Geophys. J. R. astr. Soc. (1985) 82, 439-459
The effects of sedimentation and compaction on oceanic
heat flow
Iain Hutchison * Bullard Laboratories, Department of Earth Sciences,
University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 OEZ
Accepted 1985 January 11. Received 1985 January 11; in original form 1984 April 4
Summary. The estimation of environmental effects forms an important part
of the interpretation of oceanic heat flow measurements. In particular, the
perturbations associated with sedimentation and surface temperature changes
must be taken into account. Analytical solutions can be obtained only for
individual, simplified versions of these problems, whereas any real example is
complicated by the process of sediment compaction which changes the bulk
thermal properties with depth. A physical model is developed which uses
sediment porosity trends to predict the thermal parameters and material
advection rates for an evolving sediment/basement system. These values are
then used in a numerical solution to the heat flow equation to give estimates
of the perturbed surface heat flux through time. In addition to variations in
sedimentation rate, sediment type, radioactive heat production and surface
temperature changes are considered. Heat flow corrections may vary by up to
a factor of 2 according to sediment type while radioactive heat production
can offset the effects of sedimentation by as much as 40 per cent. The results
also indicate that alterations determined from simple analytical models tend
to over-estimate the true perturbation to the flux.
1 Introduction
Measurements of oceanic heat flow have played an important role in the development of
lithospheric models, not only for the ocean basins but also marginal seas and, indirectly,
continental sedimentary basins (for example: McKenzie 1967; Parsons & Sclater 1977;
Sclater, Jaupart & Galson 1980). Recent advances in instrumentation and measurement
techniques have led to greater accuracy in the determination of the regional flux, in particular
through the development of multiple-penetration ‘pogoprobe’ surveys and the in situ
measurement of sediment thermal conductivity (Lister 1979; Hyndman, Davis & Wright
1979; Hutchison 1983). With care, errors in the individual heat flow measurements can now
be reduced to a few per cent with survey mean values determined within 10 per cent (see,
e.g. Von Herzen et al. 1982; Hutchison et 01. 1985). However, to be interpreted properly,
*Present address: BP Petroleum Development Limited, Britannic House, Moor Lane, London EC2Y 9BU.
440
I. Hutchison
heat flow measurements must first be corrected for any environmental effects which perturb
the geothermal flux. In particular, in many oceanic and deep marginal basins, the rapid
deposition of cold sediment blankets the basement, leading to a reduction in the observed
surface flux. Against this reduction, radioactive heat production in thick sediments can
provide a significant contribution to the heat flow, Additional perturbations can be introduced by changes in the seafloor water temperature. Jaeger (1965) and Von Herzen & Uyeda
(1 963) describe a series of analytical solutions to these and other, related environmental
problems. Although the results have been used extensively, these models make severe assumptions in the assignment of bulk physical properties to the sediments and only simple depositional histories can be modelled. These problems also affect analytical models which describe
a lithospheric evolution which is directly linked with sedimentation (Turcotte &
Ahern 1977; Langseth, Hobart & Horai 1980).
Benfield (1949) was the first to consider the thermal effects of uplift and denudation. His
results apply equally to the case of subsidence and deposition. In the Benfield model, the
surface flux, Q,(t) is calculated through time ( t ) for a uniform half-space moving with
constant velocity V away from the surface z = 0, assuming a uniform initial heat flux, Qo, at
depth (Fig. 1). Q,(t) is then given by:
Q,(t) = Qo * [4i2erfc (Vt1'2/2~1'2)]
(1.1)
where 4i2 erfc is the second integral of the complementary error function (Carslaw & Jaeger
1959) and K is the thermal diffusivity of the half-space. Langseth et al. (1980) provide a
nomogram for the reduction factor as a function of time and V / K ~based
"
on this model.
Z=O ,T=O
Z
Iv
00
Figure 1. Simple sedimentation model: a uniform half-space moves away from z = 0 at velocity V, with
the surface temperature T, = 0. The initial equilibrium geothermal gradient (t = 0) is modified (t = r ' ) in
the way described by Von Herzen & Uyeda (1963).
The question arises as to which values should be assigned to V and K in equation (1 .I),
since, because of sediment compaction, the 'velocity' decreases and the thermal diffusivity
increases with depth of burial. Hutchison et al. (198 1 ) estimate the variation of V / K "with
~
depth for clay sediments in the Gulf of Oman; for depths greater than 2 km the argument of
the error function (equation 1.1) is reduced to approximately one-quarter of its surface
value. Furthermore, the model cannot account for changes in sedimentation rate through
time, so the application of expression (1.1) is limited in areas where the deposition rates
have varied appreciably. As a result of these uncertainties, the corrections calculated by the
Benfield and other analytical models are often inaccurate and poorly defined, with the result
that much potential information in present high accuracy heat flow surveys is not realized.
The analysis described in this paper has been developed in an attempt to overcome these
problems and provide a more detailed insight into the effects of sedimentation, compaction
Oceanic heat flow
441
and related pore fluid movement. Analytical models are used to estimate the variations of
physical properties, particle and fluid velocities with depth in the sediment column. These
provide the basis for a numerical solution to the heat flow equation which, in addition to
sedimentation, allows calculation of perturbations from radiogenic heating and seafloor
temperature changes.
2 The physical model
In its simplest form, the sediment system can be represented by a two layer model in which
the horizontal dimensions extend to infinity. A simplified 1-D equation can then be formulated and solved for the vertical heat flux. Fig. 2 shows a unit area section through this
system, with the upper layer consisting of sediment grains and pore water distributed according to the porosity function $ ( z ) , while the lower region is taken as non-porous basement.
The upper surface ( z = 0) represents the sea/sediment interface and defines the reference
frame for the following analysis. Sediment is input to the system at z = 0 with a rate Vo, as
measured by the individual sediment particles.
::$
x4
Figure 1. Simplified two layer model through the sediment/basement column. Sediment is added at
z = 0 with a rate V,. The porosity is described as a function of z, @(z) with the surface value ribo. The
basement interface occurs at depth B. Elements of depth dz at Z and Z' within the sediment and basement
layers are discussed in the text; the difference in flux in across u 1 and out across u 2 plus the radioactive
heat production, A, are related to the rate at which heat is gained or lost from dz.
The desired heat flow equation within the sediment is obtained by considering an element
dz at depth Z (Fig. 2). The rate at which heat Q (z, t ) is gained (or lost) from dz is given by
the difference in heat flux, F ( z , t ) across surfaces u1 and u2, added to the internal heat
production, A ( z ) :
= - a,F
dz + A .
(2.1)
In these, and subsequent equations. the depth and time arguments of parameters such as
heat flow, heat production etc. are omitted for simplicity. Thus, F ( z , t ) has been denoted as
F, Q ( z , t ) as Q and A ( z ) as A.
2
I. Hu tchison
442
The flux, including conductive and advective components is given by:
a,r+[p,C,v,~
F = -K
t p,c,v,(1
-
-
41 T
where
= temperature at depth z, time t,
T(z, t )
V, (z, t ) = pore fluid density, heat capacity and velocity,
p,, ,,c
p,,
= sediment particle density, heat capacity and velocity,
c,, V,(z, t )
= composite thermal conductivity at depth z.
K(z, t )
The rate at which heat is lost from dz is related to the temperature T(z, t ) by:
atQ = c * atT
(2.3)
where c(z, t ) = specific heat capacity of element dz:
(2.4)
c = PWCWd + PSC, (1 - d).
Equations (2.1), (2.2), (2.3) and (2.4) are combined to give the heat flow equation:
-
a , ( ~ a , ~-) a f ~ p W ~ , ~ , ~ + ~ , c , ~@)I
, ( 1Tn + A = [ ~ , ~ w ~ + ~ , ~a,7:
, ( ~ (2.5)
- ~ ) ~
A similar analysis for the element dz’ at 2‘ in the basement layer yields the equation:
(2.6)
KBaiT - PBCBVBa,T= P B C B ~ ~ T
where KB, pB and c B are the basement conductivity, density, heat capacity (all assumed
constant) and V B ( t )the basement velocity.
It remains to find expressions for 4, K , V,, Vw and VB. Assuming normally pressured
sediments, the porosity can be written as:
(2.7)
d (z) = d o exp (- z/X)
where I$, = surface porosity [do= 6 (O)], X = compaction constant, both of which are
characteristic of the sediment type (Rubey & Hubbert 1960).
The thermal conductivity of a composite of water and sediment is derived from Budiansky
(1970) and given as a function of the porosity:
where a = 3@(k,- k,) + k , - 2k,, with sediment and water thermal conductivities k,, k,.
Possible alternative expressions for K are given by Woodside & Messmer (1961) and
Robertson &Peck (1974).
The expression for V, is obtained from the conservation of mass within element dz. Since
the porosity (and so the mass of sediment) within dz is a function of z alone and does not
change through time, we require the mass flux of the solid fraction to be constant:
a2 h
K(1
-
dl1 = 0.
On expansion this gives:
aV
,,
d
+
X(1 - $1
~
*
1
V, = 0, since, from (2.7), ilz@= - - @
x
which has the solution:
P
v,= (1 - 4’
0 = constant.
(2.9)
Oceanic heat flow
443
We require V,(0) = Voand I$ (0) = Go, thus:
P = Vo(l
- 40)
so:
(2.10)
Since porosity is a function of depth alone, we see that, for a constant Vo,the sediment
particle velocities similarly depend only on z, i.e. V,(z, t ) = V,(z).
As the porosity falls with increasing depth of burial, the velocity of the sediment grains
with respect to the seafloor is reduced; at great depths (* A), the porosity is effectively zero
and the sediment velocity is given by V, Vo(1 - GO), the surface velocity multiplied by the
fraction of solids in the surficial sediment.
Applying a similar argument allows the evaluation of the water velocity, V, (2. t ) . Conservation of fluid mass at depth z requires:
^I
a, ( P W V W 4 )
=0
with @ ( z )given from (2.7) we can obtain:
V, exp (- z/X) = y
where y is a function of t alone and incorporates the constants $0 and pw from the left side
of the equation.
Thus, writing the equation for the water velocity in full gives:
V, ( z , t ) = Y ( t ) exp (z/X).
(2.11)
This gives the unusual result that as z + m so V, -+m. However, this can be reconciled by
considering the boundary conditions on the water movement. At the basement interface
(z = B ) we require the sediment, basement and pore water velocities to be equal, i.e.
V,(B) = V,(B, t ) = Vs(r).
Setting z = B in equation (2.1 1) and equating with the expression for the sediment velocity
at depth B from (2.10) gives:
which allows us to evaluate y:
(2.12)
Thus, with (2.1 I), we obtain the general expression describing the fluid velocity:
(2.13)
The pore water velocity is thus related to both the position within the sediment column and,
through the basement depth, B, to the total thickness of the sediments. The velocity across
the seafloor ( z = 0) decreases as B increases; for very thick sediments with B s h, V,(O, t ) is
negligible and no further fluid enters the system. Within the sediments, the pore fluid
velocity increases with depth up to z = B, where it necessarily matches the basement rate.
444
I. Hutchison
Since V, (B, t ) = VB ( t ) ,we can evaluate the basement velocity:
Vo(1 - $0)
VB =
(2.14)
[1 - $0 exp (- B/h)l.
Thus, as the basement depth increases, the basement velocity, VB decreases.
Expressions (2.13) and (2.14) for V, and V, are implicit functions of the time, t, related
through the basement depth, B. B ( t ) is found from the kinematic condition, i.e.:
VB( t ) = a,B
thus:
Vo(1 - $ 0 )
=d*B
(1 - $0 exp (- B/VI
which, after integration, gives:
Vo(1 - Q0)t = B ( t )+ $o X exp (- B/h) + E .
The constant
E
E
is found by requiring B = 0 at t = 0, thus:
= - $OX
so that:
Vo(l-&,)t=B(t)+$oX
[exp(-B/h)-l]
(2.15)
which can be solved by iteration.
The relations for sediment, pore water and basement velocities from equations (2.10),
(2.13) and (2.14) are illustrated in Fig. 3, showing their relative values for different depths
(expressed in units of the sediment compaction constant, A) and times (in terms of basement
r:
0.0
V/Y
v/v,
6.2
6.4 0.6 0 . 8
1.0
V/Y
Figure 3. Evolution of sediment (S), pore water (W) and basement ( B ) velocities as a function of the
surface sedimentation rate, V,, for different basement depths ( h = compaction constant of the sediments).
Oceanic heat flow
445
depths) after commencement of sedimentation. The velocity of the pore fluid relative to the
individual sediment particles is given by the difference between the 'water' and 'sediment'
branches of Fig. 3. Initially, the two velocities are similar and the fluid moves with the
sediment. As the column develops, the differences increase and the pore fluid appears to
migrate upwards past the individual sediment particles. The maximum relative velocity is
VO,occurring at the surface when B s A. Notice, however, that the water velocity is always
positive relative to the seafloor, so that no pore fluid can escape from the sediment system;
rather, the sediment appears to 'fall' through a surficial water layer.
Note that, within the sediments, the total mass flux of water, V, @, at a given time is
constant with depth, i.e.
-
V,$ =
VO@O(l
-
@o)exP(- B/X) = constant
[ 1 - $0 exp (- B/X)I
and that, as stated above, as B + 00 so the total water flux + 0. It is interesting that for basement depths greater that 2h(X- 1-2 km for most sediments) the rate at which water is
drawn into the system(i.e. across the seafloor) is less that 1/10 of the observed sedimentation
rate.
It is informative to examine the full form of the advective term of the heat flow equation
in the sediment region (2.5); making the substitutions for @, V, and V, gives the term:
(2.16)
Comparing the water (W)and sediment (S) contributions then gives:
W -_
s
PWC,
PSC,
[exp
$0
(m)
-
(2.17)
@Ol
Given typical physical properties (Table l), at B < h the water contribution dominates, at
B - X the heat advected by pore fluid is only - 50-80 per cent of that carried by the
sediment and for B > h the water contribution is negligible.
The analysis has used the exponential porosity depth law (equation 2.7) to allow us to
derive analytical expressions for the various sediment properties as functions of depth. This
relationship is used extensively in modelling compaction effects (e.g. Sclater & Christie
1980; Royden & Keen 1980), and generally forms a reliable approximation for normally
pressured muds, shales and carbonate sediments (e.g. Magara 1980; Schmoker & Halley
1982). Alternative porosity functions have been proposed, including linear reductions in
porosity with depth in sands (Magara 1980) and a quadratic relationship for deep sea muds
(Hamilton 1976). In principle, these could also be used to derive expressions for the fluid,
sediment and basement velocities. However, since we are concerned primarily with oceanic
environments where sediments are predominantly clays and mudstones, the use of the exponential law is well justified. Results also rely on the assumption ot a normally pressured
sediment column, i.e. we assume that the sediment permeability is sufficient to allow the
calculated fluid/sediment flow rates without causing significant pressure build-up.
Data for the permeabilities of deep sea sediments are sparse, but assuming values of about
10-9ms-' (see, e.g. Abbot et al. 1981) with a fluid/sediment flow rate of about 10-"ms-'
(from Fig. 3 , typical of sedimentation at 100-1000mMyr-') we obtain excess pressure
gradients of about
times hydrostatic. Thus, except in conditions of very high rates and
low permeabilities, the effects of overpressuring are likely to be minimal and the relationships linking porosity, sedimentation rate and fluid velocity (equations 2.7, 2.10, 2.15 and
446
L Hutchison
Fig. 3) are likely to form a reliable approximation of the properties for a real sediment
column.
3 The numerical model
The heat flow equations for the system described in Section 2 have been solved numerically
using an implicit finite difference scheme developed from the methods described by
t=O
Figure 4. Outline of the numerical model of the sedimenting system: On the left, characterization of the
solution in terms of the discrete points Tm, in z--t space. The lower region represents the basement,
buried by sediments to a depth B (t). To the right, assignment of physical properties and temperature
pomts used in the difference equations.
Table 1. Basement, sediment and pore water physical properties.
Basement
Density,
=
3330 kg m-3
Thermal Conductivity,
=
3.10 Wm-'K-'
Specific Heat Capacity,
=
1160 J k8-IK-l
=
0.0 p ~ m - 3
(Constant volume)
Heat Production,
Pore Water
Density,
=
1030 kg m-3
Thermal Conductivity,
=
0.67 Wm-lK-'
Specific Heat Capacity,
=
4180 J k g - I K - l
Heat Production,
=
0.0 p ~ m - ~
Sediments
Matrix Density,
Sandstone
Shale
Limestone
Salt
2650
2700
2710
2160
62
60
24
0
0
(kg m-3)
Surface Porosity,
(%)
Compaction Constant,
2.78
1.54
6.25
4.18
1.88
2.93
(km)
Thermal Conductivity,
(wm-
5.06
I K-I )
Specific Heat Capacity,
1088
837
1004
854
0.84
0.0
(J kg-IK-l)
Heat Production,
(p~m-~)
0.84
1.05
Oceanic heat flow
441
Beaumont, Keen & Boutilier (1982). The differential terms in equations (2.5) and (2.6) can
be approximated by expressions involving the finite differences Az in depth and A t in time;
the model solution in z - t space is then represented by the temperature values Tm, on the
grid of points ( m A z , n A t ) m = 1, M; n = 1, N (Fig. 4). The initial temperatures, boundary
conditions and physical parameters are set at t = 0, then the model allowed to evolve by time
stepping through the A t i . At each iteration, the numerical model is updated in accordance
with predictions of the physical model.
Before describing the numerical methods used, some general consideration of the model
configuration is required, particularly with respect to boundary conditions and geometry.
Following the analytical models, the lower boundary condition was initially set as constant
conductive flux at depth, so that the temperature gradient at the bottom of the model
should remain constant through time. Consequently, there is a minimum allowable depth at
which the conductive heat flow would remain essentially unaffected by the processes at the
upper surface. This requirement has been investigated analytically, using solutions for a
uniform half-space with the thermal properties of oceanic basement (Table 1) moving away
from z = 0 at 100 mMyr-' to give the profiles of heat flow versus depth plotted in Fig. 5. It
can be seen that by setting Z,,
greater than 150km the thermal transient will, for all
practical purposes, be contained within the model, since the flux at this depth is altered by
less than 1 per cent after a sedimentation event duration of 100Myr. However, because of
the sediment cover in a real model, the thermal properties will change rapidly over the upper
140
-
160 -
180
-
200
0.75
I 1 I
1
~
0.80
1
1
1
0.85
I
~
1
1
1
0.90
1
~
1 I1 l 1l 1
0.95
~
1 . ~ 0 1.05
O/O,
Figure 5. The fractional alteration to the geothermal flux as a function of depth and time (0-100 Myr),
calculated from the analytical solutions for the temperatures within a uniform half-space with basement
physical properties ( K = 8 X lo-' m zs-') moving at velocity 100 m Myr-' away from z = 0.
I. Hutchison
448
lOkm of the system, so it is also desirable to employ a small grid spacing, Az. Thus, a
conflict exists between the need to keep Az small and yet to have a large value of Z,,,,
since an excessive number of depth elements would result in unreasonably long computation
times. With this factor in mind, the numerical model was developed including the option of
making the coordinate transform:
y = z'l"
Most models were calculated withn = 2, so that the effective box size increased quadratically
with depth.
Details of the numerical solution of the heat flow equation are given in the Appendix. It
can be shown that the temperatures of the model grid points at time t are related to the
temperatures at the previous time, t - At by M-2 simultaneous equations, where M is the
total number of discrete depth points used (normaliy, M was set at 50 or 100). The coefficients of these equations are determined by the average values of conductivity, heat capacity
density, radioactive heat production and advection velocities obtained from the physical
model for the time interval t - At, t. In practice, the simultaneous equations were solved by
expressing them in the form of a tri-diagonal matrix of rank M-2 which was then reduced
using a Crout factorization method, available as a standard routine on the Cambridge
IBM 3081 computer. By repealing the computation N times, temperature profiles of the
system are obtained for the individual times n A t , n = 1, N. At each step, the matrix coefficients are recalculated according to the predictions of the physical model. In practice, results
proved to be largely insensitive to the discrete way in which these parameters were changed.
The numerical model was tested using two configurations with known analytical solutions.
By assigning @o= 0 and the same thermal properties to the sediment particles as to the
basement, the uniform half-space described by the Benfield model was simulated. The
analytical and numerical solutions generally agreed to within 1 per cent for suitable choices
of Ay and At (e.g. for Vo=IOOmMyr-' Ay =
and At = 1 Myr gave errors in the
surface flux of 0.2 per cent at t = 1 Myr and 0.03 per cent at t = 10Myr). Secondly, setting
Vo= 0 with uniform physical properties allowed the comparison of numerical models with
the relationship of heat flow with age for a cooling oceanic plate given by Parsons & Sclater
(1977). The initial temperatures for the model were calculated from the analytical solution
for a cooling half-space at an age of 1 Myr and the top and bottom boundary conditions set
as T, = 0 and T , = constant = 1333°C respectively. The computed values agree to within 1
per cent of the Parsons & Sclater approximation over the range in which it is valid (i.e.
5-100Myr).
-
4 Model results
4.1
D E P E N D E N C E O N $,
X k,
c,, p,; Vo=C O N S T A N T
Fig. 6 shows the heat flux as a percentage of the initial equilibrium value calculated by fixing
k , and p, as 2.5 W m-' K-l and 2700 kg m-3 and varying the parameters Go from 50 to 80 per
cent and X from I to 4 km. The pore water and basement properties are assigned in Table 1
(after Beaumont et al. 1982) and the surface sedimentation rate held at 5 0 0 m Myr-'. The
corresponding curve for a uniform half-space of basement thermal properties and a velocity
of 5C)OmMyr-' is also plotted. While the initial alteration to the flux is greater for the
compacting sediments, the long-term reduction is considerably less than that predicted by
the analytical model. The magnitude of the reduction is primarily determined by the parameter
with the higher porosity sediments affecting the flux least; the compaction
constant X plays a secondary role, the larger values giving a slightly increased alteration. The
Oceanic heat flow
449
0.95
--
0.90
E-
0.85
0.80
- -
=I
0.75
h=l
-
b=eo%
F
0.70
0.65
0.60
0
20
40
88
60
100
TIVE I V R t
Figure 6. Dependence of the heat flux alteration on the compaction constants @I,,,h. The parameters V,,,
K,, c,, p , are fixed as 500 m Myr-' , 2.5 W m-' K-', 900 J kg-' K-' and 2700 kgm-'. Also shown is the
analytical solution for a basement half-space with the same 'sedimentation' rate. All curves terminate after
deposition of 10 km of material.
results shown in Fig. 6 clearly demonstrate the influence of the advective term in the heat
flow equation, as discussed in Section 2 ; after initial drawdown of pore water, the net mass
flux into the system is primarily proportional to 1 - &(equation 2.16) and the surface flux
is reduced accordingly.
In addition to @,, and h, the grain thermal conductivity, k,, heat capacity, c, and density,
ps also vary with sediment type. Consequently, the alteration to the flux shows a significant
dependence on the nature of the sediments. The parameters listed in Table 1 have been used
to calculate heat flow through time for deposition of salt, sandstone, limestone and shale
(Fig. 7). The curves each account for the accumulation of 10 km of sediment. Least alteration occurs with the higher conductivity salt and sandstones, while the largest reductions
result from the low porosity, low conductivity limestones. For the same depth of deposited
1 .@@
0.95
0.90
0.85
0.80
2
1
0.75
0.70
0.65
0.60
\
Basement
0.55
1
0
"
"
l
~
10
"
'
I
'
'
"
I
"
"
20
I
30
"
~
'
I
40
50
TIVE I M R J
Figure 7. Comparison of the alteration to the surface heat flow due to the deposition of salt, shale,
sandstone and limestone for V, = 500 m Myr-'. Sediment properties are assigned in Table 1. The corresponding values for a basement half-space are also shown. Each curve accounts for 10 km of deposited
material.
I
450
I. Hu tchison
-
0
20
60
40
1 0 0 m/Ys
100
80
TINE (PIRI
Figure 8. The variation of heat flux through time for deposition of shales with V, = 10, 100, 250,500 and
1000 m Myr-'. The dashed lines link points of equal basement depth.
material, the corrections for salt and sandstone are approximately 50 per cent of those
found for limestone or calculated from the equivalent analytical solution for a half-space
with oceanic basement thermal properties.
4.2
Vo
D E P E N D E N C E ON S E D I M E N T A T I O N R A T E ,
The most important contributions in determining the sediment corrections are the rate and
history of deposition. These factors have been investigated in detail for shales (see Table l),
1.1
,
L
L
1 .E
0.9
x 0.8
3
I
1
L
0.7
0.6
0.5
c
0 .a
z
1.0
g
2.0
W
ZI
m
3 .E
a
;
\
l
;
;
;
h
c
g
200
0
E .E
.
20.0
,
40 -0
.
TINE I N R I
,
.
60.E
,
1
80.0
Figure 9. Example of the changes in surface flux due to a variable sedimentation history. The depth of
burial of the basement and the deposition rate are plotted beneath the calculated heat flow (expressed as
a fraction of Qo).
Oceanic heat flow
45 1
with Fig. 8 showing the calculated flux through time for sedimentation rates up to 1000 m
Myr-'. The dashed lines link points of equal sediment thickness. Up to SOOmMyr", the
alteration is primarily a function of the sedimentation rate; only for very early times (less
than 10Myr) are the curves strongly dependent on the duration of the sedimentation event.
However, at very high sedimentation rates (in excess of 500 m Myr-'), significant alteration
to the flux continues through time. Since the physical properties assigned to shales are
typical of most deep ocean sediments, the curves plotted in Fig. 8 can be used as a simple
nomogram to allow the estimation of sedimentation corrections for most oceanic environments. It is interesting to note that as little as 1 km of sediment, even when deposited at
slow rates, can give a significant reduction in the flux.
As an example of the variations in heat flow due to changes in the sedimentation rate,
Fig. 9 shows the results of a model of shale deposition at 100 mMyr-' for a period of lOMyr,
increased to 500 m Myr-' over the following lOMyr, after which time sedimentation ceases.
The surface flux responds rapidly to the local influence of the sediment, but after the
deposition stops, the system recovers slowly, with a time constant which is characteristic of
the thermal response of the entire lithosphere.
4.3
CORRECTIONS F O R A COOLING PLATE
A primary aim of calculating sedimentation corrections is to allow a comparison between
measured heat flow values and those predicted by lithospheric thermal models. However, the
oceanic plate model requires fixed initial temperatures at depth and a constant temperature
rather than constant flux lower boundary condition to account for the observed heat flow
and lithospheric thermal time constant. Thus, it is useful to investigate the sedimentation
effects for alternative boundary conditions and initial temperature structures. By initiating
the model with temperatures calculated for a cooling plate of given age and fixing a constant
temperature lower boundary condition, the behaviour of the surface heat flux in a sedimented ocean basin can be evaluated. To compare the results from a plate model with those
using the constant flux boundary condition, the calculated surface heat flows are expressed
as a percentage of the flux found for the same plate configuration but without sedimentation. Fig. 10 gives the modelled heat flow alterations for deposition at 250mMyr-',
commencing at crustal ages of 7, 27 and 48Myr, as well as that calculated for the equili-
3
LL 0.85
x
]L
E
20
40
60
80
100
TIME ( M R 1
Figure 10. Heat flow corrections calculated for shale deposition at 2 5 0 m Myr-' from the initial temperature structure of a cooling plate at 7 , 27 and 48 Ma in addition to the constant flux model.
452
I. Hutchison
1 .OO
0.95
0.90
0.85
0.80
x
3 0.75
L
0.70
0.65
0.60
0.55
0.50
i
0
I
20
'
"
'
I
~
'
J
40
60
"
'
'
80
'
~
1
~
~
~
~
I
100
TIME I M R I
Figure 11. Surface heat flux calculated for shales at V , = 100, 500 and lOOOmMyr-'. (a) Neglecting radiogenic heat production (solid lines) and (b) allowing for a sediment heat production of 1 0 - 6 W m - 3on a
background flux of SO mW m-2 (dashed lines).
brium plate model (i.e. constant flux). While the general trends are the same in all four cases,
the flux is attenuated less by deposition on the younger crust. For most practical cases the
differences between these alternative models are sufficiently small that they may be
disregarded.
4.4
CONTRIBUTION FROM RADIOGENIC HEATING
Most sediments contain small quantities of radioactive materials which contribute a component of internal heating to the geothermal flux. By assigning a heat production of
W m-3 and a background flux of SO mW m-2, the sedimentation curves for shale including
radiogenic heating have been calculated. In Fig. 11 these values are compared with the shale
relationships from Fig. 8 which were estimated neglecting internal heat production. Since
the radiogenic energy supply is fixed, the relative correction depends on the background
flux. Thus the curves shown in Fig. 11 represent the range of possible values; for a very high
geothermal flux, the radiogenic component is negligible and the corrections follow the lower
curves, whereas at low fluxes (e.g. 50mWm-2) the internal heating can offset up to 40 per
cent of the sedimentation effect.
4.5
V A R I A T I O N S IN S U R F A C E T E M P E R A T U R E ,
T,
Normally, the bottom water temperatures in ocean basins remain stable over long periods of
time and can be considered constant for the purposes of heat flow analysis. However, there
are occasions when this is not applicable; for example, in restricted basins where large
increases in temperature can accompany the deposition of salts and evaporites. This effect is
demonstrated in Fig. 12, where the sediment history used in Fig. 9 has been recalculated
assigning a 20°C higher surface temperature during the 500 m Myr'.' sedimentation event.
The sudden changes in surface conditions give sharp transients in the calculated flux and the
mean value is slightly lower than the constant temperature model during the period
10-20 Ma. Immediately following the return to normal surface temperatures, the flux is
enhanced, but after a further 20Myr the two models give indistinguishable results. While the
1.1
Oceanic heat flow
,
453
L
1.0
0.9
x 0.8
3
lL
-.I
0.7
0.6
+
z
w
W
x
2
0.5
0.0
1.0
2.6
m
3.0
- 1600
U
E
-=
\
w
t
g
800
600
466
260
0
t
1
0.0
t
20.0
40.0
60.0
80.0
T I M E IMFII
Figure 12. The effect of variations in surface temperatures for the sedimentation model used in Fig. 9.
The surface temperature is increased by 20°C during the period of rapid deposition between 10 and 20 Ma.
Background flux has been taken as 100 mW m-'.
deposition of sediment affects the temperature structure throughout the entire lithosphere,
varying conditions in the bottom water only affect the near-surface temperatures and, consequently, the perturbations show much shorter decay times. Thus, the heat taken up by the
crust during a time of high surface temperatures is lost over a similar period after the
temperatures are returned to normal.
5 Examples
5.1
THE G U L F OF OMAN
The results given in Section 4 have been used to estimate the perturbation to heat flow
measurements caused by the 7 km thick sediment pile overlying the oceanic basement in the
Gulf of Oman (Hutchison et al. 1981). While the present-day sedimentation is of silty clay
deposited at rates of 200-400 m Myr-', the detailed sedimentation history is unknown.
Taking the shale relationships including radioactive heat production (Fig. 1 1) to represent
the sediments of the Gulf of Oman gives the lower bound of the present heat flux as 80 per
cent of the value in the absence of deposition, assuming that the higher sedimentation rate
of 400mMyr-' has continued since 40Ma. The minimum alteration can be estimated by
assuming the lower sedimentation rate and a longer event duration. At these low rates, the
correction is determined primarily by the total sediment thickness (Fig. 8), giving an alteration to approximately 90 per cent for sedimentation which has continued over a period in
excess of 60 Myr. These values are less than estimated from analytical solutions which give
reductions of 20-40 per cent, depending on the values of 1' and K which are assigned.
454
5.2
I. Hutchison
T H E T Y R R H E N I A N SEA
Hutchison et al. (1985) describe a detailed heat flow survey located in the west central
Tyrrhenian Sea, within lOkm of the DSDP hole 132. The sediment section at the site is
dominated by 1-2 km thick layers of salt which were deposited when the Mediterranean was
partly desiccated during the Messinian (6.6-5.2 Ma) and the water level was 2-3 km lower
than at present. Assuming a lapse rate of 6°C km-', the ambient temperature at the sea
surface would have been about 15°C higher than during the periods of normal water level
before and after the salinity crisis. This effect is included in the thermal model as an
increased upper boundary temperature for the time covering the deposition of the salt. The
post-Messinian sediments consist of approximately 190 m of pelagic ooze with the physical
properties given from measurements at site 132: Vo= 38 m Myr-l; @o= 65 per cent; X = 2 km;
ps = 2700 kg m-3; k , = 2.5 W m-l R1.
The total salt thickness is not accurately known, but probably lies between 0,s and
1.5 km. Thus, since the duration of the salinity crisis is known, models for salt deposition at
rates of 500 and 1000 mMyr-' have been considered.
1.02
1 .OO
0.98
0.96
0.94
x
3
0.92
LL
0.90
0.88
0.86
0.84
0.82
3
i;w m
1,!I,,
I
I , ,
6
, ,
I , ,
5
, ,
I , ,
4
, ,
p
1000 m/Ma
, , I , ,
3
2
, ,
I , ,
1
, ,
[
0
TINE I N R I
Figure 13. Fractional variation of heat flux through time due to sedimentation and surface temperature
effects at Shackleton 3/81 survey D2 in the western Tyrrhenian Sea (Hutchison et al. 1985). The curves
are for Messinian salt deposition rates of 500 and 1000 m M y i ' with a surface temperature anomaly of
+ 15"C, followed by Plio-Quaternary sedimentation at 38 m Myr-I.
The calculated alterations are shown in Fig. 13; the most important factor is the rapidly
deposited Messinian salt, with the lighter Plio-Quaternary sediments giving little further
change in the flux. The heat flux-time plot is complicated by the effects of the surface
temperature changes at the beginning and end of the salinity crisis, however, they have a
short time constant and, consequently, do not affect the present heat flux significantly.
Calculations omitting the surface temperature changes gave virtually the same results for the
present-day heat fluxes. The corrections were also calculated using plate model initial conditions; the curves follow the same pattern, although the final fluxes are 1-2 per cent higher.
A further two western Mediterranean heat flow surveys reported by Hutchison et al. (1985)
gave alterations to 83-87 per cent (Balearic basin) and 92-96 per cent (southern Tyrrhenian
Sea); these compare with the values of 33-60 per cent for the Balearic and 60-90 per cent
for the Tyrrhenian estimated by Erickson & Von Herzen (1978) from an analytical approach.
Oceanic heat flow
455
6 Discussion and conclusions
The detailed variation of surface heat flow due to the processes of sedimentation, compaction
pore water movement, radioactive heating and surface temperature changes have been
investigated using a combined analytical and numerical approach. By fixing the reference
frame for the system at the seafloor, the relative motions of pore water, sediment particles
and basement material can be quantified. Equating the changes of total pore space within
the sediment system to the water flux across the seafloor has revealed the effect of dewatering with compaction; the water always moves into the sediment system, but the rate of
drawdown declines as the total sediment thickness increases.
The analysis has been developed for a normaily pressured, plane layer section whereas,
within a real sediment column, lateral variability and overpressuring may occur. In practice,
these approximations are unlikely to affect the validity of the predictions for deep oceanic
environments. Localized deviations from the computed results may occur, but the analysis
remains valid for the bulk properties of the water and sediment volumes, assuming that some
generalized porosity depth function can be defined. Clearly, a negative (i.e. upwards) pore
fluid velocity can only be achieved by non-equilibrium effects, such as hydrothermally
forced vertical flow or transient reduction of overpressuring. Conversely, our model suggests
that if relative fluid/sediment particle flow rates of the order of the sedimentation rate
cannot be achieved because of poor permeability, overpressuring must result.
Large-scale horizontal advection of heat by fluid movement, either within the sediments
or the underlying basement, have not been considered. In view of the low permeability and
lateral pressure gradients of most deep marine deposits this is unlikely to be a significant
omission within the sediment layer of the model. Hydrothermal circulation within the
upper fractured and porous part of the oceanic basement may be more important, however,
this is likely to affect only areas of young crust.
The techniques described could be extended to include thermal effects in shallow sedimentary basins, the present results being representative of such systems in the absence of
ground water circulation. The numerical analysis would require further additions to cover
those cases where high permeability sediments, such as sandstones, may have acted as
aquifers allowing the lateral advection of heat by fluid movement.
The analysis presented in this paper has several direct consequences for the interpretation
of lithospheric thermal models. For the simplest, the oceanic plate model (Parsons & Sclater
1977), we see that even low sedimentation rates and moderate sediment thicknesses will
affect the measured heat flows. For example, l00Ma crust with 1 km of sediment will show
a surface flux which is 5-10 per cent less than that expected in the absence of sediments.
Qualitatively, we would expect uncorrected measurements on old crust to be systematically
low. Clearly, if heat flow is to be used to constrain lithospheric properties, we must ensure
proper correction for sedimentation.
A number of models have been proposed to describe the thermal evolution of sedimentary basins. McKenzie (1978) calculates the heat flow and subsidence for a basin caused by
simple stretching of the lithosphere. From the outset, the model ignores the effects of radioactive heat production in sediments and the underlying continental crust, as well as any
effects of sedimentation. For example, the present-day heat flow in the central North Sea
averages about 60 mW m-', whereas the contribution from lithospheric stretching alone lies
between 35 and 40mWm-'. Sclater & Christie (1980) modified the McKenzie model to
include a radioactive continental crust, giving predicted heat flows of 59-62 mW m-'. However, 3-6 km of sediments (mostly mudstones) have been deposited since the Late JurassicEarly Cretaceous stretching event. From Figs 8 and 11 we might expect a sedimentation
correction (including sediment heat production) of 10-20 per cent. The corrected heat flow
456
I. Hutchison
is then nearer to 70 mW m-2, giving a discrepancy of 35 mW m-’ from that predicted simply
from the basic lithospheric heat flow; this difference must be accounted for by alterations to
the heat production of the continental crust which underlies the sediments, Thus, although
the sedimentation effects are less than the crustal heat production, they are nonetheless
significant. Several variations have been proposed following the simple McKenzie model, e.g.
two layer stretching (Royden & Keen 1980), stretching with finite rifting times (Jarvis &
McKenzie 1980); it remains that in each case, the observed heat flows are likely to uiiderestimate the model values unless correctioils for the effects of sedimentation are made.
Turcotte & Ahern (1977) obtained similarity solutions for heat flow in a sedimentary
basin formed by passive infill of thermally driven basement subsidence. In these models,
bulk values of conductivity and density were assigned t o the sediments and any transient
effects of sedimentation were neglected. The sedimentation rates are predicted to fall as
(age)”2 with sediment thicknesses up to 10 km at l00hla; from Fig. 8 , we can estimate that
surface heat flows predicted by this model may exceed the observed values by as much as
25 per cent. Langseth et al. (1980) refined the Turcotte & Ahern model to calculate the
transient effects of sedimentation. However, the limitations of constant physical properties
and fixed subsidence history remained. From the present results, it is clear that these are
severe limitations and the model is unlikely to provide realistic estimates except in a very
few cases.
To recap, the specific conclusions from the present work are presented below:
(1) For a given rate, the alteration to the surface flux depends on the type of sediment
being deposited; sediments with a low porosity and large compaction constant (e.g.
limestone) result in the greatest reductions. High conductivity salts and evaporites affect the
flux less than any other type of sediment.
(2) The correction depends on the rate and history of deposition, although for slow rates
and long durations, the alteration can also be related to the total sediment thickness.
(3) Sedimentation history plays an important role in determining the present geothermal
flux, although the influence of any event is diminished through time following its conclusion.
After the sediment input ceases, the system recovers slowly as the entire lithosphere
temperature structure readjusts.
(4) Radioactive heating within the sediments can contribute a significant component to
the geothermal heat flow; for a low background flux this effect may offset the correction
due to sedimentation by 30-40 per cent.
(5) Bottom water temperature changes give rise to rapid alterations of the surface flux,
these variations are generally short lived and so their effects are localized to the upper few
kilometres of the crust. Consequently, the anomalies decay with a proportionally short time
constant, so that only very recent or very large temperature changes need to be considered.
The numerical methods developed allow a more comprehensive and accurate determination
of the environmental effects on the surface heat flux than can be obtained from pre-existing
analytical models. Fewer ad hoc assumptions are used and the physical parameters of the
system are better defined, giving results which are more directly applicable to the real crustal
configuration. In particular, the analysis shows that the corrections obtained from simple
analytical models (e.g. Langsetb et al. 1980; Erickson & Von Herzen 178) tend to
over-estimate the alteration of geothermal flux by prolonged sedimentation.
Acknowledgments
I would like to thank Dan McKenzie for his encouragement and many helpful comments
during the development of this work. I am also grateful for the support of a Shell Inter-
Oceanic heat flow
457
national Oil Company Research Studentship during my time in Cambridge and to BP
Petroleum Development Limited for their assistance in the production of this manuscript.
University of Cambridge, Department of Earth Sciences contribution no. ES5 12.
References
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Benfield, A. E., 1949. The effect of uplift and denudation on underground temperatures,J. app/. Phys.,
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Jaeger, J. C., 1965. Application of the theory of heat conduction to geothermal measurements, in
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geophys. Res., 85, 3711-3739.
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Turcotte, D. L. & Ahern, J. L., 1977. On the thermal and subsidence history of sedimentary basins, J.
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Appendix: Numerical solution of the heat flow equation
Writing the simplified form of the heat flow equation for uniform thermal properties gives:
natT = KagT
va,T + A .
-
(All
This applies to regions within each model element Azi (Fig. 4), with the parameters defined
as :
Az,
77 = density * heat capacity,pc, for the element
K = composite thermal conductivity,
v = total advective component, pcu,
A
= radioactive
heat production
Allowing for a coordinate transform
= z l / n ,.
ag=~,a,+~~a;
a,=F,a,;
with
1
F1=--y('-");
(1 - n) y ( 1 - 2 n ) .
F2= ___
>
n2
I?
F
- - y12 ( 1 - n )
3-
n2
This equation is approximated by finite differences by expanding the temperatures as a
Taylor series about the grid points q,i.
Thus:
Ti-
l,j=
Ti, j
-
-
3, Ti, j A y + aY2 TZ .J J
,
*
~
AY2
2 ...
AY2
Ti+I,j= T i , i + a y T i , i . A y + a $ T i , j . - . . .
2
In regions of uniform physical properties expressions (A4) can be substituted directly into
(A3) to give a set of difference equations. However. within the sediments and at the basement interface the physical properties vary from element to element; this requires the
Oceanic heat flow
459
further condition of continuity of heat flow at the boundary (Fig. 4):
~ i - ~ T i , jK-i _ 1 a , T i , j l i - = V i T i , j - K i a , ~ , j I i +
or
('45)
~ i - 1T i , j - K i - i F t a y T i , j I i - = ~ i T i , j - K i F ~ a y T i , j l i + .
Using expressions (A4), d; q, li- can be found as a function of i ' Similarly, a$ Ti, li+ can be expressed in terms of Ti+l,fi q,j and
Substituting both forms into (A3) gives:
Ti,i and aye, j li-.
a,,c,j l i + .
where
2 F3
Ay2'
ff=-
yl=
[-+---
F2 f f A Y vi-1
FI FI Ki-1
1,
%-",
Y2=[""'+
F1
Ki
FI
In total, M - 2 simultaneous equations of the form (A7) are found for the temperature points
c,i, i = 2 , M - 1. Boundary conditions are imposed by specifying the temperatures at i = 0
and i = M , i.e.
'0, j
= Ts(t)
and
T M , ~ T=M - l , j + Q B ( Z M - Z ~ - l ) / K M - l
(constant flux)
or
(constant temperature).
TM,j = T'
The values 01 qi, ui and Ki for the individual elements are estimated from the physical model
outlined in Section 2. The configuration of the physical system is calculated for times t and
t + A t and the forward averaged values of the physical parameters are supplied as input to
the difference model. After each iteration the values are recalculated and updated before
input to the next time step.