Following the approach of Atkinson (1983), we estimate the flux of nuclides due to advection, F A between two points in the rock, A' and B', at (Ra, n/2) and (R~ +6, n/2) (such that the line between them is
F A -~ (volume flux between the two points) • C2(R0- 2 - 3nU°R1C25 2 2 '
~C21
F° = --4nR1D2 ~-r r=R~ R2=RI +8
(this is the diffusive flux of nuclides through a unit length of the backfill surface). Thus, 8D 2
6-- 3UoC2(RI)
c3C2
cGr r=Rl R2=Rt+6
For sufficiently large 6, we may use the condition that is infinite but for faster streams we cannot solve
R2
Migration processes in a deep nuclear waste repository
153
Concontrotiocontours n 6 -
7.7800E-08 3.2100E-08 7.9100E-09 1.0000E-10
-----------x~
- - ...... . . . . ~ z ~
5.5800E-08 1.4300E-08 1.0000E-09 1.0000E-11
4-
j--
<"
BACKFILL 0
AND
8
~
---____
_~
-6
~'~
__
WASTE
>
I
-4
]Ill'
ix|
.!~lyif7
_ _
I
I
-2
0
I
2
I
4
Distonce (m)
Fig. 16. Steady-state plot of lines of constant concentrationin rock, with the same model as in Fig. 15(advection and diffusion).
directly for 6. Instead, by choosing values for 6, the corresponding Uo may be calculated numerically using the Laplace transform techniques from Section 3. We next compare approximate diffusive and advective fluxes at a fixed point in the rock at (R, n/2), where R < (R1 + 6). Considering a small volume e3 surround-
ing the point, we estimate F D = _~2D2 ~C2 ur
and FA = EC(R + e/2)-2 CiR - e/2)] UoeR
6 -
7
E >,
-~ 3 E
× 1 u_ 0
~
8-0
500
1 1000
O=r 1500 Time (yr)
I 2000
I 2500
Fig. 17. Comparison of the flux and nuclides at two points on the surface of the backfill, described in Fig. 15, against time. 0 = 0 represents the point directly exposed to the oncoming stream (advection and diffusion).
( R3), 1 + ~-g
and compare them numerically over a wide range of Uo and 6. We find that advection of nuclides is the dominant transport process for Darcy velocities of greater than approx. 1 x 10 - l ° m s -a and diffusion dominates for fluid speeds of less than 1 x 10- t3 m s - t. The fluxes obtained using this approach are not very useful in their present form, since the approximation is only valid for one direction, i.e. 0 = n/2. More satisfactory results may be obtained using the analytic approximation in a cylindrical geometry described earlier since this gives a single flux in each direction. This analysis is also extremely useful since it is valid in the velocity regime between the limits stated above. A typical groundwater flow would have a velocity in this intermediate region.
154
S.M. SHARLANDand P. W. TASKER
5.4. Summary
6. C O N C L U S I O N S
1. An analytic approximation is obtained for the concentration of nuclides in the rock, for the situation of flow perpendicular to the axis of an infinite impermeable cylindrical backfill. The concentration shows directional as well as radial and temporal dependence. An expression is derived for the flux from the backfill surface. A comparison with the flux of nuclides from the backfill sphere considered in Section 3 (where diffusion only was studied) shows a difference of about two orders of magnitude (the diffusive flux being smaller). The flux from the cylinder settles to a steady state after about 750 years (for a typical granite surrounding the backfill), compared with 10a years as seen in Section 3. 2. The loss of nuclides from a finite length cylinder in a flow normal to its axis will be greater than that from the cylinder with flow parallel to the axis, if its length is approximately greater than its diameter. 3. Diffusion is the dominant transport process by which radionuclides can escape from the wasteform when there is a flow of groundwater with a Darcy velocity of less than approx. l x l 0 13 m s -I. Advection dominates for velocities greater than about 1 x i 0 - lO m s- 1.
10-z5
P l = 10-12m2 s'-l@1:0.1 K I : 0 D2 = I 0-13m2 s-I '~2:0-01
c~ I
C B = 8 x 10-8 M
E
T
1. Considering migration of nuclides from the wasteform by diffusion only, we conclude that fluxes and total losses may be estimated with reasonable accuracy using the simple steadystate formula given in Section 3.4. However, a full solution of the governing equations is needed to obtain concentration profiles across the repository, which are necessary if the detailed chemistry of the system is to be included. 2. Over the time scales of interest a steady state will not be reached--about l0 s years is necessary. 3. The results are most sensitive to variations in the diffusive properties of the constituents of the repository, but do not show as much dependence on the repository size. 4. Considering migration of nuclides by advection in a fast stream, we may identify a distinct region of the repository, extending from the wasteform into which nuclides will be carried. 5. If the wasteform acts as a permanent source the concentration at a point in this region will be constant once the concentration front has passed. Thus, the time necessary to establish the steady state is considerably less than that in the case of static groundwater. 6. Estimates of the losses of nuclides are
lo -8`o
E ~,
10- 8.5
C vection
10 -9"° a) (n 10-9'5
8 O3
Diffusion controlled
l o -1°.° lO-16
l 10-15
I 10-I';
I
I
I
I
l
10-13
10-12
10-11
10-10
10. 9
...
Dorcy speed (rn s-1)
Fig. 18. An estimate of variation in fluxper unit area through the surfaceof the backfillwith the Darcy velocity of the groundwater.
Migration processes in a deep nuclear waste repository comparatively insensitive to the relative permeabilities of the backfill and rock. However, there is some advantage in having the permeability of the backfill one or two orders of magnitude less than that of the rock. 7. Combining the migration processes, we obtain an analytic approximation for the concentration of nuclides in the rock, surrounding an impermeable cylindrical backfill, with a groundwater flow normal to its axis. We derive a expression for the flux of nuclides from the backfill surface. This flux is larger than the diffusive flux from the backfill sphere. Generally the faster the stream is, the larger the flux will be. Also, we conclude from our analysis that the steady state is reached sooner with faster flows through the repository. 8. The loss of nuclides from a finite length cylinder in a flow normal to its axis will be greater than that from the cylinder with flow parallel to the axis, if its length is approximately greater than the diameter, i.e. the loss will be greater when the flow is parallel to the shorter overall dimension. 9. We estimate losses of nuclides due to diffusion and advection over a wide range of Darcy velocities and conclude that in a granitic rock, diffusion is dominant for velocities of less than approx, l x l 0 -13 m s -1 and advection is dominant when the fluid flows faster than about 1 × 10 -1° m s-1. Similar estimates for other
155
geologies and repository sizes may easily be made. Figure 18 shows an estimate of the variation in the flux per unit area from the surface of the backfill with Darcy velocity, for an impermeable 'dry mix' cement embedded in a granitic rock. For velocities less than about 5 x 1 0 -13 m s -1, the losses are diffusion controlled and independent of flow speed. For flows faster than this, the flux increases with the square root of fluid velocity. However, this increase will be checked at Darcy speeds greater than about 10-s m s-1 (Atkinson, 1983), since leaching in the wasteform now becomes important. In this regime the groundwater can no longer equilibrate with the contents of the source, and the flux again becomes independent of flow rate.
REFERENCES
Atkinson A. (1983) Report AERE-11077. Atkinson A., Nickerson A. K. and Valentine T. M. (1983) Report AERE-10809. Chambr6 P. L. (1982) Report UCB-NE-4017, Univ. of California, Berkeley, Calif. Clayton P. R. and Rees J. H. (1983) Report AERE-10988. Lever D. A. and Bradbury M. H. (1983) Report AERE TP1004. Lever D. A., Bradbury M. H. and Hemingway S. J. (1982) Report AERE-10614. Talbot A. (1979) J. Inst. Math. Applic. 23, 97.
APPENDIX
A
Solution of 8Ct r t
Dt
62
(')=~-~r2[rC,(r,t)]
~C2
(r,t)
D2 62 [rC2(r,t)]
= 7
~~r
a~r<<.R 1
(A.I)
RI <~r <~ R 2
(A.2)
with boundary conditions (i) (ii) (iii) (iv) (v)
Cl(r, 0 ) = 0
a<~r<~Rt, C2(r,O)=O Rt<<.r<~R2, C l ( a , t ) = C o t>lO,
C2(R2,t ) = 0
t/>0,
Ct(Rl, t ) = C2(Rx,t ) t >! 0
and ~C1
~C2
(vi) Dl~-r (Rl, t ) = D 2 ~ r r (Rl,t)
t >lO.
Taking Laplace transforms of equations (A. 1) and (A.2) and using boundary conditions (i) and (ii), one obtains D1 62 PCl(r'P) = r
-
~r 2 [rCt(r'P)]
(A.3)
156
S . M . SHARLAND and P. W. TASKER
and
pC2(r,p) D2 02
(A.4)
= -7- ~ r ~ [rC~(r, p)].
Transforming conditions (iii)-(vi) : (iii) (iv) (v)
~'~(a,p) = Col p,
(vi)
Dt ~-r (Rt,p) = D2~r-r (Rl,p).
C2(R2,p) = 0,
~t(Rt,p) = Cz(Rt,p)
and
0C1
~C2
From equation (A.3), rCl = A exp [(p/DOt/2r] + B exp [-(p/D1)l/2r].
(A.5)
Using condition (iii) and letting st = (p/D1) t/2
rC° P
A exp(sla)+ B e x p ( - s l a ) .
Thus equation (A.5) becomes
rCx
=
--aC_~
exp [sl(r -
a)] -
2B exp ( - sla) sinh [sx(r -
a)].
P From equation (A.4), rC2 = E exp [(p/D2)l/2r] + F exp [(p/D2)l/2r].
(A.6)
Using condition (iv) and letting s 2 = (P/D2) 1/2, 0 = E exp(s2R2)+F exp(-s2R2)
and
E = - F exp(-2s2R2).
Thus equation (A.6) becomes r C 2 = 2F exp (-s2R2) sinh [s2(R 2 - r ) ] .
Using condition (v),
aCo
--exp P
[sl(R 1 - a ) ] - 2 B exp(-saa) sinh [sl(R t - a ) ] = 2F exp (-s2R2)sinh [s2(R2-RO].
Using condition (vi),
2DIBexp(-sla)(~
1
DlaCo c°sh[sl(Rl-a)]t+~-l (sl-~)exp[s'(R'-a)] 4F exp (-s2R2)D2 R~
{;~
sinh [s2(R2-RO]+s 2 cosh[s2(R2-RO]
Solving for B and F and substituting into equations (A.5) and (A.6) we obtain
aCo
Cl(r,p) = - -
rp
exp [ s l ( r - a ) ]
/
sin
1
/ z
and
C2(r'P) = -
DlslCoa sinh [s2(R2-r)] rpz sinh [s2(R2-Rx) ] '
where
z=is,.,.~Sl,.~_a,~{,o, :2, s~o~oth~s:~-..~}-cosh ~.:~-~,~,s~o.1 and
sl = (p~q/DO 1/2 and s z = (po~JD2) 1/2.
•
}
.
Migration processes in a deep nuclear waste repository
157
APPENDIX B The flux of nuclides through a unit area of the surface of a sphere of radius R is given by
f(r,t) = --D~r r=R' where D is the intrinsic diffusion coefficient. Taking the Laplace transform with respect to t,
aC f(r,.) = - o ~ ; ( r , p )
/"
I,=R"
t, T(R, t), is given by
The total amount leached through a sphere of radius R after time aC
,
2
T(g,t)= - JoD~r (R,t)'4nR dr'. Denoting the Laplace transform of f(t) by - ~ [ f ] we use the result
~
f(t') dt'
= £~o[f]
P to obtain
T(R, p) -
4rcR2D OC (g, p). p dr
For a sphere at radius R > R~,
dC2 f ( R , p) = - D 2 ~-r (R, p).
Denoting
I
sinh [si(R l - a ) ] (~!D'-D2) R1
s2D2 coth [s2(R2_R,)J }_cosh [sl(Rl_a)J(slD,)l
as z, =
DzDxslCoa ! . - s 2 cosh [s2(R2-R)] zp sinh [s2(R2--R1)] ( R
sinh
[s2(Rz-R)].~ R2 J
and
T= zp2 4nD1D2sC l[s2(R °asnih 2 -- R1) ] { - Rs2 c°sh [s2(R2- R)]-sinh [s2(R2- R)]t"
APPENDIX C Solution of Darcy's equation in a spherical geometry with a boundary at radius R 1 separating regions of different permeabilitics :
Vpi= -~
/ V2p'=O
i = 1,2,
(C.1)
V'u~ = 0 where p~ and u I denote the pressure and velocity of fluid in the pores of the backfill/waste, and P2 and u 2 denote the pressure and velocity of fluid in rock pores; # 1,2 and K 1.2 denote the viscosities and permeabilities in the two regions. The boundary conditions arc (i) p is continuous at r = R~, (ii) u is continuous at r = R~ and (iii)
u ~
U o as r ~
oc.
158
S . M . SHARLAND and P. W. TASKER
Transforming equation (C. 1) to spherical coordinates,
1L(r2OPi~+ l ~(sinoOPi~=O. r 2 Or \ Or/ r 2 sin 0 80J
(C.2)
Assume p = R(r)O(O)for some R and O, then p = r - "- ip. (cos O) or
p = r"P. (cos 0),
where P.(O) is the nth Legendre polynomial, p must remain finite at r = O, therefore
Pl = ~ A.r"P.(cos 0). n=O
Now P2 = 0
o r cos 0 as r -~ 0% thus /~Uo P2 =
r cos O+ ~ B.r-"-lP.(cos 0).
K2
n=O
Using condition (i),
~o pU ° ~ ~., A.R]P.(cos O)=~-2 Rt cos 0 + Z B.R;" 1pn(cos 0). n=O
(C.3)
n=0
Using condition (ii), o~
K2~
K1 ~" nA.R{"-lP.(cos O) = - U o cos 0 + - -
(n+l)B.R'~"-2p,(cosO).
,u n=o # n=o But A. = B. = 0 for h ~ 1, thus, solving equations (C.3) and (C.4) for A1 and B1 and using P1 (cos 0) = cos 0, we find
3l~Uor cos 0 Pl--
2K2+K1
and
P2
#Uo_ cos OFr + R 3 ( K 2 - K , ) ] K2 [_ r2 (2K2+K1) J.
Hence, U1
3UoK1
- - ( C O S
2K2 + Kx
0, sin 0,0)
and =Uo(_COS0[ U2
1
2R~ , K 2 - K , , ] r~ ( ~ J '
sin0[l+R
3 (K2-K,)]0). ra (2K2+K0J'
(C.4)
0306-4549/86$3.00+ 0.00 Pergamon Press Ltd
MIGRATION PROCESSES IN A MODEL DEEP REPOSITORY FOR THE DISPOSAL OF INTERMEDIATE-LEVEL N U C L E A R WASTE S. M. SHARLAND a n d P. W. TASKER Theoretical Physics Division, AERE Harwell, Oxon OX11 0RA, England (Received 17 April 1985) Abstract--It is difficult to conceive Of radionuclides escaping from a repository by any means other than migration in groundwater. Simple models of the repository are constructed and various migration processes are identified and assessed, according to the flow speed of water through the repository. Diffusion in static water and advection in fast flows are considered separately initially, but later we examine the effect of slow flows in which both these processes contribute to the removal of radionuclides. Concentration profiles across the repository, fluxes of nuclides and total losses are obtained from the analysis. We investigate the time scales necessary for the steady state to be achieved in the repository and conclude that flow speed is roughly inversely proportional to this time scale, i.e. faster flows establish a steady state sooner than slow ones. We also assess the sensitivity of the results to the physical properties of the components of the repository.
NOMENCLATURE a= Co = C1 = C2 = C8 = D1 = D2 = /5 = F = G= K1= K2= Kd =
Pe = RI = R2 =
Uo = u= v= w= ct = 2= 4, = p = ~b1 = ~'2 =
Radius of wasteform (m) Concentration of nuclides in wasteform (M) Concentration of nuclides in backfill (M) Concentration of nuclides in rock (M) Concentration of nuclides at backfill surface (M) Intrinsic diffusion coefficient in backfill (m 2 s-1) Intrinsic diffusion coefficient in rock (m 2 s- 1) Apparent diffusion coefficient (m 2 s- 1) Groundwater volume flux ( m 3 s - 1) Groundwater volume flux (m 3 s- t) Permeability of backfill and wasteform (m 2) Permeability of rock (m 2) Mass-based equilibrium distribution coefficient (m 3 kg- 1) Peclet number Radius of backfill sphere or cylinder (m) Radius of sphere in rock (m) Darcy velocity of groundwater (m s-1) Darcy speed (m s- t) Darcy speed (m s- 1) Velocity of fluid in pores (m s- 1) Capacity factor Radioactive decay constant Porosity Density of wasteform (kg m - 3) Stream function of flow in backfill/waste (m 3 s- 1) Stream function of flow in rock (m 3 s-1)
1. INTRODUCTION It is envisaged t h a t long-lived intermediate-level radioactive waste (ILW) will be buried in deep u n d e r g r o u n d sites a n d s u r r o u n d e d by a n u m b e r of engineered barriers. T h e repository itself a n d the s u r r o u n d i n g geology are referred to as the near field 141 ANE 13:3-E
while the r e m a i n i n g rock structure c o n t a i n i n g routes to the surface is k n o w n as the far field. The aim of the 'source t e r m ' studies is to develop a realistic model of the repository from which the loss of radionuclides from the near field m a y be calculated, using experimental data. In considering the possible m e c h a n i s m s by which the radionuclides m a y be t r a n s p o r t e d from the source, the model m u s t i n c o r p o r a t e such details as the content of the waste, the form a n d physical properties of its packaging a n d other engineered barriers a n d the type a n d structure of the s u r r o u n d i n g geology. Careful a t t e n t i o n m u s t be paid to the chemical conditions present in the repository since factors such as the a d s o r p t i o n a n d solubility of the various constituents will have a considerable effect o n the overall loss rates. A repository will consist of the waste c o n t a i n e d in a solid matrix, p r o b a b l y concrete. This is c o n t a i n e d in a metal can which in t u r n is s u r r o u n d e d by a backfill material, again p r o b a b l y concrete. B e y o n d t h a t is the s u r r o u n d i n g rock whose mineral c o m p o s i t i o n depends on the particular chosen site. It is difficult to imagine any loss of radioactive material from this site other t h a n by m i g r a t i o n t h r o u g h groundwater. O n e of the barriers to loss of waste material is the choice of a dry site. However, we will assume the p e n e t r a t i o n of the repository by water as a prerequisite to the leaching of the waste. Nevertheless, some corrosion of the canister m a y already have t a k e n place by the action of residual pore water in the backfill. The c o m p o s i t i o n of the g r o u n d w a t e r t h a t will penetrate the site is determined by the n a t u r e of the geology of the region. As it
142
S.M. SHARLANDand P. W. TASKER
permeates the repository it will equilibrate with the backfill. This will lead to important changes in composition, particularly in pH. Ionic diffusion will occur between the backfill and outer minerals, leading to changes in the water chemistry in both regions over a fairly long time scale. No loss of nuclides will occur until the metal canister surrounding the wasteform has been penetrated. One part of the source term analysis will, therefore, be an assessment of the lifetime of the can under the chemical conditions of the repository. Once water has penetrated the concrete wasteform, dissolution of the radioactive elements will occur. These elements are then lost by diffusion and if there is a flow of water by convection. Inclusion of the many complex processes involved in the evolution of the near field of an ILW repository is likely to necessitate a large numerical model. Initially, however, we consider a simple idealized model that is, nevertheless, capable of incorporating all the necessary phenomena as data become available. We construct this model with a view to obtaining analytic solutions. In this way, we can not only determine the magnitude of concentrations, fluxes etc. but can also ascertain the time scales involved in their progressions. In Section 2, we set up our idealized model of the waste repository. In Section 3, we consider diffusion of the nuclides from the wasteform in the stationary groundwater within the repository. Section 4 considers the effect of a fast flow through the model and in Section 5 we consider a situation in which both advection and diffusion of the nuclides take place. 2. THE M O D E L
The idealized model considered initially comprises concentric spheres representing the wasteform, backfill and surrounding rock, as illustrated in Fig. 1. This is not intended to represent the complete repository, but rather the behaviour ofa wasteform near the edge of the site. We have assumed that water has penetrated and that the canister is no longer intact. We also assume that the groundwater has equilibrated with the backfill and wasteform and that the concentration of radionuclides within the wasteform is determined by the equilibrium solubilities of the species and is constant throughout. At time t = 0, we imagine that the concentration outside the wasteform is zero and the concentration within the wasteform is held constant throughout time, i.e. it acts as a permanent source of nuclides. Thus we are ignoring the leaching rate, which is clearly a pessimistic assumption. The source should stop when all the nuclides have been released and this can be incorporated when the exact inventory of the contents is known. Initially we consider only one species of
~
Rock
(FIL
I
\o,
ROCK
Z Fig. 1. Schematicillustration of the simple idealizedmodel of the waste repository.
radionuclide, although the extension to several noninteracting species is trivial. The results presented here have also neglected radioactive decay. This can be readily included but the results for the very long-lived isotopes will not vary significantly. 3. DIFFUSION 3.1. Governing equations We consider loss of nuclides from the source by diffusion only. The equation governing the concentration in the repository is given by
~i O t (r, t) = DiV 2 Ci(r, t ) - 2~iC~(r,t) + Q(r, t), i = 1,2,
(1)
where Cl(r,t) represents the radionuclide concentration in the water contained in the micropores of the backfill and C2(r, t) represents the concentration in the water in the rock. Diffusion in a sorbing, porous medium is described by two parameters, the intrinsic diffusion coefficient, D and the capacity factor, ~. The capacity factor reflects both the physical retention of the nuclide in the porosity (~b)and the bulk chemical retention due to equilibrium sorption. Thus, o~ = c~ + pKd,
where p is the density of the wasteform and K d is the mass-based equilibrium distribution coefficient for the combined solid phases in the wasteform (Lever et al., 1982). The term Q(r,t) represents the phenomenon of leaching from the wasteform. This term may be
Migration processes in a deep nuclear waste repository reasonably ignored if we assume that the water has equilibrated with the wasteform. Clearly we are taking a pessimistic view of the situation in neglecting this. The term 2aiC,(r, t) represents the radioactive decay of the radionuclide under consideration; 2 is the radioactive decay constant and is related to the half-life t~ by 2 = In (2)/t~.
143
where Co is the concentration of nuclides in the wasteform. Conditions (v) and (vi) constrain the concentration and flux of nuclides to be continuous at the backfillrock boundary. We solve equations (2) and (3) with boundary conditions (i)--(vi)using a Laplace transform technique. Details are given in Appendix A. By transforming in the variable t and solving for C~(r,p), C2(r, p) we obtain
CI(r'P) = aC° rp exp [ s l ( r - a ) ] I1 and
C2(r,p) =
DlslCoa sinh [s2(R2-r)] rpz sinh [SE(R2-R1)]
where
and S1 ----(pctl/D1)t-
and
Initially, for simplicity we consider a nuclide with a sufficiently long half-life that its decay will not be significant over the time scales of most interest, and we may neglect this term. However we shall note later on that this phenomenon may be easily included in the analysis. We transform equation (1) to a spherical geometry, coordinates (r, 0, tk) and obtain
~CI (r,t) ~t
Di ~2 = ~ - ~ r 2 [rCt(r,t)]
a <~ r <~ R 1
c3t
De ~2
r cgr2 [rC2(r't)]
R1 <~ r <~ R2, (3)
assuming symmetry in 0 and ~b. The parameter /31.2 is the effective diffusion coefficient of the media and is given b y / ) = D/ot. The boundary conditions of the problem are (i) Cl(r,O) = 0 a <~ r <~ RI, (ii) C2(r,0 ) = 0 R1 ~< r ~< R2, (iii) Cl(a ,t) = C O t >l O, (iv) C2(R2, t) = 0 t/> 0, (v) CI(RI, t ) = C2(RI, t) t >1 0 and (vi) D I ~ ( R I ,
t)= D2~L(RI,
t)
t~>0,
Note that inclusion of the radioactive decay term in equation (1) will merely have the effect of multiplying Cl(r, t) and C2(r, t) by e - l t The flux per unit area of escaping nuclides is given by
~Ci fi(r, t) = -- D i ~-r
i = 1, 2
(2)
and
dCz(r,t)
s2 = (PCtz/Dz) ½.
and an expression for the Laplace transform, f~(r, p) may be readily obtained from the above. Similarly, an expression may be derived for the transform of the total amount of nuclides passing through the sphere after time t, T(r, p), where
T~(r,t) = - |
30
~C, D i-~-r (R,t )A dt'. ?
A is the surface area of the sphere radius R. Details of the derivations and some examples of the transforms are given in Appendix B. Analytic inversion of all these transforms would be extremely difficult, so we have inverted them numerically using an approach used by Talbot (1979) which involves integrating the inversion integrals along the steepest descent contour of the function g(t) = lit.
144
S.M. SHARLANDand P. W. TASKER
3.2. Results Parameters used for initial runs. The parameters chosen for the repository size were a = 0.5 m,
1.8--
?E 1.5-
R~ = 2 m
T , 1.2-
and ~0,9--
%
R2 at infinity. Initially we consider a fairly dry cement as backfill (with a water/cement ratio at mixing of about 0.3), with intrinsic diffusion coefficient Dt = 10 -~2 m 2 s -~ (Atkinson et al., 1983). We take cq = q~l = 10-1 for such a cement (i.e. consider the progress of a nonsorbing ion such as 1291). We assume the rock to be of granitic type and take D 2 = 10 -13 m 2 s -~ and ~t2 = q~2 = 10-2 (Lever and Bradbury, 1983), and we take the source concentration Co = 10- 7 M which is a typical limiting solubility of a species in the wasteform. Clayton and Rees (1983) derive values for the limiting solubilities of the actinides under disposal conditions in the range 10 -22 to 6 x 10 -6 M. However, we note that the analytic expressions obtained for the Laplace transforms of the concentrations etc. depend linearly on Co so an alternative choice of this parameter simply rescales the results. Results from initial runs. Using the data obtained from inverting the Laplace transforms, we have plotted concentration profiles across the repository at various times and obtained graphs showing the variations in flux and total loss with time. Figure 2 shows a profile plot at time 3000 years. The change in slope at r = 2 m reflects the condition that the flux of nuclides across the
WASTEFQRM BACKFILL 1.O D=10-12m 2 s-1
ROCK D=10-13m2 s-1
=o.1
"7 o.6o 0.3-I 500
1000
1 5 O O 2000 Time (yr}
2500
3000
Fig. 3. Flux of radionuclides per unit area through a sphere of radius 3 m against time (loss by diffusion only).
boundary is continuous. For the parameters used we obtain a change of a factor of 10 (increasing on the rock side). Figure 3 shows a plot of flux/unit area with time. The graph will eventually level out to a steady value. The time needed to achieve the steady state is studied in more detail later. Figure 4 shows a plot of total loss through a sphere of radius 3 m against time.
3.3. Sensitivity to parameters By changing the parameters individually or in pairs, one can assess the sensitivity of the results to the conditions present in the repository. (i) Concentration at source, Co. Since C~(r, p) depends linearly on Co then changing Co will only have the effect of rescaling the results. (ii) Diffusion coefficient in rock,/)2- Results for the granitic rock were compared with those obtained from
~ =OOl
0.8
1.25
1.00
x 0.6
~o 0.7.5
~
0.4
?,
o
0.50
0.2
~- 0.25
I 0
0.5
1.0
1.5
2.0
2.5
3.0
I ~z_...J 3.5 " 4.0
Radius (m) Fig. 2. Concentration profile across the repository surrounded by granite after 3000 years (considering loss by diffusion only).
O
30
60
90 120 Time (x 103 yr)
150
Fig. 4. Total loss of nuclides through a sphere at radius 3 m against time (diffusion only).
Migration processes in a deep nuclear waste repository WASTEFORM BACKFILL 1.0 D ,10-12m2 s-1
145
ROCK D : 2 x I O -11 m2 s-1 ¢ =0.146
@=o.1
0.8 _o x
0.6
8 0
20
40
~ 0.4
80 100 120 (x 103 yr)
Time
140
I
160
Fig. 6. Total loss of nuclides through the surface of a sphere in the surrounding sandstone at radius 3 m against time (diffusion only).
o 0.2
0
60
05
1.0
1.5
2.0
Radius
,
2.5 (m)
i
3.0
i
5.5
,,,_2..
5.5
greater than those through the granitic rock. Although Fig. 5 shows a shallower gradient at R = 3 m, D 2 is sufficiently large that the flux through this point and hence the total loss will be greater with a sandstone rock surrounding the backfill. (iii) Diffusion coefficient in backfill, D 1. We consider cements with higher water/cement ratios at mixing, e.g. D1 = 10 T M m 2 s -I and D 1 = 10 -1° m 2 s -1 and compare with the case of D 1 = 10-12 m 2 s - 1, keeping D 2 = 10 -13 m 2 s -1 and ~b2 = 10 -2. Figure 7 shows a plot of the concentration profiles for each of the three cases after 1000 years. As expected, the profile in the backfill is flatter for higher diffusion coefficients. Figure
Fig. 5. Concentration profile across the repository surrounded by sandstone after 3000 years (diffusion only).
using D 2 = 2 x 10 -11 m 2 s -1 and ~b2 = 0.146, which represents a typical sandstone (Lever et al., 1982). Figure 5 shows a concentration profile after 3000 years. The slope at the backfill-rock boundary decreases by a factor of about 10 in this case. Figure 6 shows a plot of total loss of nuclides against time. Comparing with Fig. 4, we see that the total losses are about four times
Z)l = 10-1o m2 s-1 1.00 ~
!1 D2 =I0-13m2 s-i
7,-
0.75
= -
-
x
~_ cJ 0.50
\o,=,o-,2::,
o
0.00
5
1.0
1.5
\
2.0
2.5
3.0
3.5
I
I
4.0
4.5
Radius (m)
Fig. 7. Comparison of the concentration profiles after 1000 years for three different cement mixes as backfill (diffusion only).
146
S.M. SHARLANDand P. W. TASKER 1.8
2
30
1.5
~25
r~O 1.2
620
0.9
~5 ~o
-6 0.6 I~ 0.5
~5 I 20
I 40
I I I 60 80 100 Time (x 103 yr)
I 120
I 140
300
600 Time
900
1200
1500
(xlO 3 yr)
Fig. 8. Total loss through a sphere at radius 3 m (in granite) against time, with a 'wet mix' cement backfill (diffusion only).
Fig. 10. Comparison of total losses from diffusion models with the outer boundary at infinity, 5 m and 3.5 m (calculated at radius 3 m) (diffusion only).
8 shows the total loss at radius 3 m for the case of D~ = 10-lo m 2 s-1. The total is slightly greater than for a 'drier mix' concrete reflecting the larger concentration gradient at r = 3 m (as shown in Fig. 7). (iv) Outer boundary condition distance, R2. So far we have considered the case of zero concentration at infinity. Now we investigate the situation of finite Rv This represents the possibility of having groundwater flow some distance from (but not directly through) the wasteform. All the radionuclides are imagined to be carried away in the stream, hence constraining C2 to be zero at this radius. Taking R2 = 5 m and 3.5 m and examining the short-time concentration profiles, we
find there is little difference between these cases and the 'R2 at infinity' situation. Figure 9 shows the steadystate profiles and Fig. 10 shows the variation in total losses for each of the three cases. Over the time scales of most interest these variations are not significant. F r o m the above analysis, we conclude that the concentrations and fluxes present in the repository are most sensitive to variations in the diffusive properties of the backfill and geology but are not adversely affected by changes in the boundary conditions of the problem. We may also conclude that the analysis is readily adapted to represent a wide range of physical conditions that may be present in the repository.
1.0
BACKFILL
I
ROCK
I
I 0.8
b-
'o ~ 0.6
.~_ E
~
0.4
0.2
o.o 0.5
] .1.0
I 1.5
2.0
I 2.5
I 3.0
5.5
4.0
4.5
I 5.0
Radius of westeform (m)
Fig. 9. Comparison of steady-state concentration profiles for the outer boundary of the diffusion model at infinity, 5 m and 3.5 m (diffusion only).
Migration processes in a deep nuclear waste repository
147
8
3.4. Comparison with the steady state In the steady state we may derive a simple formula describing the flux of nuclides from the repository. The flux at the surface of a sphere radius r is given by
A i
-
"
J = - - 4 n r 2 D i ~r • For the steady state J is constant over r, therefore integrating over r we obtain J
1 dr
D(r) 47zr2
dC
2 0
I
300
600
I
I
900
1200
I
1500
Time(xlO3yr)
_j[-1 (1 LDl
Rl--al #
~O2
= 4r Co
Fig. l 1. Flux of nuclides through a sphere at radius 3 m (in granite) against time, to show how it settles to the steady-state value represented by the dashed line (diffusion only).
and
4nCoDtD2Rla J = [D2(R1-a)+Dla]"
(4)
The fluxes obtained from the Laplace transform method agree well with this expression at long times. For the initial set of parameters considered it was found that the steady state was achieved after approx, l0 s years. Although periods of time of this order are not directly relevant, it is perhaps interesting to note how fluxes of nuclides from the wasteform settle to the steady-state value (Fig. 11). The sharp peak represents the passing of the concentration front and the associated steep concentration gradient at that time. After about 7000 years the steady-state approximation gives an optimistic value for the flux of nuclides. However, after such a long time many of the assumptions made during the mathematical analysis may no longer be valid. For example, the integrity of the concrete backfill can no longer be relied on. Over shorter time scales the steady-state approximation is a pessimistic view of the situation and since the results only differ by about 10-20%, it is reasonable to use this formula to estimate losses from the repository. Nevertheless, a full solution of the equations is needed to obtain the concentration profiles and their evolutions which are necessary if the detailed chemistry cf the problem is to be included.
3.5. Summary 1. Solutions to the governing equation are obtained using a Laplace transform method and numerical inversion, ignoring leaching and radioactive decay. Concentrations, fluxes and total losses from the repository are obtained. 2. Initial calculations using parameters for a typical dry cement embedded in a granitic geology,
indicate losses of about 104 x Co mol/150,000 years (assuming a permanent source of concentration C OM), with time scales to reach a steady state of the order of 10 a years. 3. The results are most sensitive to variations in the diffusive properties of the backfill and rock, but are not as dependent on the boundary conditions of the problem, i.e. the repository dimensions etc. 4. The analysis may be easily adapted to incorporate a wide range of physical parameters that might describe the conditions in the repository. 5. The fluxes (and total losses) from the repository may be approximated using a simple formula derived from considering the steady-state situation, despite the extremely long times necessary for this to be achieved. However, a full solution to the governing equations is necessary to obtain the concentration profiles and their progression with time. 4. ADVECTION
4.1. Governin# equations We now set up a simple model to investigate the effect of water flow through the repository. As before, we regard the waste, backfill and geology as concentric spheres, but now we assume there is a flow through the system. Flow through a porous medium is governed by Darcy's equation : Vp(r)=---
/re(r) and K
V'u(r)=0,
(5)
where p(r) represents the pressure field, and u(r) represents the Darcy velocity field;/~ is the viscosity of
148
S.M. SHARLANDand P. W. TASKER
the fluid and K is the permeability of the medium. We assume that the permeability of the wasteform and backfill are equal. We solve V2p = 0 in a spherical geometry with the boundary conditions (i) u is continuous at r = R 1, (ii) p is continuous at r = R1 and (iii) u ~ U o as r--* oo, where Uo is the velocity of the fluid far from the wasteform, and obtain the pressure field and hence velocity field (details are given in Appendix C) : 3UoK1 ul = ( 2 K ~ l j ( - c o s
0, sin 0,0)
and
o2
. f =
~1-
2R 3 ( K 2 - K 0 q
(6)
ok-cos V-- 7-
s m o / 1-4 -- - - / , L r (2K2 + K 1 ) J
o/ /
where Ul and u2 are the velocity fields in the backfill/wasteform and rock, respectively, and K t and K 2 are the permeabilities of the two regions. From these velocity fields the stream functions ~bl(r, 0) and ~//2(r,/9) may be derived : 3 U ° K 1 sin2 r2
/
2(2K2 + K1 )
~bl(r,O ) -
/
and
] [- (7)
~b2(r, 0) =
U° sin2 Or2 2
1
2R 3 (K2--K1)
1 w(r)=~u(r),
r 3 (2K2 + K 1 ) J ' J
The stream function, ~b, is derived from the mass conservation equation for an incompressible fluid which in the spherical geometry under consideration reduces to
1 0
flow (i.e. lines of constant stream function) for the cases : (i) K I = 10 -19 m 2, K 2 = 10 -18 m 2, i.e. backfill less permeable than rock; and (ii) K 1 = 10 -14 m 2, K 2 = 10-16 m 2, i.e. rock less permeable than backfill. These parameters were chosen as typical for the type of cements and geologies that may be chosen as repository components (Atkinson, 1983). We now consider transport of radionuclides due to fluid flow only, which will be a reasonable approximation for fast flows. We neglect loss due to diffusion completely. Since we are considering a steady flow, the path of a material element of fluid coincides with a streamline. The escaping nuclides will only follow the paths of those elements that have actually passed through the wasteform itself, since we are ignoring the possibility of any diffusion into the stream. Thus the region of contamination in the repository is bounded by the streamlines which coincide with the edge of the wasteform. This is schematically illustrated in Fig. 14. The concentration behind the concentration front will be constant in this region and at worst equal to the limiting solubility of the source Co. If the flow is too fast for chemical equilibrium to be reached between the groundwater and wasteform then the concentration will clearly be less than Co. The progression of the concentration front through the shaded region of Fig. 14 can be easily measured since the fluid velocity is known everywhere. For example, the time, T, taken for the front to reach a point at radius R may be evaluated by integrating 1/[w(r)lalong the streamline through that point, where w(r) is the velocity of the fluid in the pores ; w(r) is related to the Darcy velocity a(r) by
where q~ is the porosity of the medium. In general T approximates to T-
Re Uo'
1
c3 r 2 0~ (r2Ur) + (sin 0 uo) = O. r sin 0 O0
since the streamlines are reasonably parallel for most choices of K 1 and K 2.
We obtain ~k from
1 Ur
0$
r 2 sin 0 00
1 and
uo
0~b
r sin 0 Or'
where ur and u 0 are the radial and tangential components of u, respectively. Physically the stream function may be interpreted as follows : the flux of fluid volume across a curve joining two points is given by the difference in the stream functions evaluated at each point. Figures 12 and 13 show the streamlines of the
4.2. Sensitivity to parameters
To investigate the effect of the relative permeabilities of the backfill/wasteform and the rock we first calculate the flux density across the backfill, using the stream functions. The volume flux between two points is equal to the difference in stream function evaluated at the points. Consider two points A, B at the edge of the backfill, such that the line joining them is perpendicular to the flow. For K 2 < K 1,i-e.the rock less permeable than
Migration processes in a deep nuclear waste repository
/
149
ROCK K2=1~18m 2
c o ,m
-2 ROCK
-4
I
-8
-6
I
--4
]~
I
-2
Distance
F i g . 12.
~
0
I
2
4
J
6
(m)
Streamlines through the repository with the backfill and waste less permeable than the surrounding rock.
the backfill, the flux across the backfill, G is given by G = ffl(A)-~kl(B)
3UoK1R 2 (2K2+K0"
(8)
For K1 < K2, from equation (7), the flux across the backfill, F is given by
3UoK2R 2 F = ~2(A)-~,2(B) = (2K2+K~)'
(9)
In the case of a highly permeable backfill G approximates to G = 3Uo R2 and in the limit of an impermeable backfill, F = 3 UoR2/2. Atkinson (1983) uses these extremes to make estimates of the loss rates. In each case, he also includes an estimate for the diffusive loss into the stream and concludes that limits of extremes of permeabilities only result in about a 20-fold change in the release rate of nuclides. He also shows that the release rate is
4--
" ~
BACKFILL
8c o
A,'2=10-16 m2 0
121
-2
-4
I
-
-6
; ~ -4
I -2
~
I Distonce
F i g . 13.
~
0
I 2
~
I 4
6
(m)
Streamlines through the repository with the backfill and waste more permeable than the rock.
150
S.M. SHARLANDand P. W. TASKER
"-'--"-
ROCK
q
J J
Z///////////~
,I
o CI
_ _
i
ROCK
-4 -8
I
I
-6
--4
1__
I
-2
~
0
2
I
I
4
6
"
Distance (m)
Fig. 14. Schematic illustration of the model indicating the region of contamination by radionuclides for fast flows (loss of nuclides by advection only). independent of the permeability of the repository when K1/K 2 <~ 10 -2 and concludes that under typical conditions the permeability of the repository is not a major factor in determining the rate of loss of nuclides but there is some advantage in having K 1 approximately one or two orders of magnitude less than K2.
4.3. Summary 1. Solutions to Darcy's equation describing flow through a porous medium are obtained in the spherical geometry of our model, assuming the wasteform and backfill are of equal permeabilities. 2. Assuming a fast enough flow that diffusion may be neglected, we may identify a distinct region of the repository into which nuclides will penetrate. The concentration in the region behind the concentration front is independent of time assuming the source concentration remains constant. 3. Estimates of the losses due to advection with some diffusion are fairly insensitive to the relative permeabilities of the backfill and rock but it is preferable to choose the permeability of the backfill to be one or two orders of magnitude less than that of the rock. 5. A D V E C T I O N
AND
DIFFUSION
5.1. Governing equations The equation governing the concentration of radionuclides in the respository in the presence of a
slow water flow is now
cq~-[-(r,t)+u,~r,t)'VCi(r,t) = DiV2Ci(r,t)
i = 1,2. (10)
An analytic solution of this equation in a spherical geometry is difficult since the concentration now has angular dependence. However, one can obtain an analytic approximation in a cylindrical geometry following Chambr6 (1982). He assumes an impermeable infinite cylindrical wasteform embedded in a porous medium. We shall assume an infinite impermeable cylindrical backfill. Although, it is unlikely that a chosen backfill material would have zero permeability, this approximation can still provide a reasonable solution to the problem as shown in Section 4.2. Initially we consider flow normal to the axis of the cylinder but we shall later compare this with flow parallel to the axis. By solving Darcy's equation in a cylindrical geometry and setting K1 = 0, we obtain the flow field u(r, 0) given by u(r, 0) = U 0 [ - ( 1 -R~'~r2 ] cos 0 , ( 1 + r~-)sin 0, 0 ] = (u, v, 0).
(11)
In cylindrical coordinates equation (10) transforms to dC
8C
OC
~ +u(r,O)T/+v(r,O)So = D['O2C
1 8C
1 c~2C'~
( x ~ - - ~ - + r N + r - ~ 8 0 2 ],
(12)
Migration processes in a deep nuclear waste repository where we now drop the subscript 2 from the concentration, diffusion coefficient etc. denoting the properties in the rock. The boundary conditions are (i) C(RI,0, t) = C a 0~<0~<2r~ t~>0, (ii) C(@,O,t)=O 0~<0~<2r~ t~>0 and (iii) C(r,O,O)=O
Rl <~r<~ oo O<~O<<.2n,
where Ca denotes the concentration at the backfill surface which we assume to be constant. We assume the backfill cylinder has radius R1. We introduce the non-dimensional variables z=
Uot ~tR,
and
UoR1
Pe -
D
,
Both the concentration and flux now depend on angular position as well as radial position and time. The concentration is smallest at 0 = 0, where the flux is largest. Figures 15 and 16 show the concentration contours in the rock at 500 and 10,000 years, using parameters for a granitic rock with the concentration at the backfill surface, CB, equal to 8 x 10 -8 M and Uo = 1 x 10 -12 m s -1. The flow direction is from the right in each case. Figure 17 shows the progression of the flux with time at 0 = 0 and 0 = ~. The flux is greatest in the direction of the approaching stream since the concentration gradient is largest here. Chambr6 estimates the time necessary to establish the surface mass flux to 9 9 ~ of its steady value as T, given by
Uo T ctR 1
and equation (12) then transforms to c3z
rE COS
Or -- l + r 2 ]
1 f02C
r
1~C
1 02C~
002 } (13)
forR 1 ~< r ~< oo, 0 ~< 0 ~< 2 h a n d z >1 0. Peis called the Peclet number for the flow and provides a measure of the relative importance of the advection and diffusion terms in the governing equations. For a large Peclet number, advection is d o m i n a n t and for a small Peclet n u m b e r diffusion is more important. The form of equation (13) suggests an asymptotic solution for a large Peclet number. To obtain this we transform equation (13) with i+
to obtain t3C
t~C dC 2R cos 0 ~-~ + 2 sin 0 c30
- 1.2.
For a flow with Uo = 1 x 10-12 m s - 1, T -'- 750 years which is considerably less than the time necessary to establish the diffusive steady state studied earlier.
c~0
=Pe\~r 2 + ~ + 7
r=R1
151
t32C i_0(Pc_ 1/2)' dR 2
5.2. Comparison with flow parallel to the axis of the cylinder The angular dependence of the surface mass flux may be removed by averaging f(R1, 0, z) over the perimeter of the cylinder and an expression may be obtained for the mass transfer from the surface, ~/ . . . . . for a cylinder of length L (in the steady state) : ....
:
4.513 CBL(UoRID) 1/2.
We now make an approximation for the mass loss in the case of flow parallel to the cylinder axis. Following Chambr6's approach, we approximate the cylinder as a flat plate of length L and width 2nR r The flow is in the direction of the length of the plate. The flux, #/~on~,from the surface may be estimated as 5)/lo,g
2xRILDCa
(14) with boundary conditions (i)-(iii). Chambr6 obtains the following solution of this : 2
where 3 is the boundary layer width, and may be taken as
. (15)
The surface mass flux/unit area, f(R1, O,z) may be derived from this expression :
\Uo/ Hence, h)/iong "~ 21tRI(DLUo)I/2.
f(R,, O,z) = - D C3~r
Consider the ratio of these two estimates :
r=Rl
_ DCB y2Pe (coth R1 / ~
2~+cos
0)]'/2.
/~/ ....
~/long
4.513 ( L y / 2
2re \R1,]
;
152
S.M. SHARLANDand P. W. TASKER Concentration contours 7.7800E-08
I -x--
- - - -
5.5800E-08
--~--
------
5.2100E-
08
. . . . .
1.43,00E-08
- - - - -
7.9100E1.0000E-10
09
. . . .
1.0000E08 1.0000E-11
s: lil)!! %\\\ £;///
Q
I',,. -2
~
- ~
Z_Z~ ~SA~ /
-4
-6 -- l
I 1 4
1
I
12
0
I
I
2
4
Distance (m) Fig. 15. Plot of lines of constant concentration of nuclides in rock, assuming an impermeable cylindrical backfillwith flow normal to axis(from the fight).This plot representsthe situationafter500 years (adv¢ction and diffusion).
~/ . . . . > M~o,, for cylinders with L > 1.938 R1, that is the mass loss from a cylinder in a flow normal to its axis will be greater than that from the cylinder with flow parallel to the axis, if its length is approximately greater than twice its radius.
perpendicular to u, assuming K l = 0).
5.3. Estimate offlow regimesfor dominant diffusion and advection
where the volume flux is calculated from the stream functions (7) as before. We equate this quantity with the diffusive flux of nuclides from the surface of the backfill, Fo, and in doing so make the assumption that all the nuclides diffusing into the region between A' and B' are swept away by the stream :
The above analysis is valid for flows with a Peclet number greater than about 10, that is with upstream speeds greater than about 5 x 1 0 - l a m s -a, for the granitic type rocks and repository dimensions considered earlier. We can now consider the flow regimes in which each type of transport process is dominant. Returning to the spherical geometry used earlier we assume a flow with speed U o in the far field and estimate the width of region surrounding the backfill sphere, 6, in which the concentration of radionuclides is non-zero for the steady state. Following the approach of Atkinson (1983), we estimate the flux of nuclides due to advection, F A between two points in the rock, A' and B', at (Ra, n/2) and (R~ +6, n/2) (such that the line between them is
F A -~ (volume flux between the two points) • C2(R0- 2 - 3nU°R1C25 2 2 '
~C21
F° = --4nR1D2 ~-r r=R~ R2=RI +8
(this is the diffusive flux of nuclides through a unit length of the backfill surface). Thus, 8D 2
6-- 3UoC2(RI)
c3C2
cGr r=Rl R2=Rt+6
For sufficiently large 6, we may use the condition that is infinite but for faster streams we cannot solve
R2
Migration processes in a deep nuclear waste repository
153
Concontrotiocontours n 6 -
7.7800E-08 3.2100E-08 7.9100E-09 1.0000E-10
-----------x~
- - ...... . . . . ~ z ~
5.5800E-08 1.4300E-08 1.0000E-09 1.0000E-11
4-
j--
<"
BACKFILL 0
AND
8
~
---____
_~
-6
~'~
__
WASTE
>
I
-4
]Ill'
ix|
.!~lyif7
_ _
I
I
-2
0
I
2
I
4
Distonce (m)
Fig. 16. Steady-state plot of lines of constant concentrationin rock, with the same model as in Fig. 15(advection and diffusion).
directly for 6. Instead, by choosing values for 6, the corresponding Uo may be calculated numerically using the Laplace transform techniques from Section 3. We next compare approximate diffusive and advective fluxes at a fixed point in the rock at (R, n/2), where R < (R1 + 6). Considering a small volume e3 surround-
ing the point, we estimate F D = _~2D2 ~C2 ur
and FA = EC(R + e/2)-2 CiR - e/2)] UoeR
6 -
7
E >,
-~ 3 E
× 1 u_ 0
~
8-0
500
1 1000
O=r 1500 Time (yr)
I 2000
I 2500
Fig. 17. Comparison of the flux and nuclides at two points on the surface of the backfill, described in Fig. 15, against time. 0 = 0 represents the point directly exposed to the oncoming stream (advection and diffusion).
( R3), 1 + ~-g
and compare them numerically over a wide range of Uo and 6. We find that advection of nuclides is the dominant transport process for Darcy velocities of greater than approx. 1 x 10 - l ° m s -a and diffusion dominates for fluid speeds of less than 1 x 10- t3 m s - t. The fluxes obtained using this approach are not very useful in their present form, since the approximation is only valid for one direction, i.e. 0 = n/2. More satisfactory results may be obtained using the analytic approximation in a cylindrical geometry described earlier since this gives a single flux in each direction. This analysis is also extremely useful since it is valid in the velocity regime between the limits stated above. A typical groundwater flow would have a velocity in this intermediate region.
154
S.M. SHARLANDand P. W. TASKER
5.4. Summary
6. C O N C L U S I O N S
1. An analytic approximation is obtained for the concentration of nuclides in the rock, for the situation of flow perpendicular to the axis of an infinite impermeable cylindrical backfill. The concentration shows directional as well as radial and temporal dependence. An expression is derived for the flux from the backfill surface. A comparison with the flux of nuclides from the backfill sphere considered in Section 3 (where diffusion only was studied) shows a difference of about two orders of magnitude (the diffusive flux being smaller). The flux from the cylinder settles to a steady state after about 750 years (for a typical granite surrounding the backfill), compared with 10a years as seen in Section 3. 2. The loss of nuclides from a finite length cylinder in a flow normal to its axis will be greater than that from the cylinder with flow parallel to the axis, if its length is approximately greater than its diameter. 3. Diffusion is the dominant transport process by which radionuclides can escape from the wasteform when there is a flow of groundwater with a Darcy velocity of less than approx. l x l 0 13 m s -I. Advection dominates for velocities greater than about 1 x i 0 - lO m s- 1.
10-z5
P l = 10-12m2 s'-l@1:0.1 K I : 0 D2 = I 0-13m2 s-I '~2:0-01
c~ I
C B = 8 x 10-8 M
E
T
1. Considering migration of nuclides from the wasteform by diffusion only, we conclude that fluxes and total losses may be estimated with reasonable accuracy using the simple steadystate formula given in Section 3.4. However, a full solution of the governing equations is needed to obtain concentration profiles across the repository, which are necessary if the detailed chemistry of the system is to be included. 2. Over the time scales of interest a steady state will not be reached--about l0 s years is necessary. 3. The results are most sensitive to variations in the diffusive properties of the constituents of the repository, but do not show as much dependence on the repository size. 4. Considering migration of nuclides by advection in a fast stream, we may identify a distinct region of the repository, extending from the wasteform into which nuclides will be carried. 5. If the wasteform acts as a permanent source the concentration at a point in this region will be constant once the concentration front has passed. Thus, the time necessary to establish the steady state is considerably less than that in the case of static groundwater. 6. Estimates of the losses of nuclides are
lo -8`o
E ~,
10- 8.5
C vection
10 -9"° a) (n 10-9'5
8 O3
Diffusion controlled
l o -1°.° lO-16
l 10-15
I 10-I';
I
I
I
I
l
10-13
10-12
10-11
10-10
10. 9
...
Dorcy speed (rn s-1)
Fig. 18. An estimate of variation in fluxper unit area through the surfaceof the backfillwith the Darcy velocity of the groundwater.
Migration processes in a deep nuclear waste repository comparatively insensitive to the relative permeabilities of the backfill and rock. However, there is some advantage in having the permeability of the backfill one or two orders of magnitude less than that of the rock. 7. Combining the migration processes, we obtain an analytic approximation for the concentration of nuclides in the rock, surrounding an impermeable cylindrical backfill, with a groundwater flow normal to its axis. We derive a expression for the flux of nuclides from the backfill surface. This flux is larger than the diffusive flux from the backfill sphere. Generally the faster the stream is, the larger the flux will be. Also, we conclude from our analysis that the steady state is reached sooner with faster flows through the repository. 8. The loss of nuclides from a finite length cylinder in a flow normal to its axis will be greater than that from the cylinder with flow parallel to the axis, if its length is approximately greater than the diameter, i.e. the loss will be greater when the flow is parallel to the shorter overall dimension. 9. We estimate losses of nuclides due to diffusion and advection over a wide range of Darcy velocities and conclude that in a granitic rock, diffusion is dominant for velocities of less than approx, l x l 0 -13 m s -1 and advection is dominant when the fluid flows faster than about 1 × 10 -1° m s-1. Similar estimates for other
155
geologies and repository sizes may easily be made. Figure 18 shows an estimate of the variation in the flux per unit area from the surface of the backfill with Darcy velocity, for an impermeable 'dry mix' cement embedded in a granitic rock. For velocities less than about 5 x 1 0 -13 m s -1, the losses are diffusion controlled and independent of flow speed. For flows faster than this, the flux increases with the square root of fluid velocity. However, this increase will be checked at Darcy speeds greater than about 10-s m s-1 (Atkinson, 1983), since leaching in the wasteform now becomes important. In this regime the groundwater can no longer equilibrate with the contents of the source, and the flux again becomes independent of flow rate.
REFERENCES
Atkinson A. (1983) Report AERE-11077. Atkinson A., Nickerson A. K. and Valentine T. M. (1983) Report AERE-10809. Chambr6 P. L. (1982) Report UCB-NE-4017, Univ. of California, Berkeley, Calif. Clayton P. R. and Rees J. H. (1983) Report AERE-10988. Lever D. A. and Bradbury M. H. (1983) Report AERE TP1004. Lever D. A., Bradbury M. H. and Hemingway S. J. (1982) Report AERE-10614. Talbot A. (1979) J. Inst. Math. Applic. 23, 97.
APPENDIX
A
Solution of 8Ct r t
Dt
62
(')=~-~r2[rC,(r,t)]
~C2
(r,t)
D2 62 [rC2(r,t)]
= 7
~~r
a~r<<.R 1
(A.I)
RI <~r <~ R 2
(A.2)
with boundary conditions (i) (ii) (iii) (iv) (v)
Cl(r, 0 ) = 0
a<~r<~Rt, C2(r,O)=O Rt<<.r<~R2, C l ( a , t ) = C o t>lO,
C2(R2,t ) = 0
t/>0,
Ct(Rl, t ) = C2(Rx,t ) t >! 0
and ~C1
~C2
(vi) Dl~-r (Rl, t ) = D 2 ~ r r (Rl,t)
t >lO.
Taking Laplace transforms of equations (A. 1) and (A.2) and using boundary conditions (i) and (ii), one obtains D1 62 PCl(r'P) = r
-
~r 2 [rCt(r'P)]
(A.3)
156
S . M . SHARLAND and P. W. TASKER
and
pC2(r,p) D2 02
(A.4)
= -7- ~ r ~ [rC~(r, p)].
Transforming conditions (iii)-(vi) : (iii) (iv) (v)
~'~(a,p) = Col p,
(vi)
Dt ~-r (Rt,p) = D2~r-r (Rl,p).
C2(R2,p) = 0,
~t(Rt,p) = Cz(Rt,p)
and
0C1
~C2
From equation (A.3), rCl = A exp [(p/DOt/2r] + B exp [-(p/D1)l/2r].
(A.5)
Using condition (iii) and letting st = (p/D1) t/2
rC° P
A exp(sla)+ B e x p ( - s l a ) .
Thus equation (A.5) becomes
rCx
=
--aC_~
exp [sl(r -
a)] -
2B exp ( - sla) sinh [sx(r -
a)].
P From equation (A.4), rC2 = E exp [(p/D2)l/2r] + F exp [(p/D2)l/2r].
(A.6)
Using condition (iv) and letting s 2 = (P/D2) 1/2, 0 = E exp(s2R2)+F exp(-s2R2)
and
E = - F exp(-2s2R2).
Thus equation (A.6) becomes r C 2 = 2F exp (-s2R2) sinh [s2(R 2 - r ) ] .
Using condition (v),
aCo
--exp P
[sl(R 1 - a ) ] - 2 B exp(-saa) sinh [sl(R t - a ) ] = 2F exp (-s2R2)sinh [s2(R2-RO].
Using condition (vi),
2DIBexp(-sla)(~
1
DlaCo c°sh[sl(Rl-a)]t+~-l (sl-~)exp[s'(R'-a)] 4F exp (-s2R2)D2 R~
{;~
sinh [s2(R2-RO]+s 2 cosh[s2(R2-RO]
Solving for B and F and substituting into equations (A.5) and (A.6) we obtain
aCo
Cl(r,p) = - -
rp
exp [ s l ( r - a ) ]
/
sin
1
/ z
and
C2(r'P) = -
DlslCoa sinh [s2(R2-r)] rpz sinh [s2(R2-Rx) ] '
where
z=is,.,.~Sl,.~_a,~{,o, :2, s~o~oth~s:~-..~}-cosh ~.:~-~,~,s~o.1 and
sl = (p~q/DO 1/2 and s z = (po~JD2) 1/2.
•
}
.
Migration processes in a deep nuclear waste repository
157
APPENDIX B The flux of nuclides through a unit area of the surface of a sphere of radius R is given by
f(r,t) = --D~r r=R' where D is the intrinsic diffusion coefficient. Taking the Laplace transform with respect to t,
aC f(r,.) = - o ~ ; ( r , p )
/"
I,=R"
t, T(R, t), is given by
The total amount leached through a sphere of radius R after time aC
,
2
T(g,t)= - JoD~r (R,t)'4nR dr'. Denoting the Laplace transform of f(t) by - ~ [ f ] we use the result
~
f(t') dt'
= £~o[f]
P to obtain
T(R, p) -
4rcR2D OC (g, p). p dr
For a sphere at radius R > R~,
dC2 f ( R , p) = - D 2 ~-r (R, p).
Denoting
I
sinh [si(R l - a ) ] (~!D'-D2) R1
s2D2 coth [s2(R2_R,)J }_cosh [sl(Rl_a)J(slD,)l
as z, =
DzDxslCoa ! . - s 2 cosh [s2(R2-R)] zp sinh [s2(R2--R1)] ( R
sinh
[s2(Rz-R)].~ R2 J
and
T= zp2 4nD1D2sC l[s2(R °asnih 2 -- R1) ] { - Rs2 c°sh [s2(R2- R)]-sinh [s2(R2- R)]t"
APPENDIX C Solution of Darcy's equation in a spherical geometry with a boundary at radius R 1 separating regions of different permeabilitics :
Vpi= -~
/ V2p'=O
i = 1,2,
(C.1)
V'u~ = 0 where p~ and u I denote the pressure and velocity of fluid in the pores of the backfill/waste, and P2 and u 2 denote the pressure and velocity of fluid in rock pores; # 1,2 and K 1.2 denote the viscosities and permeabilities in the two regions. The boundary conditions arc (i) p is continuous at r = R~, (ii) u is continuous at r = R~ and (iii)
u ~
U o as r ~
oc.
158
S . M . SHARLAND and P. W. TASKER
Transforming equation (C. 1) to spherical coordinates,
1L(r2OPi~+ l ~(sinoOPi~=O. r 2 Or \ Or/ r 2 sin 0 80J
(C.2)
Assume p = R(r)O(O)for some R and O, then p = r - "- ip. (cos O) or
p = r"P. (cos 0),
where P.(O) is the nth Legendre polynomial, p must remain finite at r = O, therefore
Pl = ~ A.r"P.(cos 0). n=O
Now P2 = 0
o r cos 0 as r -~ 0% thus /~Uo P2 =
r cos O+ ~ B.r-"-lP.(cos 0).
K2
n=O
Using condition (i),
~o pU ° ~ ~., A.R]P.(cos O)=~-2 Rt cos 0 + Z B.R;" 1pn(cos 0). n=O
(C.3)
n=0
Using condition (ii), o~
K2~
K1 ~" nA.R{"-lP.(cos O) = - U o cos 0 + - -
(n+l)B.R'~"-2p,(cosO).
,u n=o # n=o But A. = B. = 0 for h ~ 1, thus, solving equations (C.3) and (C.4) for A1 and B1 and using P1 (cos 0) = cos 0, we find
3l~Uor cos 0 Pl--
2K2+K1
and
P2
#Uo_ cos OFr + R 3 ( K 2 - K , ) ] K2 [_ r2 (2K2+K1) J.
Hence, U1
3UoK1
- - ( C O S
2K2 + Kx
0, sin 0,0)
and =Uo(_COS0[ U2
1
2R~ , K 2 - K , , ] r~ ( ~ J '
sin0[l+R
3 (K2-K,)]0). ra (2K2+K0J'
(C.4)