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A fracture-loop thermal balance model of black smoker circulation
Tectonophysics, 122 (1986) 307 -324 Elsevier Science Publishers
307
B.V., Amsterdam
A FRACTURE-LOOP
- Printed
THEXMAL
in The Netherlands
BALANCE
MODEL
OF BLACK
SMOKER
CIRCULATION
M.R. STRENS
and J.R. CANN
of Geology, The Unk.wrsity, Newcastle tcpon Tjme, NEI 7RU (Great Britain]
Department (Received
February
25, 1985; revised version
accepted
July, 1985)
ABSTRACT
Strens.
M.R. and Cann,
J.R., 1986. A fracture-loop
thermal
balance
model of black smoker
circulation.
Tectonoph_wirs, 122 : 307-324. This
time-dependent
circulation
during
model
suppIy to black smoker Flow is modelled mid-ocean systems
systems
location.
turbulent
flow, which
to flow is P~~na~tIy
digitally,
calculated throughout
is driven
conductive
is based
on seafloor
by the buoyancy heat
flux from
of a hot spring,
Results show that this system cannot
recently-solidified particularly
Jf the circulation
thus provides
magma
favourable.
chamber,
We conclude
penetrates
and ophiolite between
upflow
tbe rock is computed
enabling obtain
magma
zone. The fault is allowed
at the rock-water
interface
through
hydrot~~ma~ the major heat
chamber. and very close to the
evidence
of black
and downflow. to propagate
analyticalfy.
smoker
temperature
Hydraulic downwards.
with the water
at every position
the exit water
the resulting sufficient
and
whether
in the model by a series of pipes carrying
difference
in the discharge
for each time step. The model
smokers.
hot rocks
in a single fault, which is parallel
flow
in the loop can be and mass flow rate
ore mass to be estimated.
heat from a thin layer of rock (such as the 1-2
km thick layer of Iavas and dykes over@ing an axial magma black
between
in order to establish
The fault is represented
so that the heat exchange
the lifespan
balance
ore deposits,
comes from the rock or from an underlying
ridge axis. This configuration and their structural
The horizontal
the thermal
of volcanogenic
as an open thermosyphon
resistance treated
examines
the formation
chamber)
a greater
ore masses of a few million tonnes that systems producing
to sustain tong-lived, ore-forming
volume
of hot rock can be formed
to the base of a if conditions
larger ore masses must tap a magmatic
are heat
supply.
The model we present here is one of a number of thermal balance models, each designed to investigate the exchange of heat between rocks and circulating hydrothermal solutions during the formation of volcanogenic ore deposits. In this paper we examine one aspect of the thermat balance of ocean-floor high-temperature hydrothermal circulation, such as gives rise to the black smoker hot springs of the East Pacific Rise. In these black smoker systems, the overall heat
30X
flux is many segment
times greater
of mid-ocean
than
is available
ridge (Macdonald
either in hot rock or in magma
on a steady-state
until it is tapped
by the hydrothermal
Here we test the efficacy of hot rock as a heat store, and extract rock into water
circulating
in a single
basis from a short
et al., 1980). Heat must therefore
fault parallel
be stored circulation.
the heat from the
to the mid-ocean
ridge, The
water flow is confined to a fault because of geological constraints which are discussed below. Our target is to produce a high-temperature (350°C) water flow for long enough to generate a sulphide deposit of 3 million tonnes on the ocean floor. Other models are possible. We have developed one in which heat is extracted from a magma chamber, as an alternative to heat extraction from rock (Cann et al., in press), and have also examined a third model in which recharge is through a porous medium rather than in a fracture. This model may be more applicable to systems with a greater area of recharge
THE CHARACTERISTICS
than seems to apply to black smoker systems.
OF BLACK SMOKER
CIRCULATION
We define a black smoker system from the characteristics of the hot vents on the East Pacific Rise, where, at 21 “N, the exit temperature of the hydrothermal fluid has been estimated at about 35O”C, with an exit velocity of 1-5 m s-’ (RISE, 1980, Macdonald et al., 1980), and, subsequently in the range 275”-35O*C, with flow velocities of 0.7-2.4 m s-’ and a mass flow rate of about 150 kg s ’ (Converse et al., 1984). At 13”N the exit temperature of fluid partially diluted with seawater is up to 33O”C, so the maximum fluid temperature will be somewhat higher. The flow velocity at 13”N is estimated at 0.5-2 m s -’ (Hekinian et al., 1983, 1984). At both locations the iron concentration in the hydrothermal fluid is about 110 ppm (Edmond, 1981; Michard et al., 1984). We assume that such systems are broadly analogous to the systems responsible for the deposition of the volcanogenic sulphide ore bodies found in ophiolite suites, although the known black smokers are thought to be very young and are associated with very small sulphide
deposits.
The efficiency of ore deposition is very variable between the observed vents. The black smokers themselves are extremely inefficient, since a high proportion of the sulphides is carried away by ocean currents. In contrast the warm vents on the Galapagos Spreading Centre are highly efficient, since sub-surface mixing with cofd water causes deposition of a high proportion of the dissolved metals as sulphides (Edmond,et al., 1979). Our goal in the model is the production of an average-sized Cyprus-type ore deposit of 3 million tonnes. We can estimate the time that this should take to form from the above data. With a mass flow rate of 150 kg s-‘, an exit water temperature of 35O”C, an Fe concentration of 110 ppm and 100% efficiency, the time required will be 3000 years, or proportionately longer for lower efficiency.
309 STRUCTURAL
CONSTRAINTS
ON BLACK SMOKER FLOW
All of the active black smokers discovered
so far are located within
a few hundred
metres of the axis of fast-spreading mid-ocean ridges, where the tectonic grain is very strongly aligned parallel to the ridge crest. The vents are invariably located on ridge-crest-parallel fissures or growth faults within the axial zone (RISE, 1980; Ballard et al., 1981; Hekinian et al., 1984), which is bounded by a series of major normal edge-crest-par~lel faults (Klitgord and Mudie, 1974; Searle et al., 1981). This close association of high-temperature hydrothermal substantiated by evidence from Cyprus, where the sulphide
upflow with faults is ore bodies and their
underlying feeder pipes are associated with faults running parallel to the ancient spreading centres (Adamides, 1980; Gass and Smewing, 1981), and, since they are mostly covered with a 100-500 m thickness of lavas, they also probably o~~nated within the narrow eruptive zone (Constantinou and Govett, 1972; Cann, 1980). There is therefore strong evidence that the hydrothermal upflow is channelled in faults. There is little direct evidence of the downflow zones, but consideration of the magnitude and structure of the permeability within the vicinity of the vents strongly suggests that the downflow is also predominantly in the faults. It can be shown quantitatively that a high proportion of the flow will be channelled by the faults since they form major discontinuities in the permeability, thus short-circuiting the convective flow which will preferentially follow the low resistance path. For example, a porous medium with a cross-sectional area of several square kilometres would require a permeabi~ty greater than lo-* m2 to carry the same mass flow as a 1 km length of a narrow, rough fault (see Appendix). There is little data on the permeability of the crust in these regions. The figure quoted by Anderson et al. (1982) from the sheeted dykes at the base of D.S.D.P. hole 504B is lo-t6 m2 in crust 6 m.y. old. In crust 7.2 m.y. old in D.S.D.P. hole 39514 near the Mid-Atlantic Ridge permeabilities
were
10-‘4-10-‘5
m2 (Hickm~
et al., 1984).
Even
if the pathways
have
become clogged with time it is unlikely that these rocks ever had a permeability as high as lo-* m*. It would therefore seem highly probable that recharge flow is also channelled by the faults. The major faults bounding the extrusive zone at a distance of about 1 km from the mid-ocean ridge axis impose a limit on the extent of the model perpendicular to the ridge axis, since these faults would have channelled previous circulation systems, thus putting a limit of about 1 km on the lateral extent of hot rock which has not already been cooled. The extent of the circulation system along the strike of the ridge also appears not to exceed 1 km from the observed spacing of black smoker vents, although this is highly variable (Ballard et al., 1981; Hekinian et al., 1984). The maximum area of hot rock from which heat might be gathered is thus about 2 km’.
310
DISCUSSION OF THE TYPE OF MODEL
The limited
structural
constraints
outlined
above,
namely
to a small area in which the fault zones, forming
permeability,
channel
the flow, mean that we cannot
as a valid representation In addition
that
smoker
flow is
major discontinuities
black
in the
use porous medium
flow theory
of this system in the model, but must turn to fracture
to the structural
constraints,
our choice of a fracture
flow.
flow system has
also been governed by the type of flow to be modelled. Porous medium flow theory is based on laminar flow, which is essentially slow and requires a well-developed permeability. These two conditions are not met for black smoker flow, which arises in rocks of low permeability (see above) and which has vent velocities of several metres per second. Several lines of evidence indicate that the flow is turbulent, as one would expect in conduits as rough as faults. Converse et al. (1984) described the type of flow in the 21’N vents as “plug” flow, which is a characteristic of turbulent flow in a circular pipe. In addition the Reynolds number of lo’-10h for the upflow (Converse et al., 1984) again implies that flow will be turbulent. We consider that the most realistic and practical means of representing flow in a very rough fault is by a series of small pipes carrying turbulent flow. We assume for simplicity that downflow occurs in the same fault as the upflow, but the model could easily be modified for downflow
in another
ridge-crest-parallel
fault.
PREVIOUS MODELS
Most existing models simulate large-scale, slow, low-temperature seafloor circulation, and none has been developed so far purely with the aim of modelling rapid, very high-temperature circulation. They have usually been designed to investigate the evolution and pattern of the convection cells over a large area in order to explain the distribution of heat flow values on the ocean floor (Ribando et al., 1976; Fehn and Cathles, 1979; Fehn et al., 1983). Since they cover large areas, the scale of the fault pattern is small relative to the total area and may be regarded as constituting a porous medium, so that porous medium theory may be used as a valid representation of the flow in this case. These models do not take into consideration any pre-existing fault pattern, although Fehn and Cathles incorporate zones of high permeability in the porous medium. The fracture-loop models of Bodvarsson and Lowell (1972) and Lowell (1975) predate the discovery of black smokers and hence do not attempt to simulate very high temperature flow. The only model so far that embraces black smoker flow is that of Lister (1974, 1982), which quantifies the evolution of the circulation system from a period of rapid flow, which causes the downward advance of a cracking front, thus creating the permeable matrix in which later, slow, closed circulation may persist for millions of years. Lister’s model predicts black smoker temperatures, but only when the permeability is as high as 10-7-10-9 m2, that is many orders of magnitude
greater
than the lO-“j
m2 measured
by Anderson
et al. (1982).
311
Our structure,
model
is therefore
the flow geometry, the constraints observed
different
since we are investigating
the above
models
balance
and then only of a high-temperature,
imposed
characteristics
THE STRUCTURE
from
only the thermal
by the geological of black smoker
structure
in both
rather
aims
and
than modelling
black smoker system within at mid-ocean
ridges
and
the
flow.
OF THE MODEL
The model examines the heat exchange between running parallel and close to the mid-ocean-ridge axis, The heat supply is maintained by horizontal conductive The model consists of an open thermosyphon within
convective flow in a fault, and the adjacent hot rocks. heat flow through the rock. a single constant-width fault,
which is treated as a series of pipes. Seawater at 0°C enters the fault and flows downward in a long zone extending along the fault, and eventually rises in a narrow upflow zone. The flow is driven by the buoyancy difference between the recharge and discharge zones, and the resistance to flow arises from the hydraulic properties of the system. Downflow and upflow velocities vary in proportion to the respective areas of recharge and discharge for continuity of mass. This model has been developed from
Fig. 1. The geometry of the model. The half-width of the fault only is shown.
312
a simpler model with the same geometry flow velocities were constant. Heat is exchanged conductive transient
at the water-rock
(Strens and Cant-r, 1982) interface
heat flow in the rock is calculated heat conduction
since it will be negligible
equation.
of the fault wall. The horizontal
analytically
from the one-dimensional
Vertical heat conduction
in the vicinity
but in which the
of a vent compared
in the rock is ignored, with the convective
heat
transfer. The convective water flow is treated digitally so that heat exchange is calculated at each level with the pencil of rock (referred to as a “block”) adjacent to each limb (Fig. 1). The two limbs of the system are assumed to be thermally independent.
Each block of rock has a different
but uniform
initial
temperature,
so
that any temperature profile may be set up. The model is run on a cyclical basis so that the contact temperature, which constrains the heat conduction, may be changed according to the new rock and water temperatures calculated at the end of each time increment. The fault is assumed to open suddenly to a given depth, and is then propagated downwards at a given rate unless the water temperature rises above a given temperature at which cracking is unlikely to continue. The fact that little is known about typical cracking temperatures, or the process of cracking at sub-seafloor pressures, necessitates this somewhat crude method. The advance of the fault is thus controlled by the fault propagation rate at lower water temperatures, but is arrested while the water temperature is high enough to inhibit cracking. Values given to these parameters
are discussed
below.
THE MODEL EQUATIONS
The solution to the one-dimensional transient heat conduction equation, constrained by a contact temperature, for a semi-infinite block of rock is given by Carslaw and Jaeger (1959): r,=T,+(T,-T,)erf
x i 2(at)“2
T,= T,
(Z=O,O
T, = T.
(x=0)
1
(1)
co)
T, is the rock temperature at distance x from the contact, Tc is the contact temperature between rock and water, T, is the initial rock temperature, x is the horizontal distance from the fault, (Y is the thermal diffusivity of the rock and t is the elapsed time. The contact temperature, q, in eqn. (1) is steady. To allow the contact temperature to be changed at the end of each cycle, while remaining steady during each cycle, eqn. (1) is modified in a way analogous to that given by Carslaw and Jaeger (1959, eqn. 3, p. 63) so that a factor is added for each change in contact
313
temperature: For n cycles: T,=
7;.,,,,+(T,-
X
7;.,,,) erf
erf i
If the heat transfer
with the water at x = 0 were perfect
the contact
temperature,
TC, would be the same as the water temperature T,. We have given some impedance to the heat flow by taking a weighted mean of the rock temperature (at x = 0.1 m) and the water temperature in inverse proportion to the thermal diffusivities of rock and water so that: TC= 0.2T,=, , + OJT,.. Other proportions were tested and the model was found to be relatively insensitive to this formulation, because the water and rock temperature are close except during a short initial period. The heat flow from the rock at x = 0 is derived as follows: Partial differentiation of eqn. (1) gives the temperature gradient at x = 0:
ar (T,-T,) Bx = ( Tat)‘/2 this means heat flow from rock at x = 0: -KA(q-c) q, =;” =
(2) ( nat)“2
where K is the thermal conductivity of the rock. The value of A, the heat exchange contact area, varies according to whether the equation is applied to the upflow zone or the downflow zone. The total heat flow, Q, up to time t is obtained by integrating eqn. (2): Q=2KA(7;-T,)
i
&
1
“I
(3)
Equation (3) is modified in the same way as eqn. (1). so that the contact temperature can be changed after each cycle. For n cycles, the conductive heat flux at the water-rock interface is: Q,,=2KA(q,,,-T,)
The heat flow from each block of rock during the nth cycle, Qcvc = Q,, - Q,,- ,, is equated to the heat removed by the water within and flowing through the fault adjacent to it, assuming the water to be a well-stirred fluid. Heat exchange equation:
Qcyc= c,,m4Ah
+ u,,Ar)(Ti
- T,.)
314
where cm. P,,, and u,,, are the specific heat, density and linear flow velocity of the water in the mth block. Temperatureand pressure-dependent values of density and specific heat are used. w is the half-width fault
and
increment,
takes
a different
i.e. thickness
temperature
value
of block,
of the fault, u is the dimension
for upflow
and
downflow,
At is the cycle length
and
along the
Ah is the vertical T: is the new water
after time At.
The mass flow through Continuit_); equation:
the system is defined
by the exit mass flow rate.
= awp,, u,, awp,,, u 111 where p_ and u,, are the exit density and flow velocity. This equation applies to both upflow and downflow provided the appropriate value for u is taken. The water temperature thus obtained at each position in the loop at the end of each time increment is then used to calculate the buoyancy force and hence the flow velocities for the next cycle. Flow equution:
where
u,. 5,. d, and f,
are the velocity,
mean
density,
friction factor for the upflow, p is the mean density the mean density of the downflow.
equivalent
diameter
and
for the whole system, and p1 is
Equation (4) is derived, as follows, by equating the total pressure drop in the system, assuming turbulent pipe flow, to the buoyancy force. The total pressure drop consists of the drops due to inertial forces at E (Fig. 2) the acceleration at A and friction in both limbs (Massey, 1979). Pressure drop = apuf + iP( U, - Up)’ + 4ph
$
? ?
Fig. 2. Diagrammatic limbs is indicative
representation of their thermal
E, and at the constriction,
:
fu’ + 2 y ’
i
of the flow system in the model. The space shown between independence.
Resistance
A, at the base of the upflow
the two
to flow arises at the entrance
to the system,
limb, as well as from the friction
in both limbs.
315
where uZ. d, and fi are the velocity,
equivalent
diameter
and friction
factor in the
downflow. Since
U, >> u2 and 2hf,/d,
B 4 for typical
values
of h - 1 km, f, - 0.01 and
d, - 0.03 m (see below) 2
Total pressure
drop
= 4$h 5
? 1
i.e. the pressure drop due to friction in the upflow pipe only. This pressure drop is equated to the buoyancy force kg(& - &) to give eqn. (4). The term 5d,/f, will be referred to as the friction constant (F). Its value is difficult to determine since no data are available on the roughness of the conduits, and reports of vent diameters range from 30 cm at the 21*N vents (Macdonald et al., 1980) to 3 cm at 13”N (Hekinian et al., 1984). Since the feeder zones seen beneath Cyprus ore deposits are composed of a network of narrow, anastomosing fissures, we have chosen to model the discharge zone as a bundle of narrow, rough pipes. If each pipe is taken to have a diameter of 3 cm and a friction factor of 0.01, corresponding to a roughness on the walls of the pipe of about a millimetre, which is consistent with geological observations, then, at a discharge water temperature of 350°C, and at a load pressure of 350 bar, it will pass fluid at a linear velocity of 2 m s-‘, which is the mean velocity observed for black smokers. A pipe of this diameter and roughness thus possesses the hydraulic resistance appropriate to black smoker conditions, and a bundle of 150 such pipes would pass 150 kg s-‘, the mass flow rate estimated for the 21°N black smoker field. The diameter, d,, and number, n, of pipes are related to the fault half-width, W, so that nndff4 = 4w2, assuming that the discharge zone is equidimensional. These values for d, and f, yield a value for F, the friction constant, of 15. CONSTRAfNTS
ON THE BOUNDARY
AND
INITIAL
CONDITIONS
The dimensions of the model are determined which the circulation occurs, which, for black mid-ocean
ridge axis. There is strong
evidence
by the structural smoker systems,
environment in is close to the
that very often, especially
on faster
spreading ridges, the eruptive zone is underlain by a magma-chamber at a depth of 1-2 km. Such evidence not only comes from seismic studies near spreading centres (Orcutt et al., 1976), but also from petrological studies on ocean-floor lavas (Bryan and Moore, 1977; Flower et al., 1977) and from geological mapping of ophiolite complexes, where thick gabbro units, clearly formed from crustal magma chambers, underlie about 1 km of lavas and 1 km of dykes (Moores and Vine, 1971; Gass and Smewing, 1981). Thus the hydrothermal circulation would be expected either to be confined to the l-2 km of hot rock overlying a magma chamber, or to extract heat from the magma itself, or to penetrate into the hot rock of a recently solidified magma chamber as far as the base of the oceanic crust. This provides clear
316
constraints circulation
on the depth of circulation in the model. We have therefore depths of 1 km and 6 km. For 1 km circulation the geothermal
taken as 1 K m-’ magma depth
chamber.
from 0°C at the seafloor
tested two gradient is
to 1000°C at 1 km depth, just above a
For 6 km deep circulation
of 1 km with the 5 km of rock below
the gradient at 1000°C
is again
1 K m‘-’
representing
to a
a recently
solidified magma chamber. Perpendicular to the ridge axis the limit of the extent of hot rock is about 1 km (see above). The extent of the mode1 in this direction is infinite, but the contribution to the heat flux from beyond a distance of 1 km is found to be negligible within the time span of several thousand
years.
ORE MASS CALCULATION
The model calculates the exit water temperature and mass flow rate for each time cycle during the life span of a hot spring. The resultant ore mass can then be calculated if one assumes that all of the dissolved iron is precipitated as Fe!&. The accuracy of this calculation depends on the accuracy of the iron solubility-temperature data. We have used line B in Fig. 3, which is based on experimental data of Mottl et al. (1979) with two points added: the measured concentrations in black smokers at 21“N and 13”N (Edmond, 1981; Michard et al., 1984), and Seyfried and Bischoff’s (1977) experimental point. Mottl et al.‘s points indicate an exponential increase in iron solubility with increasing temperature (line A). Since the system being modelled is an open system with a much higher overall water/rock ratio than Mottl et al.‘s experimental systems, we have constructed line B, which is heavily weighted towards the 21’N observational data and also Seyfried and Bischoff’s point which has a water/rock ratio of 50 : 1.
Fig. 3. Fe-solubility (Edmond,
1981),
data. Data are: e-from
q-Seyfried
Moitl et at. (1979),
@-21”N
black smoker
analysis
and Bischoff (1977). Line A is drawn through the points of Mottl et al. Line
B is the weighted line (see text) used for ore mass calculations.
317
The exponential relationship between Fe solubility and temperature shown in Fig. 3 means that the water temperature is the do~nating
factor in determining
the
resultant ore mass, and only when the water temperature is greater than 300°C is the iron solubility sufficiently high for it to contribute a significant amount of ore. The following results are based on an efficiency of deposition of 100%. RESULTS FROM THE M0DEL
Our aim, as stated above, is to test whether sufficient heat is available from the rocks to sustain the circulation system in the model, so that the vented water temperature is at least 35PC with a mass flow rate of about 150 kg s-l for several thousand years (or ~roportion~ly longer if the flow rate is less) in order to produce an iron sulphide deposit of 3 million tonnes. The values taken for the parameters required by the model are given in Table 1. In general these parameters are not well constrained, so the values taken represent the best estimate available to us for the ridge axis environment. The recharge zone length is taken as 1 km. This is based on observed spacings of black smoker fields, and on the fact that convection cells tend to be ~~dimension~. The fault propagation rate of IO m yr- 1 is based on Lister’s (1982) estimate. Other lower values were tested, including zero, but these gave lower water temperatures, thus reducing the potential for ore deposition and therefore of little interest in this analysis. The value taken for the crack propagation rate has little significance at high water temperatures which will inhibit cracking. Since we are unable to define the relative water and rock temperatures at which cracking will occur, we have tried to
TABLE 1 Values for the variable parameters in the mode8 Faranleter
Standard wfue
Recharge zone length
I km
Fault propagation
IOmyr-’
rate
Range of values
laitiai fault depth
5OQm
-
Width of fault
a.4 m
0.2-0.8
Friction constant
1.5
l-la0
m
Initial temp. field in rock circ. to 1 km
1
Km-’
10 K’m-’
to 100 m depth
10oO°C beIow zoo m ck
to 6 km
lKrnwl
tolkmdepth
and 1000°C below 1 km
-
318
make the model as realistic as possible by only allowing the crack to advance while the water temperature at the base of the circulation is under 370°C. By doing this the maximum observed
exit water
black smoker
temperatures
temperatures.
in the model
was chosen as the level at which the rock temperature estimate ashighasl
of the cracking
temperature
are similar
to the maximum
The value of 500 m for the initial
fault depth
will be below Lister’s (1982)
of 527’C, even when the geothermal
gradient
is
Km-‘.
The parameters that have highly variable, and therefore unknown values, namely the fault width and the friction constant, were tested through a range of realistic values. The fault width determines the water/rock ratio of the volumes exchanging heat, and the friction constant determines the linear flow velocity for a given water temperature. These parameters determine the mass flow rate and therefore the rate of heat removal. The standard values were chosen so that together they give a mass flow of around 150-200 kg s-’ at black smoker temperatures, which is consistent with estimates for observed black smoker fields (Converse et al., 1984). In addition to the standard initial temperature profile in the rock a “high” rock temperature profile was also tested for the 1 km deep circulation, since it is possible that black smoker circulation may arise shortly after volcanic activity. In this case the rock temperature was taken as 1000°C to within 100 m of the seafloor with a linear gradient above.
Circulation to I km depth The model was first run with the standard parameters (Fig. 4, a). After an initial burst of heat during the first year the exit water temperature never reached 3OO”C, so that the resultant ore mass would be negligible.
I
OO
500.
i 000
1500
1
Years
Fig. 4. Exit water temperatures for 1 km deep circulation. Geothermal gradient in rock = 1 K m- ‘. All parameters take standard values (Table 1) unless specified otherwise. a -standard friction (F = 1). c-narrow
fault width (0.2 m).
parameters, b-high
319
I
OO
500
Fig. 5. Exit water temperatures
1000
I
1500
for 1 km deep circulation.
Years Geothermal
100 m depth with rock at 1000°C below 100 m, to represent conditions Standard parameter values unless specified otherwise. c-narrow
fault width (0.2 m), d-high
a -standard
friction (F=l)
gradient in rock = 10 K m-’
to
shortly after volcanic activity.
parameters,
b-high
friction (F = l),
and narrow fault width (0.2 m). Potential ore
mass given in million tonnes.
The variable parameters were then tested systematically. The friction in the upflow pipe was increased to its maximum feasible value (Fig. 4, b) and the fault width was halved to 0.2 m (Fig. 4, c), but temperatures above 300°C were still obtainable only for about 30 years. These runs were repeated with the “high” initial rock temperature profile (Fig. 5, a, b, c) but temperatures above 3OO’C were still achieved for a maximum of only 130 years (Fig. 5, c). The extreme values for friction and fault width were then combined (Fig. 5, d) yielding an exit water temperature in excess of 300°C for 1000 years, but, because the prolonged high temperature was obtained by reducing the mass flow rate to about 20 kg s-l, the resulting ore mass was only 0.2 million tonnes, and very far short of our goal of 3 million tonnes.
Circulation to 6 km depth
Water temperatures of 300°C were again achieved only during the first year when the model was run with the standard set of parameters (Fig. 6, a). Once again it was necessary to reduce the mass flow rate to prolong high-temperature flow (Fig. 6, b, c) so that, with a fault width of 0.2 m for example, exit temperatures of over 300°C could be maintained for 3500 years with a mass flow rate of 50 kg s-l, potentially yielding nearly 2$ million tonnes of ore. Our goal of 3 million tonnes could clearly be reached by changing the parameters to reduce the mass flow rate further, but the system would then be rather different from the natural systems we are trying to model, since at low mass flow rates either the water temperature must be higher than 350°C or the flow must continue for many thousands of years.
320
0
0
2000
1000
Fig. 6. Exit water temperatures km depth with rock at 1000°C parameter
3000
for 6 km deep circulation. below 1 km, to represent
values unless specified
row fault width (0.2 m). Potential
otherwise.
a -standard
Years
Geothermal a recently
gradient
solidified
parameters,
b -high
in rock = 1 K m
magma
chamber.
friction
’ to 1
Standard
(F = l), c -nar-
ore masses given in million tonnes.
CONCLUSIONS
The foregoing results show that the system modelled can produce sizeable ore deposits (> 1 million tonnes) if the circulation penetrates to the bottom of the crust, but that no significant ore deposits will result from 1 km deep circulation. These conclusions are based on the results obtained using those values for the parameters most favourable to ore deposition. Whatever reasonable parameters are taken the 1 km deep circulation will not produce large ore deposits, and the ore masses resulting from the 6 km deep circulation will be maxima unless local conditions permit a longer recharge zone. The ore masses have been calculated on the assumption that precipitation of dissolved iron as iron sulphide is 100% efficient. If the efficiency were low, as it clearly
is at the East Pacific
Rise vents,
then the total
mass flow, and hence
the
amount of heat required, would need to be several times greater to produce the same ore deposit. In this case the flow would need to continue for much longer, since the model has shown that a higher mass flow rate would result in lower exit temperatures. Alternatively large ore deposits may only be possible where local conditions cause highly efficient deposition, such as in the Galapagos warm springs field. It is clearly possible then that ore deposits of a few million tonnes could result from convective flow in a fault zone which penetrates through a newly-solidified magma chamber to the base of the crust, provided that conditions favour highly efficient deposition, but such circulation is unlikely to account for the largest ore masses, which can be lo-15 million tonnes in Cyprus. If the circulation is shallow (penetrating to only 1 km depth), the model shows that black smoker flow is very short-lived, implying that longer-lived ore-forming black smokers require a greater heat flux than can be obtained by the system in the
321
model from a 1 km thick layer of hot rock. There are three possible the system may acquire that
the flow penetrates
shallow circulation capable
through
may be sustained
to be an underlying systems
a greater heat flux. One we have already a greater
crustal magma chamber. of depositing
thickness
by an additional
of hot rock.
namely
Secondly
the
heat source which would have
The third alternative
very large ore masses
ways in which
considered,
may only
is that circulation arise in locations
where there are no major faults to channel the flow, so that heat can be extracted from an extensive area of rock by porous medium circulation, either with flow entirely in a porous medium as in Lister’s (1982) model, or as in our third model with porous recharge and fault discharge. This form of circulation would be more efficient at heat extraction than the system modelled (which relies on conduction, a slow method of heat transfer), but only if the permeability is high enough. This alternative is unlikely to be a valid representation of the system generating the Cyprus ore deposits, which are clearly associated with major faults (Adamides, 1980) close to a spreading centre. Of these three alternatives we favour the possibility that a magma chamber is the principal source of heat for the formation of large volcanogenic sulphide bodies, since the heat supply from rock alone is always limited by the volume of rock that has not been cooled by previous circulation (Cann and Strens, 1982). The heat supply from the magma chamber will mainly be in the form of latent heat, produced at a rate of lo9 J m-3 of magma crystallised. Our model of magma-driven hydrothermal circulation (Cann et al., in press) shows that a 3 million tonne ore deposit could be formed, for example, in 4000 years by a black smoker system with 70% efficiency with the heat from 30 km3 of magma, the heat being conducted through a solid boundary layer about 10 m thick. So much heat is needed to form the very largest ore deposits that they may require the total freezing of a magma chamber and its subsequent penetration by the hydrothermal circulation system. We have shown, however, with this model that it is possible for fault-dominated circulation, located close to the mid-ocean-ridge axis, to be maintained by the heat supply from the rock for long enough to produce ore deposits of several million tonnes but only when conditions are particularly favourable. Although we have used this model to examine black smoker circulation at spreading centres, it could also be applied to other environments, and appears particularly suitable for modelling flow in fault zones in an expanding sedimentary basin, and thereby providing the means of predicting the size of the resultant lead-zinc sulphide deposits.
ACKNOWLEDGEMENTS
This work was supported typed by Elizabeth Walton.
by N.E.R.C.
grant
GR3/5073.
The manuscript
was
322 Appendix: COMPARISON OF FLOW IN A FAULT AND A POROUS MEDIUM
Compare medium
the downflow
in a 1 km length
of fault
40 cm wide and a porous
of 2 km’ extent.
(1) Fracture flow Flow will be turbulent since the extreme roughness of the fault would prevent laminar flow. In addition the Reynolds numbers calculated for the 21 “N system with a mass flow of 150 kg s-’ are greater than the critical Reynolds number in all but the coolest part. The fault is considered as many small pipes. Darcy’s relation for steady, turbulent pipe flow (Massey, 1979) defines the pressure gradient due to friction in a pipe. AP
2f
2-
P
z=--p
(1)
where f is the friction jj the mean density.
factor, d is the pipe diameter,
u is the linear flow velocity and
u = J/A&i where J is the mass flow rate and A, is the total cross-sectional area of the fault. Taking d =0.03 m, f = 0.05, p = 923 kg m-’ (at 150°C, the mean downflow temperature), flow:
A, = 0.4 x lo3 m and
J = 150 kg SK’ (cf. 21”N
system).
For fault
-AP = 5 x 10m4 kg m-2 se2 AZ (2) Porous medium frow The porous medium flow equation, gradient due to viscous laminar flow. AP _=_ AZ where
also named
after Darcy, defines
the pressure
Jv
(3)
A,,pk n is the dynamic
viscosity,
A,,
is the cross-sectional
area
and
k
the
permeability of the porous medium. Taking n = 1.85 X 10m4 kg m-’ ss’ (at 15O”Q A,, = 2 X lo6 m2, ,E= 923 kg mm3 (at 150°C) and substituting from (2) the permeability necessary to carry the same mass flow as the fault can be calculated, giving k = lo-’ m2. REFERENCES Adamides, N.G., 1980. The form and environment of formation of the Kalavasos ore deposits, Cyprus. In: A. Panayiotou (Editor), Ophiolites. Proc. Int. Ophiolite Symp., Cyprus, 1979, pp. 117-127.
323
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circulation
in fault zones at mid-ocean
ridge
307
B.V., Amsterdam
A FRACTURE-LOOP
- Printed
THEXMAL
in The Netherlands
BALANCE
MODEL
OF BLACK
SMOKER
CIRCULATION
M.R. STRENS
and J.R. CANN
of Geology, The Unk.wrsity, Newcastle tcpon Tjme, NEI 7RU (Great Britain]
Department (Received
February
25, 1985; revised version
accepted
July, 1985)
ABSTRACT
Strens.
M.R. and Cann,
J.R., 1986. A fracture-loop
thermal
balance
model of black smoker
circulation.
Tectonoph_wirs, 122 : 307-324. This
time-dependent
circulation
during
model
suppIy to black smoker Flow is modelled mid-ocean systems
systems
location.
turbulent
flow, which
to flow is P~~na~tIy
digitally,
calculated throughout
is driven
conductive
is based
on seafloor
by the buoyancy heat
flux from
of a hot spring,
Results show that this system cannot
recently-solidified particularly
Jf the circulation
thus provides
magma
favourable.
chamber,
We conclude
penetrates
and ophiolite between
upflow
tbe rock is computed
enabling obtain
magma
zone. The fault is allowed
at the rock-water
interface
through
hydrot~~ma~ the major heat
chamber. and very close to the
evidence
of black
and downflow. to propagate
analyticalfy.
smoker
temperature
Hydraulic downwards.
with the water
at every position
the exit water
the resulting sufficient
and
whether
in the model by a series of pipes carrying
difference
in the discharge
for each time step. The model
smokers.
hot rocks
in a single fault, which is parallel
flow
in the loop can be and mass flow rate
ore mass to be estimated.
heat from a thin layer of rock (such as the 1-2
km thick layer of Iavas and dykes over@ing an axial magma black
between
in order to establish
The fault is represented
so that the heat exchange
the lifespan
balance
ore deposits,
comes from the rock or from an underlying
ridge axis. This configuration and their structural
The horizontal
the thermal
of volcanogenic
as an open thermosyphon
resistance treated
examines
the formation
chamber)
a greater
ore masses of a few million tonnes that systems producing
to sustain tong-lived, ore-forming
volume
of hot rock can be formed
to the base of a if conditions
larger ore masses must tap a magmatic
are heat
supply.
The model we present here is one of a number of thermal balance models, each designed to investigate the exchange of heat between rocks and circulating hydrothermal solutions during the formation of volcanogenic ore deposits. In this paper we examine one aspect of the thermat balance of ocean-floor high-temperature hydrothermal circulation, such as gives rise to the black smoker hot springs of the East Pacific Rise. In these black smoker systems, the overall heat
30X
flux is many segment
times greater
of mid-ocean
than
is available
ridge (Macdonald
either in hot rock or in magma
on a steady-state
until it is tapped
by the hydrothermal
Here we test the efficacy of hot rock as a heat store, and extract rock into water
circulating
in a single
basis from a short
et al., 1980). Heat must therefore
fault parallel
be stored circulation.
the heat from the
to the mid-ocean
ridge, The
water flow is confined to a fault because of geological constraints which are discussed below. Our target is to produce a high-temperature (350°C) water flow for long enough to generate a sulphide deposit of 3 million tonnes on the ocean floor. Other models are possible. We have developed one in which heat is extracted from a magma chamber, as an alternative to heat extraction from rock (Cann et al., in press), and have also examined a third model in which recharge is through a porous medium rather than in a fracture. This model may be more applicable to systems with a greater area of recharge
THE CHARACTERISTICS
than seems to apply to black smoker systems.
OF BLACK SMOKER
CIRCULATION
We define a black smoker system from the characteristics of the hot vents on the East Pacific Rise, where, at 21 “N, the exit temperature of the hydrothermal fluid has been estimated at about 35O”C, with an exit velocity of 1-5 m s-’ (RISE, 1980, Macdonald et al., 1980), and, subsequently in the range 275”-35O*C, with flow velocities of 0.7-2.4 m s-’ and a mass flow rate of about 150 kg s ’ (Converse et al., 1984). At 13”N the exit temperature of fluid partially diluted with seawater is up to 33O”C, so the maximum fluid temperature will be somewhat higher. The flow velocity at 13”N is estimated at 0.5-2 m s -’ (Hekinian et al., 1983, 1984). At both locations the iron concentration in the hydrothermal fluid is about 110 ppm (Edmond, 1981; Michard et al., 1984). We assume that such systems are broadly analogous to the systems responsible for the deposition of the volcanogenic sulphide ore bodies found in ophiolite suites, although the known black smokers are thought to be very young and are associated with very small sulphide
deposits.
The efficiency of ore deposition is very variable between the observed vents. The black smokers themselves are extremely inefficient, since a high proportion of the sulphides is carried away by ocean currents. In contrast the warm vents on the Galapagos Spreading Centre are highly efficient, since sub-surface mixing with cofd water causes deposition of a high proportion of the dissolved metals as sulphides (Edmond,et al., 1979). Our goal in the model is the production of an average-sized Cyprus-type ore deposit of 3 million tonnes. We can estimate the time that this should take to form from the above data. With a mass flow rate of 150 kg s-‘, an exit water temperature of 35O”C, an Fe concentration of 110 ppm and 100% efficiency, the time required will be 3000 years, or proportionately longer for lower efficiency.
309 STRUCTURAL
CONSTRAINTS
ON BLACK SMOKER FLOW
All of the active black smokers discovered
so far are located within
a few hundred
metres of the axis of fast-spreading mid-ocean ridges, where the tectonic grain is very strongly aligned parallel to the ridge crest. The vents are invariably located on ridge-crest-parallel fissures or growth faults within the axial zone (RISE, 1980; Ballard et al., 1981; Hekinian et al., 1984), which is bounded by a series of major normal edge-crest-par~lel faults (Klitgord and Mudie, 1974; Searle et al., 1981). This close association of high-temperature hydrothermal substantiated by evidence from Cyprus, where the sulphide
upflow with faults is ore bodies and their
underlying feeder pipes are associated with faults running parallel to the ancient spreading centres (Adamides, 1980; Gass and Smewing, 1981), and, since they are mostly covered with a 100-500 m thickness of lavas, they also probably o~~nated within the narrow eruptive zone (Constantinou and Govett, 1972; Cann, 1980). There is therefore strong evidence that the hydrothermal upflow is channelled in faults. There is little direct evidence of the downflow zones, but consideration of the magnitude and structure of the permeability within the vicinity of the vents strongly suggests that the downflow is also predominantly in the faults. It can be shown quantitatively that a high proportion of the flow will be channelled by the faults since they form major discontinuities in the permeability, thus short-circuiting the convective flow which will preferentially follow the low resistance path. For example, a porous medium with a cross-sectional area of several square kilometres would require a permeabi~ty greater than lo-* m2 to carry the same mass flow as a 1 km length of a narrow, rough fault (see Appendix). There is little data on the permeability of the crust in these regions. The figure quoted by Anderson et al. (1982) from the sheeted dykes at the base of D.S.D.P. hole 504B is lo-t6 m2 in crust 6 m.y. old. In crust 7.2 m.y. old in D.S.D.P. hole 39514 near the Mid-Atlantic Ridge permeabilities
were
10-‘4-10-‘5
m2 (Hickm~
et al., 1984).
Even
if the pathways
have
become clogged with time it is unlikely that these rocks ever had a permeability as high as lo-* m*. It would therefore seem highly probable that recharge flow is also channelled by the faults. The major faults bounding the extrusive zone at a distance of about 1 km from the mid-ocean ridge axis impose a limit on the extent of the model perpendicular to the ridge axis, since these faults would have channelled previous circulation systems, thus putting a limit of about 1 km on the lateral extent of hot rock which has not already been cooled. The extent of the circulation system along the strike of the ridge also appears not to exceed 1 km from the observed spacing of black smoker vents, although this is highly variable (Ballard et al., 1981; Hekinian et al., 1984). The maximum area of hot rock from which heat might be gathered is thus about 2 km’.
310
DISCUSSION OF THE TYPE OF MODEL
The limited
structural
constraints
outlined
above,
namely
to a small area in which the fault zones, forming
permeability,
channel
the flow, mean that we cannot
as a valid representation In addition
that
smoker
flow is
major discontinuities
black
in the
use porous medium
flow theory
of this system in the model, but must turn to fracture
to the structural
constraints,
our choice of a fracture
flow.
flow system has
also been governed by the type of flow to be modelled. Porous medium flow theory is based on laminar flow, which is essentially slow and requires a well-developed permeability. These two conditions are not met for black smoker flow, which arises in rocks of low permeability (see above) and which has vent velocities of several metres per second. Several lines of evidence indicate that the flow is turbulent, as one would expect in conduits as rough as faults. Converse et al. (1984) described the type of flow in the 21’N vents as “plug” flow, which is a characteristic of turbulent flow in a circular pipe. In addition the Reynolds number of lo’-10h for the upflow (Converse et al., 1984) again implies that flow will be turbulent. We consider that the most realistic and practical means of representing flow in a very rough fault is by a series of small pipes carrying turbulent flow. We assume for simplicity that downflow occurs in the same fault as the upflow, but the model could easily be modified for downflow
in another
ridge-crest-parallel
fault.
PREVIOUS MODELS
Most existing models simulate large-scale, slow, low-temperature seafloor circulation, and none has been developed so far purely with the aim of modelling rapid, very high-temperature circulation. They have usually been designed to investigate the evolution and pattern of the convection cells over a large area in order to explain the distribution of heat flow values on the ocean floor (Ribando et al., 1976; Fehn and Cathles, 1979; Fehn et al., 1983). Since they cover large areas, the scale of the fault pattern is small relative to the total area and may be regarded as constituting a porous medium, so that porous medium theory may be used as a valid representation of the flow in this case. These models do not take into consideration any pre-existing fault pattern, although Fehn and Cathles incorporate zones of high permeability in the porous medium. The fracture-loop models of Bodvarsson and Lowell (1972) and Lowell (1975) predate the discovery of black smokers and hence do not attempt to simulate very high temperature flow. The only model so far that embraces black smoker flow is that of Lister (1974, 1982), which quantifies the evolution of the circulation system from a period of rapid flow, which causes the downward advance of a cracking front, thus creating the permeable matrix in which later, slow, closed circulation may persist for millions of years. Lister’s model predicts black smoker temperatures, but only when the permeability is as high as 10-7-10-9 m2, that is many orders of magnitude
greater
than the lO-“j
m2 measured
by Anderson
et al. (1982).
311
Our structure,
model
is therefore
the flow geometry, the constraints observed
different
since we are investigating
the above
models
balance
and then only of a high-temperature,
imposed
characteristics
THE STRUCTURE
from
only the thermal
by the geological of black smoker
structure
in both
rather
aims
and
than modelling
black smoker system within at mid-ocean
ridges
and
the
flow.
OF THE MODEL
The model examines the heat exchange between running parallel and close to the mid-ocean-ridge axis, The heat supply is maintained by horizontal conductive The model consists of an open thermosyphon within
convective flow in a fault, and the adjacent hot rocks. heat flow through the rock. a single constant-width fault,
which is treated as a series of pipes. Seawater at 0°C enters the fault and flows downward in a long zone extending along the fault, and eventually rises in a narrow upflow zone. The flow is driven by the buoyancy difference between the recharge and discharge zones, and the resistance to flow arises from the hydraulic properties of the system. Downflow and upflow velocities vary in proportion to the respective areas of recharge and discharge for continuity of mass. This model has been developed from
Fig. 1. The geometry of the model. The half-width of the fault only is shown.
312
a simpler model with the same geometry flow velocities were constant. Heat is exchanged conductive transient
at the water-rock
(Strens and Cant-r, 1982) interface
heat flow in the rock is calculated heat conduction
since it will be negligible
equation.
of the fault wall. The horizontal
analytically
from the one-dimensional
Vertical heat conduction
in the vicinity
but in which the
of a vent compared
in the rock is ignored, with the convective
heat
transfer. The convective water flow is treated digitally so that heat exchange is calculated at each level with the pencil of rock (referred to as a “block”) adjacent to each limb (Fig. 1). The two limbs of the system are assumed to be thermally independent.
Each block of rock has a different
but uniform
initial
temperature,
so
that any temperature profile may be set up. The model is run on a cyclical basis so that the contact temperature, which constrains the heat conduction, may be changed according to the new rock and water temperatures calculated at the end of each time increment. The fault is assumed to open suddenly to a given depth, and is then propagated downwards at a given rate unless the water temperature rises above a given temperature at which cracking is unlikely to continue. The fact that little is known about typical cracking temperatures, or the process of cracking at sub-seafloor pressures, necessitates this somewhat crude method. The advance of the fault is thus controlled by the fault propagation rate at lower water temperatures, but is arrested while the water temperature is high enough to inhibit cracking. Values given to these parameters
are discussed
below.
THE MODEL EQUATIONS
The solution to the one-dimensional transient heat conduction equation, constrained by a contact temperature, for a semi-infinite block of rock is given by Carslaw and Jaeger (1959): r,=T,+(T,-T,)erf
x i 2(at)“2
T,= T,
(Z=O,O
T, = T.
(x=0)
1
(1)
co)
T, is the rock temperature at distance x from the contact, Tc is the contact temperature between rock and water, T, is the initial rock temperature, x is the horizontal distance from the fault, (Y is the thermal diffusivity of the rock and t is the elapsed time. The contact temperature, q, in eqn. (1) is steady. To allow the contact temperature to be changed at the end of each cycle, while remaining steady during each cycle, eqn. (1) is modified in a way analogous to that given by Carslaw and Jaeger (1959, eqn. 3, p. 63) so that a factor is added for each change in contact
313
temperature: For n cycles: T,=
7;.,,,,+(T,-
X
7;.,,,) erf
erf i
If the heat transfer
with the water at x = 0 were perfect
the contact
temperature,
TC, would be the same as the water temperature T,. We have given some impedance to the heat flow by taking a weighted mean of the rock temperature (at x = 0.1 m) and the water temperature in inverse proportion to the thermal diffusivities of rock and water so that: TC= 0.2T,=, , + OJT,.. Other proportions were tested and the model was found to be relatively insensitive to this formulation, because the water and rock temperature are close except during a short initial period. The heat flow from the rock at x = 0 is derived as follows: Partial differentiation of eqn. (1) gives the temperature gradient at x = 0:
ar (T,-T,) Bx = ( Tat)‘/2 this means heat flow from rock at x = 0: -KA(q-c) q, =;” =
(2) ( nat)“2
where K is the thermal conductivity of the rock. The value of A, the heat exchange contact area, varies according to whether the equation is applied to the upflow zone or the downflow zone. The total heat flow, Q, up to time t is obtained by integrating eqn. (2): Q=2KA(7;-T,)
i
&
1
“I
(3)
Equation (3) is modified in the same way as eqn. (1). so that the contact temperature can be changed after each cycle. For n cycles, the conductive heat flux at the water-rock interface is: Q,,=2KA(q,,,-T,)
The heat flow from each block of rock during the nth cycle, Qcvc = Q,, - Q,,- ,, is equated to the heat removed by the water within and flowing through the fault adjacent to it, assuming the water to be a well-stirred fluid. Heat exchange equation:
Qcyc= c,,m4Ah
+ u,,Ar)(Ti
- T,.)
314
where cm. P,,, and u,,, are the specific heat, density and linear flow velocity of the water in the mth block. Temperatureand pressure-dependent values of density and specific heat are used. w is the half-width fault
and
increment,
takes
a different
i.e. thickness
temperature
value
of block,
of the fault, u is the dimension
for upflow
and
downflow,
At is the cycle length
and
along the
Ah is the vertical T: is the new water
after time At.
The mass flow through Continuit_); equation:
the system is defined
by the exit mass flow rate.
= awp,, u,, awp,,, u 111 where p_ and u,, are the exit density and flow velocity. This equation applies to both upflow and downflow provided the appropriate value for u is taken. The water temperature thus obtained at each position in the loop at the end of each time increment is then used to calculate the buoyancy force and hence the flow velocities for the next cycle. Flow equution:
where
u,. 5,. d, and f,
are the velocity,
mean
density,
friction factor for the upflow, p is the mean density the mean density of the downflow.
equivalent
diameter
and
for the whole system, and p1 is
Equation (4) is derived, as follows, by equating the total pressure drop in the system, assuming turbulent pipe flow, to the buoyancy force. The total pressure drop consists of the drops due to inertial forces at E (Fig. 2) the acceleration at A and friction in both limbs (Massey, 1979). Pressure drop = apuf + iP( U, - Up)’ + 4ph
$
? ?
Fig. 2. Diagrammatic limbs is indicative
representation of their thermal
E, and at the constriction,
:
fu’ + 2 y ’
i
of the flow system in the model. The space shown between independence.
Resistance
A, at the base of the upflow
the two
to flow arises at the entrance
to the system,
limb, as well as from the friction
in both limbs.
315
where uZ. d, and fi are the velocity,
equivalent
diameter
and friction
factor in the
downflow. Since
U, >> u2 and 2hf,/d,
B 4 for typical
values
of h - 1 km, f, - 0.01 and
d, - 0.03 m (see below) 2
Total pressure
drop
= 4$h 5
? 1
i.e. the pressure drop due to friction in the upflow pipe only. This pressure drop is equated to the buoyancy force kg(& - &) to give eqn. (4). The term 5d,/f, will be referred to as the friction constant (F). Its value is difficult to determine since no data are available on the roughness of the conduits, and reports of vent diameters range from 30 cm at the 21*N vents (Macdonald et al., 1980) to 3 cm at 13”N (Hekinian et al., 1984). Since the feeder zones seen beneath Cyprus ore deposits are composed of a network of narrow, anastomosing fissures, we have chosen to model the discharge zone as a bundle of narrow, rough pipes. If each pipe is taken to have a diameter of 3 cm and a friction factor of 0.01, corresponding to a roughness on the walls of the pipe of about a millimetre, which is consistent with geological observations, then, at a discharge water temperature of 350°C, and at a load pressure of 350 bar, it will pass fluid at a linear velocity of 2 m s-‘, which is the mean velocity observed for black smokers. A pipe of this diameter and roughness thus possesses the hydraulic resistance appropriate to black smoker conditions, and a bundle of 150 such pipes would pass 150 kg s-‘, the mass flow rate estimated for the 21°N black smoker field. The diameter, d,, and number, n, of pipes are related to the fault half-width, W, so that nndff4 = 4w2, assuming that the discharge zone is equidimensional. These values for d, and f, yield a value for F, the friction constant, of 15. CONSTRAfNTS
ON THE BOUNDARY
AND
INITIAL
CONDITIONS
The dimensions of the model are determined which the circulation occurs, which, for black mid-ocean
ridge axis. There is strong
evidence
by the structural smoker systems,
environment in is close to the
that very often, especially
on faster
spreading ridges, the eruptive zone is underlain by a magma-chamber at a depth of 1-2 km. Such evidence not only comes from seismic studies near spreading centres (Orcutt et al., 1976), but also from petrological studies on ocean-floor lavas (Bryan and Moore, 1977; Flower et al., 1977) and from geological mapping of ophiolite complexes, where thick gabbro units, clearly formed from crustal magma chambers, underlie about 1 km of lavas and 1 km of dykes (Moores and Vine, 1971; Gass and Smewing, 1981). Thus the hydrothermal circulation would be expected either to be confined to the l-2 km of hot rock overlying a magma chamber, or to extract heat from the magma itself, or to penetrate into the hot rock of a recently solidified magma chamber as far as the base of the oceanic crust. This provides clear
316
constraints circulation
on the depth of circulation in the model. We have therefore depths of 1 km and 6 km. For 1 km circulation the geothermal
taken as 1 K m-’ magma depth
chamber.
from 0°C at the seafloor
tested two gradient is
to 1000°C at 1 km depth, just above a
For 6 km deep circulation
of 1 km with the 5 km of rock below
the gradient at 1000°C
is again
1 K m‘-’
representing
to a
a recently
solidified magma chamber. Perpendicular to the ridge axis the limit of the extent of hot rock is about 1 km (see above). The extent of the mode1 in this direction is infinite, but the contribution to the heat flux from beyond a distance of 1 km is found to be negligible within the time span of several thousand
years.
ORE MASS CALCULATION
The model calculates the exit water temperature and mass flow rate for each time cycle during the life span of a hot spring. The resultant ore mass can then be calculated if one assumes that all of the dissolved iron is precipitated as Fe!&. The accuracy of this calculation depends on the accuracy of the iron solubility-temperature data. We have used line B in Fig. 3, which is based on experimental data of Mottl et al. (1979) with two points added: the measured concentrations in black smokers at 21“N and 13”N (Edmond, 1981; Michard et al., 1984), and Seyfried and Bischoff’s (1977) experimental point. Mottl et al.‘s points indicate an exponential increase in iron solubility with increasing temperature (line A). Since the system being modelled is an open system with a much higher overall water/rock ratio than Mottl et al.‘s experimental systems, we have constructed line B, which is heavily weighted towards the 21’N observational data and also Seyfried and Bischoff’s point which has a water/rock ratio of 50 : 1.
Fig. 3. Fe-solubility (Edmond,
1981),
data. Data are: e-from
q-Seyfried
Moitl et at. (1979),
@-21”N
black smoker
analysis
and Bischoff (1977). Line A is drawn through the points of Mottl et al. Line
B is the weighted line (see text) used for ore mass calculations.
317
The exponential relationship between Fe solubility and temperature shown in Fig. 3 means that the water temperature is the do~nating
factor in determining
the
resultant ore mass, and only when the water temperature is greater than 300°C is the iron solubility sufficiently high for it to contribute a significant amount of ore. The following results are based on an efficiency of deposition of 100%. RESULTS FROM THE M0DEL
Our aim, as stated above, is to test whether sufficient heat is available from the rocks to sustain the circulation system in the model, so that the vented water temperature is at least 35PC with a mass flow rate of about 150 kg s-l for several thousand years (or ~roportion~ly longer if the flow rate is less) in order to produce an iron sulphide deposit of 3 million tonnes. The values taken for the parameters required by the model are given in Table 1. In general these parameters are not well constrained, so the values taken represent the best estimate available to us for the ridge axis environment. The recharge zone length is taken as 1 km. This is based on observed spacings of black smoker fields, and on the fact that convection cells tend to be ~~dimension~. The fault propagation rate of IO m yr- 1 is based on Lister’s (1982) estimate. Other lower values were tested, including zero, but these gave lower water temperatures, thus reducing the potential for ore deposition and therefore of little interest in this analysis. The value taken for the crack propagation rate has little significance at high water temperatures which will inhibit cracking. Since we are unable to define the relative water and rock temperatures at which cracking will occur, we have tried to
TABLE 1 Values for the variable parameters in the mode8 Faranleter
Standard wfue
Recharge zone length
I km
Fault propagation
IOmyr-’
rate
Range of values
laitiai fault depth
5OQm
-
Width of fault
a.4 m
0.2-0.8
Friction constant
1.5
l-la0
m
Initial temp. field in rock circ. to 1 km
1
Km-’
10 K’m-’
to 100 m depth
10oO°C beIow zoo m ck
to 6 km
lKrnwl
tolkmdepth
and 1000°C below 1 km
-
318
make the model as realistic as possible by only allowing the crack to advance while the water temperature at the base of the circulation is under 370°C. By doing this the maximum observed
exit water
black smoker
temperatures
temperatures.
in the model
was chosen as the level at which the rock temperature estimate ashighasl
of the cracking
temperature
are similar
to the maximum
The value of 500 m for the initial
fault depth
will be below Lister’s (1982)
of 527’C, even when the geothermal
gradient
is
Km-‘.
The parameters that have highly variable, and therefore unknown values, namely the fault width and the friction constant, were tested through a range of realistic values. The fault width determines the water/rock ratio of the volumes exchanging heat, and the friction constant determines the linear flow velocity for a given water temperature. These parameters determine the mass flow rate and therefore the rate of heat removal. The standard values were chosen so that together they give a mass flow of around 150-200 kg s-’ at black smoker temperatures, which is consistent with estimates for observed black smoker fields (Converse et al., 1984). In addition to the standard initial temperature profile in the rock a “high” rock temperature profile was also tested for the 1 km deep circulation, since it is possible that black smoker circulation may arise shortly after volcanic activity. In this case the rock temperature was taken as 1000°C to within 100 m of the seafloor with a linear gradient above.
Circulation to I km depth The model was first run with the standard parameters (Fig. 4, a). After an initial burst of heat during the first year the exit water temperature never reached 3OO”C, so that the resultant ore mass would be negligible.
I
OO
500.
i 000
1500
1
Years
Fig. 4. Exit water temperatures for 1 km deep circulation. Geothermal gradient in rock = 1 K m- ‘. All parameters take standard values (Table 1) unless specified otherwise. a -standard friction (F = 1). c-narrow
fault width (0.2 m).
parameters, b-high
319
I
OO
500
Fig. 5. Exit water temperatures
1000
I
1500
for 1 km deep circulation.
Years Geothermal
100 m depth with rock at 1000°C below 100 m, to represent conditions Standard parameter values unless specified otherwise. c-narrow
fault width (0.2 m), d-high
a -standard
friction (F=l)
gradient in rock = 10 K m-’
to
shortly after volcanic activity.
parameters,
b-high
friction (F = l),
and narrow fault width (0.2 m). Potential ore
mass given in million tonnes.
The variable parameters were then tested systematically. The friction in the upflow pipe was increased to its maximum feasible value (Fig. 4, b) and the fault width was halved to 0.2 m (Fig. 4, c), but temperatures above 300°C were still obtainable only for about 30 years. These runs were repeated with the “high” initial rock temperature profile (Fig. 5, a, b, c) but temperatures above 3OO’C were still achieved for a maximum of only 130 years (Fig. 5, c). The extreme values for friction and fault width were then combined (Fig. 5, d) yielding an exit water temperature in excess of 300°C for 1000 years, but, because the prolonged high temperature was obtained by reducing the mass flow rate to about 20 kg s-l, the resulting ore mass was only 0.2 million tonnes, and very far short of our goal of 3 million tonnes.
Circulation to 6 km depth
Water temperatures of 300°C were again achieved only during the first year when the model was run with the standard set of parameters (Fig. 6, a). Once again it was necessary to reduce the mass flow rate to prolong high-temperature flow (Fig. 6, b, c) so that, with a fault width of 0.2 m for example, exit temperatures of over 300°C could be maintained for 3500 years with a mass flow rate of 50 kg s-l, potentially yielding nearly 2$ million tonnes of ore. Our goal of 3 million tonnes could clearly be reached by changing the parameters to reduce the mass flow rate further, but the system would then be rather different from the natural systems we are trying to model, since at low mass flow rates either the water temperature must be higher than 350°C or the flow must continue for many thousands of years.
320
0
0
2000
1000
Fig. 6. Exit water temperatures km depth with rock at 1000°C parameter
3000
for 6 km deep circulation. below 1 km, to represent
values unless specified
row fault width (0.2 m). Potential
otherwise.
a -standard
Years
Geothermal a recently
gradient
solidified
parameters,
b -high
in rock = 1 K m
magma
chamber.
friction
’ to 1
Standard
(F = l), c -nar-
ore masses given in million tonnes.
CONCLUSIONS
The foregoing results show that the system modelled can produce sizeable ore deposits (> 1 million tonnes) if the circulation penetrates to the bottom of the crust, but that no significant ore deposits will result from 1 km deep circulation. These conclusions are based on the results obtained using those values for the parameters most favourable to ore deposition. Whatever reasonable parameters are taken the 1 km deep circulation will not produce large ore deposits, and the ore masses resulting from the 6 km deep circulation will be maxima unless local conditions permit a longer recharge zone. The ore masses have been calculated on the assumption that precipitation of dissolved iron as iron sulphide is 100% efficient. If the efficiency were low, as it clearly
is at the East Pacific
Rise vents,
then the total
mass flow, and hence
the
amount of heat required, would need to be several times greater to produce the same ore deposit. In this case the flow would need to continue for much longer, since the model has shown that a higher mass flow rate would result in lower exit temperatures. Alternatively large ore deposits may only be possible where local conditions cause highly efficient deposition, such as in the Galapagos warm springs field. It is clearly possible then that ore deposits of a few million tonnes could result from convective flow in a fault zone which penetrates through a newly-solidified magma chamber to the base of the crust, provided that conditions favour highly efficient deposition, but such circulation is unlikely to account for the largest ore masses, which can be lo-15 million tonnes in Cyprus. If the circulation is shallow (penetrating to only 1 km depth), the model shows that black smoker flow is very short-lived, implying that longer-lived ore-forming black smokers require a greater heat flux than can be obtained by the system in the
321
model from a 1 km thick layer of hot rock. There are three possible the system may acquire that
the flow penetrates
shallow circulation capable
through
may be sustained
to be an underlying systems
a greater heat flux. One we have already a greater
crustal magma chamber. of depositing
thickness
by an additional
of hot rock.
namely
Secondly
the
heat source which would have
The third alternative
very large ore masses
ways in which
considered,
may only
is that circulation arise in locations
where there are no major faults to channel the flow, so that heat can be extracted from an extensive area of rock by porous medium circulation, either with flow entirely in a porous medium as in Lister’s (1982) model, or as in our third model with porous recharge and fault discharge. This form of circulation would be more efficient at heat extraction than the system modelled (which relies on conduction, a slow method of heat transfer), but only if the permeability is high enough. This alternative is unlikely to be a valid representation of the system generating the Cyprus ore deposits, which are clearly associated with major faults (Adamides, 1980) close to a spreading centre. Of these three alternatives we favour the possibility that a magma chamber is the principal source of heat for the formation of large volcanogenic sulphide bodies, since the heat supply from rock alone is always limited by the volume of rock that has not been cooled by previous circulation (Cann and Strens, 1982). The heat supply from the magma chamber will mainly be in the form of latent heat, produced at a rate of lo9 J m-3 of magma crystallised. Our model of magma-driven hydrothermal circulation (Cann et al., in press) shows that a 3 million tonne ore deposit could be formed, for example, in 4000 years by a black smoker system with 70% efficiency with the heat from 30 km3 of magma, the heat being conducted through a solid boundary layer about 10 m thick. So much heat is needed to form the very largest ore deposits that they may require the total freezing of a magma chamber and its subsequent penetration by the hydrothermal circulation system. We have shown, however, with this model that it is possible for fault-dominated circulation, located close to the mid-ocean-ridge axis, to be maintained by the heat supply from the rock for long enough to produce ore deposits of several million tonnes but only when conditions are particularly favourable. Although we have used this model to examine black smoker circulation at spreading centres, it could also be applied to other environments, and appears particularly suitable for modelling flow in fault zones in an expanding sedimentary basin, and thereby providing the means of predicting the size of the resultant lead-zinc sulphide deposits.
ACKNOWLEDGEMENTS
This work was supported typed by Elizabeth Walton.
by N.E.R.C.
grant
GR3/5073.
The manuscript
was
322 Appendix: COMPARISON OF FLOW IN A FAULT AND A POROUS MEDIUM
Compare medium
the downflow
in a 1 km length
of fault
40 cm wide and a porous
of 2 km’ extent.
(1) Fracture flow Flow will be turbulent since the extreme roughness of the fault would prevent laminar flow. In addition the Reynolds numbers calculated for the 21 “N system with a mass flow of 150 kg s-’ are greater than the critical Reynolds number in all but the coolest part. The fault is considered as many small pipes. Darcy’s relation for steady, turbulent pipe flow (Massey, 1979) defines the pressure gradient due to friction in a pipe. AP
2f
2-
P
z=--p
(1)
where f is the friction jj the mean density.
factor, d is the pipe diameter,
u is the linear flow velocity and
u = J/A&i where J is the mass flow rate and A, is the total cross-sectional area of the fault. Taking d =0.03 m, f = 0.05, p = 923 kg m-’ (at 150°C, the mean downflow temperature), flow:
A, = 0.4 x lo3 m and
J = 150 kg SK’ (cf. 21”N
system).
For fault
-AP = 5 x 10m4 kg m-2 se2 AZ (2) Porous medium frow The porous medium flow equation, gradient due to viscous laminar flow. AP _=_ AZ where
also named
after Darcy, defines
the pressure
Jv
(3)
A,,pk n is the dynamic
viscosity,
A,,
is the cross-sectional
area
and
k
the
permeability of the porous medium. Taking n = 1.85 X 10m4 kg m-’ ss’ (at 15O”Q A,, = 2 X lo6 m2, ,E= 923 kg mm3 (at 150°C) and substituting from (2) the permeability necessary to carry the same mass flow as the fault can be calculated, giving k = lo-’ m2. REFERENCES Adamides, N.G., 1980. The form and environment of formation of the Kalavasos ore deposits, Cyprus. In: A. Panayiotou (Editor), Ophiolites. Proc. Int. Ophiolite Symp., Cyprus, 1979, pp. 117-127.
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