MATHEMATICAL MODEL FOR HYDROTHERMAL CONVECTION AROUND A RADIOACTIVE WASTE DEPOSITORY IN HARD ROCK

Annalsof Nuclear Energy,Vol. 7, pp. 313 to 334
Pergamon Press Ltd 1980.Printed in Great Britain
A MATHEMATICAL MODEL FOR HYDROTHERMAL
CONVECTION AROUND A RADIOACTIVE
WASTE DEPOSITORY IN HARD ROCK
D. P. HODGK1NSON
Theoretical Physics Division, AERE, Harwell, Didcot, Oxon OXll 0RA, U.K.
(Received 16 October 1979)
Abstract--A mathematical model of thermally induced water movement in the vicinity of a hard rock
depositoryfor radioactivewaste is presented and discussed. For the low permeabilityrocks envisaged for
geological disposal the equations describing heat and mass transfer become uncoupled and linear.
Analytic solutions to these linearized equations are derived for an idealized spherical model of a
depository in a uniformly permeable rock mass. As the hydrogeological conditions to be expected at a
disposal site are uncertain, examples of flow paths are presented for a range of different permeabilities,
porosities, boundary conditions and regional cross-flows.
l. INTRODUCTION
The rock surrounding a depository for heat emitting
radioactive waste, will be subjected to elevated tem-
peratures and thermal gradients over distances of
several hundred metres for many centuries (Hodgkin-
son, 1977; Bourke and Hodgkinson, 1977; Hodgkin-
son and Bourke, 1978). Any water present in the frac-
tures of the rock mass will tend to rise because of
buoyancy effects. If the water had previously leached
away some of the radionuclides in the waste, then this
could shorten the water-borne leakage path and tran-
sit time back to the biosphere.
The purpose of this study is to make preliminary
estimates of the thermal convection currents around a
depository. In order to minimize the migration of
waste, depositories will be sited in rocks of low per-
meability where there is little water flow. Conse-
quently, the dominant heat transfer mechanism will
be heat conduction through the rock rather than con-
vective transfer due to the flow of water through the
fractures.
In this situation it is possible to simplify the calcu-
lation of thermally induced convective flows. Firstly,
the temperature profiles around the depository are
deduced from a solution of the heat conduction equa-
tion. The water flow is then determined using the pre-
viously determined temperatures as a driving force. In
this way the solution of a coupled non-linear problem
is reduced to the solution of two linear problems,
which is a much simpler task.
At the present time, very little is known about the
flow of water through highly impermeable fractured
A.ra.L 7]6--^
rock masses (GAIN Symposium Proceedings, 1978;
OECD-NEA and IAEA, 1979). Consequently, the
estimates described here make the simplest assump-
tion, namely that it can be described by Darcy flow
(Bear, 1972) through a saturated homogeneous iso-
tropic porous medium (Bourke et al., 1979).
In hard rocks the flow is dominantly through the
fracture system (Maini and Hocking, 1977), the intact
rock being impermeable by comparison. The porous
medium model should therefore be a reasonable
approximation on a scale that is large compared with
the mean separation between fractures, which is likely
to be in the range 0.I-10 m (Black, 1978). Since the
size scale of a depository is typically several hundred
metres, the porous medium approach should be suit-
able for predicting the overall far field effects of ther-
mal convection.
In the present simplified analysis, no account is
taken of the variation of the permeability and po-
rosity with space and time due to the initial state of
the rock or to the temperature and stresses induced
by the construction and operation of a depository.
Indeed, the experimental data for such an analysis is
not available. The permeability and porosity values
for deep crystalline rock may vary by many orders of
magnitude from site to site, in different directions and
at different depths (GAIN Symposium Proceedings,
1978; OECD-NEA and IAEA, 1979; Lundstrtm
and Stille, 1978; Burgess et al., 1979). This leads to
large uncertainties in the flows calculated here.
In the absence of any site specific information on
the present and future boundary condition for flow
313

314 D.P. HODGrdNSON
near the surface of the earth, two alternatives have
been considered. These are a free-surface (or water-
table) boundary condition, and a constant pressure
boundary condition as would occur, for example, if
there were a lake or sea above the depository site.
The depository is idealized as a uniform spherical
heat source with a heat output which decays with
time (Hodgkinson, 1978). This should give a reason-
able estimate of the flow around multi-level deposit-
ories such as the present Harwell conceptual design
(Beale et al., 1979). However, single level depositories
with their largely vertical temperature gradients will
have to be treated separately.
The plan of the paper is as follows. The assump-
tions on which the present analysis is based are de-
scribed in Section 2. Qualitative arguments concern-
ing heat and fluid transfer in a porous medium, and
their relevance to the present problem, are discussed
in Section 3. The simplifications to the general flow
equations that occur in low permeability rock are dis-
cussed in Section 4 and their general and specific ana-
lytical solutions are derived in Sections 5 and 6. Sec-
tion 7 describes some experimental results and theor-
etical models for the fracture permeability and po-
rosity of hard rocks. The present theory is applied to
the problem of geological disposal in Section 8, where
the sensitivity to permeability, porosity, boundary
conditions and regional cross-flow is investigated.
Some conclusions are summarized in Section 9.
2. ASSUMPTIONS
The present analysis assumes that the fractured
rock mass can be treated as an equivalent porous
medium (Bear, 1979) with constant values of the per-
meability and porosity. The fractures are assumed to
be saturated with water, which strictly speaking
would only apply after the recharge of the depository
volume has taken place (Lindblom, 1977).
2.1 Governing equations
The equations which describe heat and mass
transfer in a saturated porous medium are discussed
in many papers and books (see, for example, Bear,
1972; Combarnous and Bories, 1975). According to
Bear (1972), a sufficiently approximate mathematical
statement of the problem is
l[-
r , -~z8(rqz)1 O, (1)
rear (q) + =
8P I~
O~ + -kqz = -Pfg, (2)
-- + q, = 0, (3)
0r
(pC)~v T[ = " L r Or Or} + Oz2J
+ H(r,z,t)- ploCy q~ ~+ q~
~r OzJ'
(4)
dPz
- #ps,,, (5)
dT
where the equations have been written in cylindrical
polar coordinates with the z-axis pointing vertically
upwards. These equations have the following mean-
ing.
(1) Equation (1) expresses the steady state conser-
vation of mass of the fluid. The quantities q, and q:
are the radial and axial macroscopic velocity compo-
nents. The actual velocity of the fluid in the fractures
is
V~.z= q,,z/n, (6)
where n is the flowing porosity which is assumed to
be constant in space and time. In the application of
this theory, discussed in later sections of this paper, n
is taken to be the effective fracture porosity of the
hard rock site. In using the steady state conservation
equation (1) it is assumed that the flow transients aris-
ing from compressibility effects decay away on a
timescale that is short compared to the time evolution
of the temperature distribution.
(2) Equation (2) is the equation of motion of the
fluid (Darcy's Law) in the vertical direction. The usual
Boussinesq approximation is made, which amounts to
only keeping fluid density (ps) variations in the
buoyancy term (i.e. the right-hand side of this equa-
tion). This term is the driving force in the present
problem, whereas in isothermal flow, movement
results from externally imposed variations in the pres-
sure (P). The remaining symbols in equation (2) are
the fluid viscosity (#) which is assumed to be indepen-
dent of temperature, the acceleration due to gravity
(g) and the permeability (k) which is assumed to be
constant.
(3) Equation (3) is the equation of motion of the
fluid in the radial direction.
(4) Equation (4) describes the energy (heat) trans-
port in the saturated porous medium. The first term
on the right-hand side describes the contribution of
heat conduction, the second is the heat production
rate and the third represents the heat carried by con-
vection. This final term can be neglected in situations
of interest to geological disposal (precise conditions

Mathematical model for hydrothermal convection 315
for this to be true are given later). The average ther-
mal parameters are defined by:
(pC),, = nPsoCs + (1 - n)p,. C,, (7)
Fay = n F/ + (1 - n)F,, (8)
7., = ra,/(pc)~,, (9)
where Pso, Cj-, Fs (p,, C~, F,) are the density, specific
heat the thermal conductivity of the fluid (solid), and
T,, is the average thermal diffusivity. For the fracture
dominated flows considered in this paper, the po-
rosities are so small that the average quantities
defined above are essentially equal to the values for
the intact rock. The quantity, H(r,z,t), is the heat pro-
duction rate per unit volume.
(5) Equation (5) expresses the assumed linear
change of fluid density (Pl) with temperature rise. The
average fluid density is pyo and fl is the volumetric
thermal expansion coefficient. The temperature
dependence of Pl is only made use of in the buoyancy
term in equation (2), elsewhere the constant value PSo
is used (Boussinesq approximation).
A complete list of symbols with their units (S.I.) and
definitions, is given at the end of the paper.
The flow at any given time is described by the
stream function, ~O(r,z,t),which is defined by
q' = r 0z (10)
1 94,
qz r Or (11)
The streamlines (lines of constant stream function) are
a measure of the direction and quantity of flow at a
given time. For fixed radial coordinate (r) the amount
of water flowing through the cylindrical shell between
zl and zz is
where equation (5) has been used to change a density
derivative into a temperature derivative. While the
differential operator in this equation might at first
glance look like the Laplacian operator, it should be
emphasized that this is not in fact the case.
In terms of the stream function, ~k,the energy equa-
tion (4) can be written
(pc)., R,L; r + + H(r,z,t)
1Fc3~9 OT ~, (15)
+pro'G~LT~ ~r ~r ~z
2.2 Pathlines
The streamlines discussed in the last section de-
scribe the quantity and direction of water flow at any
given instant of time. However, as the temperature
distribution evolves in time, the streamlines will
change in magnitude and direction in response to the
altered driving force. Water will only flow along the
streamlines if the time taken for water to flow through
the depository volume is much shorter than the time
taken for the temperature distribution to evolve. This
is not generally the case in the present problem, so the
water will instead follow pathlines which are defined
by
dr Vr(r,z, t), dzdt = dt = V,(r,z, t). (16)
2.3 Initial and boundary conditions
In the absence of the heat source provided by the
depository it is assumed that there is no flow in the
rock mass. Consequently the stream function is in-
itially taken to be
f
Z2
2~ r.q,'dz = 2n[$(r, zx) - $(r, z2)]. (12)
zt
Similarly, for fixed axial coordinate (z) the volume
flow rate through an annulus between rl and r2 is
f,22r~ r'qz'dr = 2g[0(r2,z ) - 0(rl,z)]. (13)
l
Thus the difference between ~ at two points is a
direct measure of the volume rate of flow.
By cross-differentiation of equations (2) and (3),
the stream function can be shown to satisfy
(~ O~O)] oflkpfo OT1F(:32~ r O (14)
7La-U + ~ ~ ~, Or'
~0(r,z,0) = 0. (17)
In view of the lack of any site specific information,
and the uncertainty about how the conditions might
change during the millennia after disposal, two alter-
native flow boundary conditions near the earth's sur-
face are considered.
The first assumes that there is a free surface of
water (i.e. a water table) within the rock mass (Bear,
1972) whose instantaneous shape is described by
z = og(r). (18)
By definition, no water can flow across this surface, so
it is a streamline, i.e.
~O(r,ogr), t) = O. (19)

316 D.P. HOtX;rdNSON
Also, the pressure is constant on the free surface
which from equations (2), (3), (10) and (11) implies
that
cN/(r, co(r),t) y a~(r, co(r),t) k.p:~g'r]
ez L ~r + -
~r)
x = 0. (20)
~r
In keeping with the Boussinesq approximation,
p:(r, co(r), t) has been replaced by Pyo in equation (20).
Note that two conditions namely (19) and (20) must
be satisfied at the boundary. The shape of the water
surface is unknown a priori, but forms part of the
solution.
The second boundary condition examined in this
paper assumes that water at the surface of the earth
(taken to be horizontal) remains at constant pressure.
Thus
o~(r,d, t)
0, (21)
~z
where z = d corresponds to the surface of the earth.
This boundary condition could arise, for example, if
there were a lake or sea above the depository. Also it
might by a reasonable approximation if the rock near
the surface has a permeability considerably greater
than that at depth (Bourke and Hodgkinson, 1979). It
contrasts with the free surface boundary condition in
that it allows water to pass freely through the surface.
The initial and boundary conditions described
above apply to the axisymmetric flows caused by the
heat source alone. In order to examine the effect of a
regional groundwater flow, some calculations are pre-
sented which include the effect of an additional hori-
zontal pressure gradient.
~P
~x - 6" Pso"g, (22)
where x is a horizontal coordinate and b is the re-
gional pressure gradient in metres of water per metre.
From equation (3), this gives rise to a horizontal flow
k
qx = -" 6.Pso.g. (23)//
As the Peclet number is small (see Section 3) this flow
does not affect the temperature field, and the effect of
combined thermal and regional flows can be found
from the principle of superposition. It is clear that the
additional pressure gradient equation (22) cannot
satisfy the constant pressure boundary condition
(equation (21))and can only approximately satisfy the
free surface boundary conditions. However, the use of
more elaborate regional flows which satisfy the
boundary conditions is not considered to be justified
in view of the preliminary nature of the present work.
The initial temperature distribution in the rock is
assumed to be given by
T(r, z, O) = T~ - ~(z - d), (24)
where T, is the mean surface (z = d) temperature and
the second term allows for a linear increase of tem-
perature with depth due to the geothermal gradient.
An isothermal boundary condition is assumed to hold
at the surface of the earth, i.e.
T(r, d, t) = T~. (25)
3. QUALITATIVEDISCUSSIONOF HEAT
TRANSFER AND FLUID FLOW
The heat emitted by the decaying radionuclides can
be transferred through the rock mass by (i) conduc-
tion, (ii) free (i.e. thermally induced) convection and
(iii) forced convection due to the regional ground-
water flow. The heat flux per unit area for conduction
is
Fay "AT
Jcond -- , (26)
L
while for convection it is
d.... = q' P/o" Cs" AT, (27)
where AT and L are characteristic temperature and
length scales.
For free convection the macroscopic velocity, q,
arises from buoyancy forces and is therefore largely
vertical and given bv
qt,ee -- if" t" P:o" k" AT (28)
while forced convection is driven by the regional
pressure gradient and is consequently largely
horizontal and from equation (23) is given by
k.p: o .g.t~
qforced -- (29)
#
From these simple formulae, dimensionless ratios
which illustrate the importance of the various heat
transfer mechanisms can be constructed. Firstly, the
ratio of heat transferred by free convection to that by
conduction (the Rayleigh number, Ra) is, from equa-
tions (26)-(28),
k" fl'ff" p~o. Cy. L. AT
Ra = (30)
p-Fay

Taking the characteristic temperature and length
scales to be AT = 100°C and L = 250 m, together
with the physical property data listed in Table 2, the
Rayleigh number has the value 2.4 x 10-2 for a per_-
meability of 10-t6m 2. The temperature field estab-
lished by conduction is therefore not significantly per-
turbed by free convection. It should be noted however
that in safety analyses of geological disposal, it might
be thought necessary to choose a pessimistically large
value for the permeability. If this was more than two
orders of magnitude greater than 10- ~6m2 then free
convective heat transfer could become important and
the resulting flow patterns could be significantly dif-
ferent from those considered in this paper.
The importance of forced convective heat transfer
to conduction (the Peclet number, Pe) is, from equa-
tions (26), (27) and (29),
Pe = k.g.p}o.Ci.f.L
(31)
Evaluating this for a regional pressure gradient
8 = 10-3, and the rock and fluid parameters used
above, gives a Peclet number of 6.1 x 10-4. Thus the
natural water flow has an insignificant effect on the
temperature field.
The ratio of free to forced convective heat transfer
is, from equations (27)-(29),
free convection = fiAT
(32)
forced convection 6
If the regional gradient 6 = 10-3, then this ratio is
greater than unity for AT/> 2.6°C. However, in this
qualitative discussion the precise meaning of AT is
not defined.
4. LOW RAYLEIGH NUMBER EXPANSION
In the last section it was shown that the Rayleigh
number is likely to be much less than unity for situ-
ations of interest to geological disposal. It is therefore
appropriate to make a perturbation expansion in this
small quantity and keep only the leading order terms
(Hickox, 1977; Cheng and Lau, 1974).
For this purpose it is convenient to define the
dimensionless quantities
R = r/L, (33)
Z = z/L, (34)
z = t'y,,/L 2, (35)
O(R, Z, z) = T(r, z, O/AT, (36)
e(g, Z, z) = O(r,z, t)/otL, (37)
where L and AT are the length and temperature
scales, and
ct = Faff(pio. Cs), (38)
is the effective thermal diffusivity. In terms of these
quantities the equations for the stream function (14)
and energy transport (15) become
o(1 dv'~ 1 d~e go
ff~ g.~-~] + ~-. O~~ = Ra" ~,
and
~-= "~-~ R" + ~]
(39)
lrde.do g~P do] H'L 2
+RLOZgg-gR'~J + r.,.aT'
(40)
where the Rayleigh number, Ra, is defined in
equation (30).
As mentioned above, at small Rayleigh numbers
it is convenient to write
and
O(R, Z, z) = ~ Ram. Ore(R,Z, T), (41)
m=O
~(R, Z, z) = ~ Ram. ~m(R,Z,z), (42)
m=I
where by virtue of equation (17) there is no
zeroth order term in the expansion of e. Substitution
of these expansions into equations (39) and (40) and
equating terms of equal order in the Rayleigh
number, gives
& = '~ Tg/+ dz~] + r~,. A~' (43)
gO,
Llde, do0 de, Odd]
+ RI dZ g--R-- d--R dZ J' (44)
dO2
g'C " " " ~
dOo 1 d2 x Lf i
dR - R gZ2 + dRkR dR ]' (45)
got 1 d2~2 g (1 d~P2~
dR - R-" dZ~- + ~-" R" ~-], (46)
002
0R .....

318 D.P. HODGI~NSON
For the present problem, only the leading order
terms (i.e. Oo and ~) need be evaluated, although
higher order terms could be calculated if required.
The leading order contribution to the temperature
field (Oo) is determined by equation (43) which is just
the transient heat conduction equation. Thus at low
Rayleigh numbers the temperature field is determined
dominantly by conduction, as suggested by the quali-
tative arguments of Section 3.
The leading order contribution to the stream func-
tion W1) is determined by the differential equation
(45). The driving term of this equation is proportional
to the radial derivative of the zeroth order tempera-
ture. It is therefore independent of the initial tempera-
ture in the rock (equation (24)) as this only varies in
the axial direction.
The shape of the free surface can also be expanded
as a power series in the Rayleigh number, to give
f2(R) = co(r)/L = D + k Ra"'g2m(R), (47)
ra=l
where
D = d/L, (48)
and it has been assumed that the free surface is in-
itially coincident with the earth's surface. In terms of
the dimensionless free surface, O.(R), the boundary
conditions (19) and (20) become
and
where
~U(R,~(R), r) = 0, (49)
c~ I0~ I c~(R)
~Z ~- + Di'R • ~3R =0' (50)
Di - kPz° "g" L (51)
0~•,tl
is the discharge number which is a measure of the
imposed pressure forces to the viscous force (Cheng
and Lau, 1974).
Substitution of equations (42) and (47) into these
boundary conditions yields, to first order in Ra,
and
%(R, D, r) = 0, (52)
i D@I(R,D,z) _ Di. dill(R) (53)
R c3Z OR
The first condition on ~P~, expressed in equation
(52), is that the surface of the earth is a streamline and
so there is no flow across it. In other words it is an
impermeable surface to the present order of approxi-
mation. The boundary condition (49) on a surface of
unknown shape has been replaced, to the present level
of approximation, by a condition on the unperturbed
free surface Z = D. This linearisation of the boundary
condition greatly simplifies the mathematical analysis.
The second boundary condition satisfied by q'l,
equation (53), defines the leading order contribution
to the change in shape of the free surface. It can be
rewritten as
~co(r) _ ~ . _1. d~k(r,d,t)
dr k" Pfo'g r c~z
it 1 t~P(r, d, t)
- q,(r, d, t) = - - ,
k' Pzo"g Pfo'g Dr
so that (54)
P(r, d, t)
c~(r) = d + - - (55)
Pro "g
Thus the shape of the free surface is to a first approxi-
mation given by the pressure (in metres of water) that
would build up at an impermeable boundary at the
surface of the earth.
5. GENERAL SOLUTIONS FOR A SPHERICALLY
SYMMETRIC HEAT SOURCE
When the porous medium contains a spherically
symmetric heat source, the low Rayleigh number flow
equations discussed in the last section possess simple
general solutions. These are derived and discussed
below.
At low Rayleigh numbers the stream function
~,(r, z, t) satisfies the differential equation (see equa-
tion (45))
_1. a2@
r Dz2
~(1 d~b) 9.fl pro . .( 5 6 ) .-k 0T
--+~r r'~ = It ~r
It is important to note that the initial temperature
distribution (24), which is independent of r, does not
affect the driving term in equation (56) which is pro-
portional to OT/c~r. Only the temperature rise is there-
fore important in causing flow.
5.1 Infinite medium
For a spherically symmetric heat source in an infi-
nite medium the temperature rise, which is obtained
by solving the transient heat conduction equation
(43), will itself be spherically symmetric, i.e.
T = T(s), (57)

where
s = (r2 + z')~, (58)
is the spherical radial coordinate.
It is therefore convenient to solve the problem in
spherical polar coordinates (s, ~b)which are related to
the previously used cylindrical polar coordinates (r, z)
by
and
r= s.sin~b, (59)
2 = 5.COS ~b.
In this coordinate system equation (56) becomes
1 [~2q,(s,,) 1 ~q,(s,,)
s" sin2~bL Os2 + s2 ocp2
(60)
(64~ giving
lfoG(s) = s ds'" s'2" T(s'). (67)
The final result for the stream function associated
with a spherically symmetric temperature distrubution
in an infinite medium is therefore
~b(r,z) 9"fl'Pf°'k r2
= -- .G(s). (68)
s 2
By an analogous argument the differential equation
describing the pressure rise at low Rayleigh numbers
g fl OTV2P = P:o" " "~z' (69)
can be shown to have the solution
_ cot q~. O~O(s,q~)/-I_ O.fl.pso.k OT(s) (61)
s2 ,94~ J u ,~s'
By choosing a trial function of the form
qJ(s,(o)- O'fl'PY°'k G(s).sinN'~b.cosS2~b, (62)
/l
it is easily shown that the choice Nt = 2, N2 = 0
gives a particular integral to equation (61) if the
function G(s) satisfies the ordinary linear differential
equation
¢32G(s) 2 OT(s)
c9s2 sZ G(s) = s" ds (63)
The general solution to this nonhomogeneous
equation can be derived using standard techniques
(Kreyszig, 1967)to give
s 2 ff" Or(s')G(s) = - ~ ds' " Os'
1 ds'" s'3" OT(s') (64)
3s s~ Os' '
where sa and sb are arbitrary constants.
For a non-singular heat source in an infinite
medium, OG(s)/Os must be finite as s ~ 0 and s---,
in order to avoid singularities in the velocity field.
Applying these boundary conditions to equation (64)
gives
So = o0, (65)
and
sb = 0. (66)
Integration by parts may be used to simplify equation
Z
P(r,z) = Pro "O "fl'fi" G(s), (70)
in an infinite medium.
The radial and axial velocities obtained from
equation (68) using equations (10) and (11) are
g "fl'Pyo" k . G(s) (71)q, = " ~-. T(s)- ,
Iz
and
qz
g'fl'p:o'k 1 V 2
u ~f ~r .r(s)
(2z2 - r2) G(s)] (72)
+ fi
Along the vertical axis of the sphere (r = 0) the
upward velocity reduces to
q,(r = O, z) = g" fl "p :° "k 2P ~. G(z). (73)
Thus at the centre of the sphere the upward velocity
is proportional to the local temperature, i.e.
q,(r = O, z = O) - 2# "fl" P/o" k T(O), (74)
while further away from the heat source it is given by
q,(r = O,z) = g" fl" p/° "k 1
2nu(pC),, "~-" E(z), (75)
where
E(z) = dz' "4rcz'2 "(pC)., T(z'), (76)

320 D.P. HODGKINsoN
is the total heat energy in a spherical region of radius
z. Thus if z lies outside the extent of the temperature
field, the upward velocity is directly proportional to
the total heat energy in the system and inversely pro-
portional to the cube of the distance from the centre
of the heat source.
5.2 Free surface boundary condition
At low Rayleigh numbers the free surface boundary
condition is, from equation (52),
~(r, d) = 0, (77)
where z = d corresponds to the earth's surface. The
solution to equation (56) that satisfies this boundary
condition is
= ~l(r, z) -- ~/2(r, z), (78)
where, from equation (68),
qq(r,z) = 9flpf°k r2
/~ ' s~?' G(s~), (79)
with
and
Sl = S = [r 2 + Z2] 1/2 , (80)
S2 = [r 2 + (g -- 2d)2] 1/2. (81)
The temperature rise corresponding to this stream
function is, from equations (56) and (78),
T = T(sl)- T(s2), (82)
which correctly satisfies the constant temperature
boundary condition (25) at the surface.
The shape of the free surface is, from equation (54),
2fl. d
~o(r) = d + (r2 + d2)1/2 GE(r2 + d2)1/2], (83)
or
oJ(r) = d +
fl'd
2(pC),v" (r2 + d2)3/2 EE(r2 + d2)1/2]'
(84)
Even if all the heat energy released by the waste con-
sidered in Section 8 (E = 2.4 × 10~6J) were con-
tained within the rock, equation (84) predicts a maxi-
mum rise in the water table of only 0.64 m, for
d= lO00 m_
5.3 Constant pressure boundary condition
The constant pressure boundary condition (21) is
satisfied by the combination
~b= ~/t (r, Z) -F ~/2(r, 2). (85)
The corresponding temperature rise is
T = T(sl) + T(s2), (86)
which satisfies an adiabatic rather than a constant
temperature boundary condition at the surface. How-
ever, this deficiency only becomes apparent when the
temperature field reaches the surface, which is many
thousands of years for the examples considered later
in this paper. Moreover, the temperature rise at the
surface is always very small and so does not signifi-
cantly depart from a constant temperature.
T(s, t) -
where
6. SOLUTIONS FOR THE SPHERICAL
DEPOSITORY MODEL
In the last section general solutions to the low Ray-
leigh number flow equations were given in terms of a
spherically symmetric temperature distribution, T(s~
and an integral, G(s~ over this distribution defined by
equation (67). Explicit expressions for T(s) and G(s)
are given here for a particular heat source represent-
ing a radioactive waste depository.
The depository is idealized as a spherical region of
radius A with the same physical and thermal proper-
ties as the surrounding rock (Hodgkinson, 1978). Heat
is released uniformly throughout this volume at a rate
which decays with time. The total power output of the
depository is written as a sum of exponential terms,
Q(t) = ~ Q,, "exp(- 2,, .t), (87)
m
where 2m is the decay constant (half-life = ln(2)/Am).
Thus the power output per unit volume is
H(r, z, t) - 3Q(t) h(A - s), (88)
4hA 3
where h(x) is the Heaviside unit step function.
The temperature rise T(s, t) is obtained by solving
the transient heat conduction equation
dT
(pC)av " ~- -- Fay" V2T + H, (89)
in an infinite region with H given by equation (88).
The solution to this problem was given in a previous
paper (Hodgkinson, 1978) and so only the result is
quoted here:
3
Q.. v(a, ~, ~.), (90)
47tEar" A
= s/A (91)
z = T,v- t/A 2 (92)
~,, = 2,. A2/~,v. (93)

Mathematical model for hydrotherraal convection 321
The function G(s) defined in equation (67) can be
written as (Hodgkinson, 1978)
3" s2
G4s) = 4nr,, .------A~. Q'" U(~, ~, ~,,), (94)
where
U(a, "r,2) = da'" #2. V(a', r, 2). (95)
expressions for V(a,r,X) and U(a,T,X) inGeneral
terms of complex error functions (Abramowitz and
Stegun, 1965; Faddayeva and Terent'ev, 1961) are
given in Appendix 1 together with some simplified
formulae valid for a ~ 1 and a ,> 1.
An elementary model of three dimensional fracture
flow (Serafim and del Campo, 1965) has proved useful
in obtaining a qualitative understanding of experi-
mental results. This considers the flow through three
sets of parallel fracture planes intersecting at right
angles with the flow along individual fractures
assumed to be laminar. The permeability (k) and por-
osity (n) for this model are related to the mean spac-
ing between fracture planes (A) and the mean aperture
of a fracture (e) by
e 3
k = ~-, (96)
and
7. PERMEABILITY AND POROSITY
At the present time, the values of permeability and
porosity to be expected at a depository site are uncer-
tain (Holmes, 1977). However, some relevant experi-
mental results and theoretical models are available,
and are discussed in this section.
Firstly it can be assumed that the intergranular
permeability is negligible compared with that
arising from the interconnecting network of near
planar fractures (Maini and Hocking, 1977) that
pervade hard rocks such as granite. Black (1978) con-
riders that the mean spacing between fracture planes
at a depository site could lie between 0.1 m and 10 n~
The mean fracture aperture might be anything from
1 mm down to 1/am or less. As the length scale of the
depository and the flow path back to the surface will
be large compared with the mean fracture spacing, it
should be reasonable to treat the rock mass as an
effective porous medium.
3e
n- A' (97)
where A and e are assumed to be independent of pos-
ition and direction.
The hydraulic conductivity (K) is sometimes used
to characterize the flow. This is related to the per-
meability (k) by
K - g" PI" k, (98)
#
and is therefore temperature dependent .through the
viscosity (g) and density (Ps) of water. Hydraulic con-
ductivity has the dimensions of velocity while per-
meability has the dimensions of area.
The permeability, porosity and hydraulic conduc-
tivity corresponding to fracture spacings (A) in the
range 0.1-10 m and fracture apertures (e) in the range
0.1 #m-1 ram, are shown in Table 1.
Table 1. Permeability (k, m2) porosity (n) and hydraulic conductivity (K, ra s- ~)corre-
sponding to fracture spacings (A) between 0.1 m and 10ra and fracture apertures (e)
between 0.1/am and 1ram, calculated from equation (96)-(98). The physical properties
of water at 20°C were used in the calculation of hydraulic conductivity
e•• 0.1 ra 1ra 10m
k = 1.7 x 10 -21 k = 1.7 x 10 -22 k = 1.7 x 10 -23
0.1 #m n = 3 x 10-6 n = 3 x 10-7 n = 3 x 10-8
K = 1.7 x 10 -z4 K = 1.7 x 10 -13 K = 1.7 x 10 -16
10/~m
1 mm
k = 1.7 x 10-xs k = 1.7 x 10-16 k = 1.7 x 10-17
n=3 x 10-4 n=3 x 10-5 n=3 x 10-6
K = 1.7 × 10-a K = 1.7 x 10 -9 K = 1.7 x 10-l°
k = 1.7 x 10 -9 k = 1.7 x 10 -1° k = 1.7 x 10 -11
n = 3 × 10 -2 rl = 3 x 10 -3 n = 3 x 10 -'t
K-- 1.7 x 10 -2 K= 1.7 × 10 -3 K= 1.7 x 10 -4

322 D.P. HODGKINSON
With these wide limits, the permeability could be any-
thing from 1.7 x 10-23-1.7 x 10-9m 2 (K = 1.7 x
10-~6-1.7 x 10-2ms-~), and the porosity values
range from 3 × 10-a to 3 x 10-2.
Experimental values for the hydraulic parameters of
a potential hard rock depository site are scarce, as
most previous research has concentrated on zones
with the highest permeability which are therefore not
suitable for the disposal of radioactive waste. The
most relevant experiments are those recently per-
formed in Sweden as part of the KBS project
(LundstriSm and Stille, 1978; Burgess et al., 1979;
Lindblom 1977; Hansson et al., 1978).
Hansson et al. (1978) found an average permeability
of 6.5 x 10-17m2 (K = 6.5 x 10-1°ms -1 at 20°C)
in a vertical borehole between 410 m and 880 m below
the surface. In a smaller scale experiment at the 360 m
level of the Stripa mine, Lundstr~Sm and Stille (1978)
found permeabilities of 5.3 x 10-is m2 (K = 0.4 x
10- lo m s- 1) at 10°C falling to 1.5 x 10-18 m2 (K =
0.2 x 10-Wms -1) at 35°C. These authors also
measured an effective porosity of 1.2 x 10-4 using a
tracer technique.
The pre-existing hydraulic parameters discussed
above could be significantly modified by the high
temperatures and induced stresses during the opera-
tional phase of the depository, and also by the stresses
induced during the construction of the depository. If
the thermal stresses are compressive then the fractures
would tend to close up thus lowering the per-
meability. This effect has been observed by
LundstrSm and Stiile (1978) as discussed above, and
should occur towards the centre of a depository.
However, in the cooler regions of rock surrounding a
depository, induced tensile stresses (Hodgkinson and
Bourke, 1978; Hodgkinson, 1978) tend to reduce the
compressive stress state in the rock which could in-
crease the aperture of fractures (Witherspoon et al.,
1977) and thereby increase the permeability. Also
shear stresses could cause movement along the frac-
tures (Stephanson and Leijon, 1979; Cook and With-
erspoon, 1978) but it is not clear what effect this
would have on the permeability.
As the permeability and porosity are uncertain to
within many orders of magnitude, the calculations
presented in the next section are necessarily exemp-
lary, and emphasise the sensitivityof the results to the
hydrogeological parameters.
8. RESULTS AND DISCUSSION
In this section the present theory is evaluated for a
representative radioactive waste depository, and the
sensitivity to the hydrogeological parameters is exam-
ined.
The quantity of waste considered for disposal is
that arising up to the year 2000 from a U.K. nuclear
power programme rising to an installed capacity of
about 40 GW(e) by that date (Roberts, 1978). For
illustrative purposes it is assumed to arise from pres-
surized water reactors and be reprocessed 4½yr after
removal from the reactor. If this waste were vitrified
by the Harvest process then it would fill about 3500
standard canisters (Beale et al., 1979; Griffin et al.,
1979) (length 2 in, diameter 0.45m) each of which
would have the time dependent power output shown
in Fig. 1 (Barton, 1979).
In the standard depository model considered here,
the heat output from this waste is assumed to be uni-
formly distributed over a sphere of radius 190 m and
buried at a mean depth of 1000m. This amounts to
the same power density as would be obtained by
spacing the 3500 canisters by 20 m in all directions
(Hodgkinson, 1977). Furthermore the waste is
assumed to be stored for 70 yr prior to disposal by
which time the heat output is 1kW for each block
(see Fig. 1) and 3.5 MW for the whole depository.
The values of the various rock and fluid parameters
used in the calculations are listed in Table 2.
Temperature profiles along the centreline (r = 0) of
the depository are shown in Fig. 2 for 50, 150, 1000,
5000 and 10,000yr after disposal. The temperature
rise at the centre of the depository reaches a maxi-
mum value of 70°C after about 150 yr and then slowly
decays as the heat is distributed over an ever increas-
ing volume of rock. Even after 10,000 yr, most of the
heat given out by the decaying radionuclides is con-
tained within the rock mass. As will be shown later,
10_1
2-
o
10-' ~
"3
13.
*5
o
10-3
PWR w(]s fe
r epr oce~sed
Length = ~0m
Diometer = 0./*Sm
~0 10 "~ 10 3 10 4 10.~
T,me out of reoclor (years)
Fig. l, Power output from a Harvest block as a function of
time.

Table 2. Physical properties of granite and water (at 40°C)
Physical quantity Symbol Value
Density of granite p,
Specific heat of granite C,
Thermal conductivity of granite F,
Density of water Plo
Specific heat of water C/
Thermal conductivity of water F r
Viscosity of water #
Volumetric thermal expansion fl
coefficient of water
Average thermal diffusivity 7~
for n <~1
Effective thermal diffusivity
for n <~l
this heat has the potential to cause buoyancy flows
long after the temperature rise at the centre of the
depository has fallen to a small fraction of its maxi-
mum value.
Figures 3-7 show isotherms (on the left-hand side)
and streamlines (on the right-hand side) in a vertical
plane through the centre of the depository for times of
50, 150, 1000, 5000 and 10,000yr after disposal. Both
2.60 x 10a kg m- 3
8.79 x 102Jkg-t° C-1
2.51Wm -1 °C-I
9.92 x 102kgm -3
4.18 x 103Jkg-l°C -1
6.23 x 10-1Wm-l°C-1
6.53 x 10-4kgm-ls -1
3.85 x 10-4°C -1
1,10 x 10-6m2s -1
6.05 x 10- 7 m 2 S- z
-- 100
P
8o
¢:
o, 60
o
._~ 40
20
E
o
0 0"5 1"0
Distance below surface {kin)
1"5
r
Fig. 2. Temperature profiles along the centreline of the
depository for various times after disposal.
0 150 years
a f tar disposal
o~ -s00
o
0~ - 1000 30
~0
~ -1500
Prmeobd 1),=10 m
-2000
1
-1000 -500 0 .500 IO00 1500 2000
Distance tram cenlreline of sphere (m)
Fig. 4. Isotherms (°C) and streamlines (m3/yr) at 150 yr
after disposal (free surface boundary condition).
isotherms and streamlines are of course axisymmetri-
cal.
A free surface (water-table) boundary condition has
been assumed to hold for flow near the earth's sur-
face. For the low Rayleigh number flows considered
here, this is approximately equivalent to having an
impermeable boundary at the surface.
The streamlines are the paths of water flow at a
0
8 -s00
o
-1000
c
-1500
Q
-2000
r , I
50 years i ~ >
ofier disposal
~er~bility =104En"
I I i0~00 J-1000 -500 500 1500 2000
Oisionce from centreline of sphere (m)
Fig. 3. Isotherms (°C) and streamlines (ma/yr) at 50 yr after
disposal (free surface boundary condition).
1000 years
after disposal / / ~ ~
-500
o~ -1000
?,
~ -1500
o
-2000 - --- • - v
-1000 - 500 0 500 1000 1500 2000
Olsfonce from cenlreline of sphere (m)
Fig. 5. Isotherms (°C) and streamlines (m3/yr) at 1000yr

324 D.P. HODGKINSON
0 i
5000 y'ears
otter~disposal~ ~ ~
o~ -500
o -1000
i -1500
.~_
-2000
-1000 -500 0 500 1000 1500 2000
Oistonce from cenlreline of sphere (m}
Fig. 6. Isotherms (°C) and streamlines (m3/yr) at 5000yr
0
10000years-- ofler disposal
-500
-I000
c
-1500
o eH.meobddy=lO_16m2
-2000 i
-1000 -500
I , ,
500 1000 1500 2000
Distance from cenlreline of sphere (m)
Fig. 7. Isotherms (°C)and streamlines (m3/yr) at 10,000yr
particular instant in time. Thus in effect, Figs 3-7 are
a series of snapshots of the flow patterns taken at
successive times. The numbers marked on the stream-
lines are the value of the stream function in m3/yr.
This is independent of the porosity and directly pro-
portional to the permeability. Consequently the shape
of the streamlines does not depend on the porosity or
permeability.
As discussed in Section 2.1, the difference between
the stream function at any two points, multiplied by
2 rt, is equal to the volume rate of flow. For example,
Fig. 5 shows that about 30 m3 water per year flows
upwards through the depository area at 1000 yr after
disposal if the permeability is 10-16 m2. The flow
rates for other permeabilities are found by a trivial re-
scaling of the numbers shown on the streamlines.
The streamlines show a pattern in which water is
convected upwards through the hot depository, and
then cools and falls forming a thermal convection
cell. The centre of this cell moves outwards from the
depository as the heat becomes more spread out. This
picture is to some extent misleading as it takes no
account of the water velocity. If the water were mov-
ing very fast then it would indeed follow the stream-
lines. However, as can be seen by comparing Figs 3-7,
the streamlines change in shape and magnitude in
response to the driving force provided by the evolving
temperature distribution. In this time dependent flow
field, the water moves so that it always follows the
instantaneous streamline.
The paths followed by water in the fractures are
known as pathlines (Bear, 1972), and are discussed in
Section 2.2. The pathlines of relevance to geological
disposal are those that originate inside the depository
at a particular time. This is because they are the paths
that would be followed by a non-sorbed radionuclide
leached from the waste.
The pathlines that originate in the mid-plane of the
depository at times of 150, 1000 and 10,000yr after
disposal are shown in Figs 8-10. These depend on the
-- 0-25
E
o
"~ 0"5
$ o7s
_g
._~
o 1-0
1-25
-t0 _0 75
i 1 r ,,~
::';;:;";7;2'?°°" l- p.,-,o'"",.--,,o,
t ~
-0~ -0"/25 0 0'25 0-5 0-75 1.0
Oistclnce from centreLine at sphere (kin)
Fig. 8. Thermal pathlines for release at 150yr after disposal (freesurface boundary condition).

-~ 025
~ OS
0 75
c
2
o 10
[ i i
Releose lime = 1000 yeors
RegionoI grod~ent = 0
i i i
permeobility =10_1l m2
porosity
I
0
,.~s , , • , , ,
-,0 -0.,s -0.s -0.2s o.2s 0s- 0.,s ,0
Oistonce from centreline of s~here {kin)
Fig. 9. Thermal pathlines for release at 1000yr after disposal (free surface boundary condition).
permeability and porosity through their ratio, which
is proportional to the mean square aperture of the
fractures (see Section 7). For illustrative purposes this
ratio has been taken to be 10-12 m2 (e.g. permeability
= 10-16m 2, porosity---10-4). It is seen that the
water does not recirculate as might have been inferred
from the streamlines shown earlier. Instead, water
rises several hundred metres above the depository
while simultaneously moving outwards. This buoyant
rise will clearly reduce the isolating effect of deposi-
tory depth. However, its full significance can only be
gauged from a comprehensive safety analysis (Hill
and Grimwood, 1978).
It is clear from Figs 8-10 that thermally induced
flows continue for at least 10,000 yr after disposal, by
which time the temperature near the centre of the
depository will have almost returned to its ambient
value.
The times marked along the pathlines are the tran-
sit times (in years) for water to move from the deposi-
tory to these points. The water is seen to rise at the
rate of about 0.1-1.0 m/yr although, as will be seen
later, this is highly dependent on the permeability and
porosity.
The streamlines and pathlines described above are
due solely to the heat output from the depository. At
a disposal site there is also likely to be a largely hori-
zontaJ regional groundwater flow driven by a pressure
gradient of about 10- a m water/m (Lindblom, 1977;
Ratigan et al., 1977; Office of Waste Isolation, 1978;
Burgess, 1977). The combined effects of these thermal
and regional flows is shown in Figs 11-13 for times of
0 r r i ~ r i i
,~ ~ m m m m
i O5
075
t3 i 0 Re 2
1,2.e" I .015 2 ' I ' '
-1,0 -075 -0, 5 0 0.25 05 075 10
Distonce from cenfreline of sphere (krn)
Fig. lO. Thermal pathlines for release at 10,000yr after disposal (flee surface boundary condition).

I 1 I r
-- 025
E
o
"c 0.5
.o 075
o
o 1"0
1 i
permecibildy = 10.1z m2
porosity
Re~eose time = 150 years
Regional gradient =10 -3
326 D.P. HODGrJNSON
1-25 i i
-,-o -o!~ -A -o~s o o-,s o'.s o.Ts ,.o
Oi$1once trem cenlrellne .el =l~ere Ikml
Fig. 11. Pathlines for simultaneous thermal and regional flow for release at 150 yr after disposal (free
surface boundary condition).
0 w i i i v" i
permeability : 10_1z rn~
-~ 025 porosity
3 35
~ m
~ 0.75
o
Release t,me : 1000 years
a 10 Regionol grodient : 10 -3
1 2 5 I I .0.i25 l ~ .I
-1-0 -0-75 -0'5 0 0-25 0 5 0-75 1.0
Distclnce from centreline of sphere Ikm)
Fig. 12. Pathlines for simultaneous thermal and regional flow for release at 1000yr after disposal (free
0 ! '! ! 1 ...... i i
-- 025
E
x
o
05
u~
"~ 075
o f'O
1 25
- 1.0 -0[75
permeability = 10_12 rnz
porosity
ReLeoe m 0000y ....
Regional gradient : 10-3
-0 5 -0-125 0 0-25 0!5 075 10
Distance from centreline of sphere (kin)
Fig. 13. Pathlines for simultaneous thermal and regional flow for release at 10,000 yr after disposal (free

-- 0.25
E
o
"~ 05
o'
075
tc
o 10
permeabdlty = i0_1C mZ
porosify
Release time = 100C years
Regional gradient = 10-3
t2s " !~ o ' ' ' ' '
-,o -o, -, -o. o 02, o, o,, ,o
Distance from cenfteline of sphere (kin)
Fig. 14. Pathlines for simultaneous thermal and regional flow for release at 1000yr after disposal (free
surface boundary condition). The ratio of permeability to porosity is 100 times larger than in the
standard example.
150, 1000 and 10,000 yr after disposal. It is seen that
the average rise of water above the depository level is
not drastically changed by the presence of a regional
cross-flow, except perhaps at 10,000 yr after disposal
(Fig. 13).
The sensitivity of these results to permeability and
porosity is illustrated in Figs 14 and 15 where the
combined thermal and regional pathlines starting at
1000 yr are shown for permeability to porosity ratios
a hundred times larger and smaller than used pre-
viously. The travel times are approximately inversely
proportional to this ratio. Thus in Fig. 14 there is a
rise of a few hundred metres in about l0 yr. However,
in Fig. 15 the water velocity is so low that it does not
rise very far during the time when there is a signifi-
cant thermally induced driving force. Improved data
on the permeability and porosity of hard rock sites is
clearly required (Bourke et al., 1979) to differentiate
between these possibilities.
A further uncertainty in the hydrogeology of a dis-
posal site is the boundary condition for flow near the
earth's surface. It should be possible to determine this
at the time of disposal but the changes brought about
in the future by natural causes and human actions
will always remain uncertain. Consequently a com-
prehensive safety analysis of geological disposal
should examine the sensitivity to the choice of flow
boundary condition. As a first step in this direction
the flows described previously in this section for a
free-surface boundary condition have been recalcu-
E 0"25
z¢
o
~ 0"5
o-75
u
c
a
~ 1.0
permeability = I0_14m2
porosity
Release time = 1000 years
Regional gradient : I03
I •25 - 3,175 I
-1.0 -0"5
N
-0 25 0 0'25 0 5 0.75
Distance from centrellne of sphere (kin}
1.0
Fig. 15. Pathlincs for simultaneous thermal and regional flow for release at 1000yr after disposal (free
surface boundary condition). The ratio of permeability to porosity is 100 times smaller than in the
standard example.

328 D.P. HODGKaUSON
0
8 -5ooo
°~ -1000
o
.~ - 1500
D
-2000
1 j i _i
50 years
afte¢ disposal
30
?0
~rmeablldy:lO-I1$ml
-tO00 -SO0 500 1000 1500 2000
Distance from centreline of sphere Ira)
Fig. 16. Isotherms (°C) and streamlines (ma/yr) at 50yr
after disposal (constant pressure boundary condition).
lated for a constant pressure boundary condition.
This might be a good approximation if there was a
considerable body of water on the surface above the
depository, or if there was a surface layer of much
higher permeability than the underlying rock (Bourke
and Hodgkinson, 1979). The major difference is that
this boundary condtion allows water to pass freely
through the surface.
Figures 16-28 are repeats of Figs 3-15 with the
flow boundary condition changed as discussed above.
The streamlines (Figs 16-20) and thermal pathlines
(Figs 21-23) show water rising perpendicularly
through the surface above the depository. For a per-
meability to porosity ratio of 10-t2 m2 it takes some
thousands of years for water to travel from the de-
pository to the surface. In general the travel time to
the surface is inversely proportional to this ratio.
The regional cross-flow (Figs 24--28) distorts the
flow paths but water could still reach the surface
under certain conditions. The major factors which
control this are the magnitude of the regional press-
ure gradient and the total amount of heat given out
by the waste.
Finally, it should be noted that the examples dis-
cussed here apply to one particular example of a de-
pository. The numerous depository design parameters
(size, shape, depth, heat output, type of waste, etc.)
will clearly affect the flows in many ways.
r r ~ i
150 years
otter disposal
-5oo
.o 5u 1
' N-1000 ~
c
o -1500
._e
a ,ermeobilify=lOJ6m2
- 2000 i i t
-tOO -500 500 1000 1500 2000
Distance ttom centreline at sphere ira)
1000 years T / ~ ,
"E otter disposal ~ ~ ~
G, ~ 500
v~
10D0
-_=
0 ~rmeobility:I0-16mz
-2000 J ~ ,~'~-- ,
-1000 -500 0 500 1000 1500 2000
Distance from centre(ine at sphere (rn)
Fig, 18. Isotherms (°C) and streamlines (ma/yr) at 1000yr
o ~ . I I'II I'I '
-soo
o
o,-,°o°i
c
o -1500
o
-2000 permeability :10"16mZ
- tOO -500 0 500 tO00 1500 2000
Distance from centretine of sphere JmJ
0 i
10000 years
after disposot
-500
3
moo
c
0 "150 O
-6
D ermeobi i y=lO e
-2000 L
-1000 -500 500 1000 1500 2000
Oi&tonce ram centreline Of sphere (m)
Fig. 20. Isotherms (°C) and streamlines (m3/yr) at 10,q00

Mathematical model for hydrothermal convection
0-25
.x
•~ 0"5
g
o
,o 0.75
¢
c3 1"0 Rele 12 |
Regionel gradient = 0 ~ +~ p ~ = IU m
1.25 J I | J , . J
-to -o,5 -o5 -o,5 o 0,5 o,, ,o
Distance from centreline of sphere (kin)
Fig. 21. Thermal pathlines for release at 150 yr after disposal (constant pressure boundary condition).
329
-- 0"25 '°=
E
"C 0"5
o.7s
Release time = I000 ),ears perrneobillty= tO_lZ rn2
c~ 1 "0 Regional gradient : 0 porosity
1.25 I I I I [ 1
-1.0 -0-75 -0.5 0"25 0 0"25 0'5 0"75 1 0
Distance from centretir~e of sphere (krnJ
Fig. 22. Thermal pathlines for release at 1000 yr after disposal (constant pressure boundary condition).
0 i i
__ 0"25
E~g
"C 0'5
o
0.75
g
.c3 1.0
1 '25 I I I L
-1-0 -0.75 0"5 0.75 1,0
L
Release time =10000 years permeability 10_1~m2
- Regional gradient : 0 porosity
I I i
-0-5 -C'25 0 0'25
Distance from centreline at sphere (kin)
Fig. 23. Thermal pathlines for release at 10,000 yr after disposal (constant pressure boundary condition).
A.N.F. 7/6--B

330 D.P. HODGKIN$0N
0 , ,
-- 0'25
E
8o
05
o~
0"/5
c
o
o 1-0
1' 25
permeability = 10_12 m2
porosity
IRelease time =150 years
-1-0 - 01"75 -0t-5 -0'125
I I
o 0'~5 o!5 o.,5 ,.0
Distance from centreline of sphere (kml
Fig. 24. Pathlines for simultaneous thermal and regional flow for release at 150 yr after disposal (con-
stant pressure boundary condition).
-- 0.25
E
o
0"5
0"75
c
2
o
1.0
1 ) i r
perrneobility : 10_10 rn2
porosity
Release time = 1000 years
- Regional gradient : 10 -3
1.25 1 l, | I I I I
-1'0 -O'TS -0'5 -0"52 0 0'25 0 •5 0,75 10
Distance from centreline at sphere (kin)
Fig. 25. Pathlines for simultaneous thermal and regional flow for release at 1000yr after disposal
(constant pressure boundary condition).
0 i i i i i ~ i
"~ 0'25
3 0.5
~ 0"75
.g
.~_
o 1.0
permeability = 10-~z m2
Release time : 10000 yetars
Regionot gradient : 10-3
125 I L
-I-0 "0!75 -0"5 -01'25 0 0125 01-5 0175 1"0
Distance from oentretine of sphere (kin)
Fig. 26. Pathlines for simultaneous thermal and regional flow for release at 10,000 yr after disposal
(constant pressure boundary condition).

P
-- 0'25
E
~t
0.7s
"1.0
0 1 r
permeability = 10-12m2
porosity
Release time = t000yeors
1'25 I i
I i I 1. I
-1'0 -0'75" -0"5 -0"25 0 0"25 0.5 0"75 1'0
Distance from centreline Of sphere (kmJ
(constant pressure boundary condition). The ratio of permeability to porosity is 100 times larger than in
the standard example.
0 ) l ) i I I i
-- 0"25
E
'~ 0"5
.o 0'75
g
Ol. 0
permeability = 10_1~,m2
porosity
Release time =1000 years
Regional gradient = 10-3
1 '25 I I I I !
-1.0 -0.75 -0!5 -0!25 0 0-25 0'5 075 1"0
Distance from centreline of sphere I km )
(constant pressure boundary condition). The ratio of permeability to porosity is 100 times smaller than
in the standard example.
9. CONCLUSIONS
Thermal convection of water in the vicinity of a A
depository could be at least as important as the re- CI
C,gional flow for several thousand years after burial, d
Water continues to flow upwards through the deposi- D
tory long after the temperature rise in the depository D~
has fallen to a small fraction of its maximum value, e
This buoyant rise reduces the isolating effect of de- E(s)
pository depth although the timescale, which depends 0G(s)
prim~ily on the ratio of permeability to porosity, is h(x)
uncertain. Whether or not this water reaches the H
earth's surface depends largely on the magnitude of
the regional groundwater flow and the nature of the ~*effc(×)
flow boundary. In order to adequately assess these Jco.d
problems improved hydrogeological data is required J~o,,
from sites deemed suitable for geological disposal.
LIST OF SYMBOLS
Radius of depository (m)
specific heat of fluid (J kg- 1°C- 1)
specific heat of solid (J kg- l °C- l)
mean depth of sphere (m)
dimensionless mean depth of sphere
discharge number
mea~ fracture aperture (m)
heat energy in sphere of radius s(J)
acceleration due to gravity (m s- a)
radial part of stream function (m2°C)
Heaviside unit step function
heat production rate per unit volume
(W m- 3)
repeated integral of the error function
heat flux per unit area for conduction
(W m- 2)
heat flux per unit area for convection
(W m- 2)

332 D.P. HODGKINSON
k
K
L
n
P
Pe
qr, z
qx
qfree, forced
Q(t)
r
R
Ra
S
t
T
T~
U(cr,~,1)
V(~r,T,l)
V,.z
W(x + iy)
X
Z
Z
Greek
Ot
~av
F/
F~
Fay
6
A
AT
e
0
0,,
2,
#
P/
Pro
Ps
(pC).,
t7
%
q,
~v
~r)
~(R)
permeability (m2)
hydraulic conductivity (m s-1)
length scale (m)
effective flowing porosity
pressure (Pa)
Peclet number
macroscopic radial and axial velocity
(ms -1)
maxcroscopic horizontal velocity (m s- ~)
macroscopic free and forced convective
velocity (m. s- l)
power output of the depository (W)
radial cylindrical coordinate (m)
dimensionless radial cylindrical coordinate
Rayleigh number
spherical radial coordinate (m)
time (s)
temperature (°C)
average surface temperature (°C)
function defined in Appendix 1
function defined in Appendix 1
radial and axial fluid velocity in fractures
(m s- 1)
complex error function
horizontal coordinate(m)
axial coordinate (m)
dimensionless axial coordinate.
Effective thermal diffusivity (m2 s-1)
volumetric thermal expansion coefficient of
fluid (°C- t)
average thermal diffusivity (m2s- 1)
thermal conductivity of fluid (W m- 1°C- 1)
thermal conductivity of solid (W m- 1
oc-I )
average thermal conductivity (W m-
°C- l)
regional pressure gradient (m water m- 1)
mean fracture spacing (m)
characteristic temperature difference (°C)
coefficient of temperature increase with
depth (°C m -1)
dimensionless temperature
coefficients in Rayleigh expansion of tem-
perature
radioactive decay constant (s-i)
dimensionless decay constant
fluid viscosity (kg m- 1s- 1)
density of fluid (kg m -~)
density of fluid at reference temperature
(kg m- 3)
density of solid (kg m- 3)
average value of density times specific heat
(J m -3 oc-a)
dimensionless spherical radial coordinate
dimensionless time
spherical polar angular coordinate
stream function (m3s- 1)
dimensionless stream function
coefficients in Rayleigh expansion of
stream function
shape of free surface (m)
dimensionless shape of free surface
coefficient in Rayleigh expansion of free
surface.
Acknowledgements--I would like to thank Pat Bourke and
John Rae for many useful discussions, and Mair Williams
for her invaluable help with computing and preparation of
the figures. Funding from the Commission for the Euro-
pean Communities as part of the European Economic
Community programme of research into underground dis-
posal of radioactive waste, is gratefully acknowledged.
REFEREN CES
Abramowitz M. and Stegun |. A. (1965) Handbook of Math-
ematical Functions. Dover, New York.
Barton H. (1979) private communication.
Beale H., Bourke P. J. and Hodgkinson D. P. (1979) Ther-
mal Aspects of Radioactive Waste Disposal in Hard
Rock, Proc. IAEA & OECD-NEA Syrup. Underground
Disposal of Radioactive Wastes, Helsinki, Finland. IAEA-
SM-243/26.
Bear J. (1972) Dynamics of Fluids in Porous Media. Ameri-
cal Elsevier.
Black J. H. (1978) Some Aspects of the Hydrogeology of a
Crystalline Rock Repository. I.G.S. Hydrogeology Dept.
Rep. N. 17K-79-1.
Bourke P. J. and Hodgkinson D. P. (1977) Granitic De-
pository for Radioactive Waste: Size, Shape and Depth v
Temperature, AERE-M2900.
Bourke P. J. and Hodgkinson D. P. (1979) Assessment of
Thermally Induced Water Movement Around a Radio-
active Waste Depository in Hard Rock. Proc. Workshop
on Low-Flow, Low Permeability Measurements in Largely
Impermeable Rocks, Paris, 19-21 March 1979. OCED
NEA & IAEA.
Bourke P. J., Gale J. E., Hodgkinson D. P. and Withers-
poon P. A. (1979) Tests of Porous Permeable Medium
Hypothesis for Flow Over Long Distances in Fractured
Deep Hard Rock. Ibid.
Burgess A. (1977) Groundwater movements around a
repository: Regional Groundwater flow analyses, KBS
TR 54:03.
Burgess A. S., Charlwood R. G., Skiba E. L., Ratigan J. L.,
Gnirk P. F., Stille H. and Lindblom V. E. (1979) Ana-
lyses of Groundwater Flow Around a High-Level Waste
Repository in Crystalline Rock. Proc. Paris Workshop,
Ibid.
Cheng P. and Lau K. H. (1974) Steady state free convec-
tion in an unconfined geothermal reservoir. J. geophys.
Res. 79, (29) 4425-4431.
Combarnous M. A. and Bories S. A. (1975) Hydrothermal
convection in saturated porous media. Adv. Hydro-
science 10, 231-307.
Cook N. G. W. and Witherspoon P. A. (1978) Mechanical
and Thermal Design Considerations for Radioactive
Waste Repositories in Hard Rock. LBL-7073/SAC-06.
Faddeyeva V. N. and Terent'ev N. M. (1961) Tables of
Values of the Function W(z) for Complex Argument. Per-
gamon Press, Oxford.
GAIN Symposium Proceedings (1978) Geotechnical Assess-
ment and Instrumentation Needs for Nuclear Waste
Isolation in Crystalline and Aroillaceous Rocks, Berkeley,
California, 16-20 July 1978. LBL-7096.
Griffin J. R., Beale H., Burton W. R. and Davies J. W.
(1979) Geological Disposal of High Level Radioactive
Waste: Conceptual Repository Design in Hard Rock.
IAEA-SM-243/93.
Hansson K., Alm~n K. E. and Ekman L. (1978) Hydrogeo-
logical Investigations in Two Boreholes in the Stripa
Test Station. Proc. OECD-NEA Seminar In Situ HeatinO

Experiments in Geological Formations, Stripa, Sweden.
Hickox C. E. (1977) Steady Thermal Convection at Low
Rayleigh Number from Concentrated Sources in Porous
Media. SAND 77-1529.
Hill M. D. and Grimwood P. D. (1978) Preliminary Assess-
ment of the Radiological Protection Aspects of Dis-
posal of High Level Waste in Geologic Formations.
NRPB-R69.
Hodgkinson D. P. (1977) Deep Rock Disposal of High
Level Radioactive Waste: Transient Heat Conduction
from Dispersed Blocks. AERE-R8763.
Hodgkinson D. P. (1978) Deep Rock Disposal of High
Level Radioactive Waste: Initial Assessment of the Ther-
mal Stress Field. AERE-R8999.
Hodgkinson D. P. and Bourke P. J. (1978) The Far'Field
Heating Effects of a Radioactive Waste Depository in
Hard Rock. Proc. OECD--NEA Seminar In Situ Heating
Experiments in Geological Formations, Stripa, Sweden,
September 1978.
Holmes D. C. (1977) Determination of Hydraulic Proper-
ties of Fractured Rock Media, Characterised by a Low
Hydraulic Conductivity; Methods and Problems. IGS
Hydrogeology Dept. Rep. 17K-77-13.
Kreyszig E. (1967) Advanced Engineering Mathematics.
Wiley, New York.
Lindblom U. (1977) Groundwater Movements Around a
Repository. Phase 1, State of the Art and Detailed Study
Plan. KBS- TR-06.
LundstrSm L. and Stille H. (1978) Large Scale Per-
meability Test of the Granite in the Stripa Mine and
Thermal Conductivity Test. LBL-7052, SAC-02.
Maini T. and Hocking G. (1977) An Examination of the
Feasibility of Hydrologic Isolation of a High Level
Waste Repository in Crystalline Rock. Invited paper at
the A. Meeting geol. Soc. Am. Seattle, 1977.
OECD-NEA & IAEA (1979) Proc. Workshop Low-Flow,
Low-Permeability Measurements in Largely Impermeable
Rocks, Paris, 19-21 March 1979.
Office of Waste Isolation (1978) Technical Support for
GELS: Radioactive Waste Isolation in Geologic Forma-
tions: Groundwater Movement and Nuclide Transport.
Y/OWI/TM-36-21.
Ratigan J. L., Burgess A., Skiba E. L. and Charlwood R.
(1977) Groundwater Movements Around a Repository:
Repository Domain Groundwater Flow Analyses.
KBS TR54:05.
Roberts L. E. J. (1978) Radioactive Waste: Policy and Per-
spectives. Lecture to the British Nuclear Energy Society,
London, 9 November 1978. Reprinted in Atom 267, 8
(1979).
Serafim J. L. and del Campo A. (1965) Interstitial Pressures
on Rock Foundations of Dams, J. Soil Mech. Founda-
tions Division. Proc. Am. Soc. cir. Engrs. SMS, 65-85.
Stephanson O. and Leijon B. (1979) Temperature Loading
and Rock Mechanics at Final Storage of Radioactive
Waste. Univ. Lulea Rel~ 01-10.
Witherspoon P. A., Amick C. H. and Gale J. E. (1977)
Stress-Flow Behaviour of a Fault Zone with Fluid Injec-
tion and Withdrawal. Univ. California Berkeley Mineral
Engng Rep. 77-1.
APPENDIX
The functions V(a, t, ;~)and U(a, z,,~)
The functions V(a,t, ~) and U(a, z, ~) were originally dis-
cussed in connection with the thermal stress analysis of a
spherical depository (Hodgkinson, 1978). General ex-
pressions for V and U in terms of complex error functions
(Ambrowitz and Stegun, 1965; Faddeyeva and Terent'ev,
1961) are reproduced in this appendix, together with some
useful special limits of the general formulae.
V(a, t,;t)
The general expression for V(a,r, ;~)is
V(a,r, 2) -- ~ 5(1 -e -;:r)-go(1 -a)+go(1 4- a)
tr<l
-gl(1 -tr)+g~(1 +a)} (A.l)
inside the heated sphere, and
1
V(a,z, 2) = -{#0(a - 1) + go(a + 1)
a>l 0"
+gl(a+ l)-gl(a- 1)I, (A.2)
outside the heated sphere. The functions gi(x) are given
by
go(x) = 1 {io erfc (x/2t 1/2) - exp(- x2/4r)
x Re W[(,tz)1/2 + ix/2tl/2]} (A.3)
and
1 1'2
gl(X) = ~ {2(,~r) : •iI erfc(x/2z 1/2)
- exp(-x2/4z) Im W[(,~t)1:2 + ix/2C/2]}. (A.4)
When cr ,~ 1, V(a, t, ~) has the series expansion
1
V(a,t,~) = ~[1 -exp(-2t)]
a,~I /,
2 " C n . ~n 1
+ ~ (a.5)
tl!n=1,3,5...
where
1
C1 = -~ [~ + i° erfc(1/2rt/2)], (A.6)
l[ l ]Cs = ~ ~ nl/2 .C/2 exp(- 1/4t) , (A.7)
ex,'-14"1,
(A.8)
with
= exp(- 1/4t){21/z. Im W[(,~r)1/2 + i/2t 1/:]
-- Re W[(~t)1/2 + i/2tl/2]J. (A.9)
At the other extreme, a >> 1, the formula for a point
source can be used, i.e.
V(a, t,;O exp(-a2/4t) Re W[(;~tp/2 + i(a/2C/2)].
~>1 3"tr
(A.10)

334 D.P. HooGgmqsorq
U(a, T,;3
The function U(a, z, ~) is defined by
lfo~U(a,z,].) = ~5 da"a '2" V(a',r,~). (A.11)
The general expression is
1 fo 3
U(~r,z,Z) a_~l aS~.~(1 -- e-Z~)- trgl(1 + a) -- ¢rg,(l -- ~)
- (1 + tr)g2(1 + o') + (1 - o")/]2(1 - o')
- g3(1 + tr) + gs(1 - o')}, (A.12)
and
1 fl ~
U(o', ~',,~)aTl ~ ( 1 --e-.)- O-gl(O--{- 1)- ag~(a - 1)
- (a + 1)g:(a + 1) + (o - 1)O2(o - 1)
- gs(a + 1) + gs(a - 1)~, (A.13)
)
where
2Z 2 / X 1
g2(x) = -- i erfc / ~7Y/21 - ;~go(X), (A.14)
and
4zs/2iaerfc( x ) 1
,qs(x) = T 2~ - 2 g~(x), (A.15)
with go(X) and Ol(x) defined by equations (A.3) and
(A.4).
When a ~ 1, U(a, r, ~) has the series expansion
1
U(~, ~, 2) .=1~ ~- [1 - exp(- ~2)]
2 "Cn
+ ~ n!(n + 2~' Oa-1 (A.16)
n= 1,3,5...
where C, are defined by equations (A.6)-(A.9). This series
expansion is useful for evaluating U(a, z, 2) with a ,~ 1
and z > 1, where use of equation (A.12) can lead to
numerical inaccuracies. When a ,> 1, use can be made of
the formula for a point source, i.e.
1
U(a, z, ~) ~1 3,~' o.3 {1 - exp(-2r) - erfc(o'/2z1/2)
+ exp(- 02/4z) •Re W[(~z)1/2 + i(a/2zl/2)]
- exp(- a2/4r).a •2. Im W[(;.r) 1/2
+ i(a/2zl/2)] I. (A.17)

MATHEMATICAL MODEL FOR HYDROTHERMAL CONVECTION AROUND A RADIOACTIVE WASTE DEPOSITORY IN HARD ROCK

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