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Problems in the extrapolation of laboratory rheological data
Tectonophysics, Elsevier
33
133 (1987) 33-43
Science Publishers
B.V., Amsterdam
- Printed
in The Netherlands
Problems in the extrapolation of laboratory rheological data M.S. PATERSON Research School of Earth Sciences, Au~alian (Received
January
National University, Cunherru, A.C.T.
21,1986;
revised version accepted
2601 (Australirr)
June 10,1986)
Abstract Paterson,
M.S., 1987. Problems
The many types of variables
in the extrapolation and deformation
behaviour
from the laboratory
particular
cases. In the case of olivine-rich
of extrapolation, independent amounts
of water. In the case of quartz-rich
any extrapolation water are probably
to geological two important
conditions variables
The problems
rocks, recent experimental
mantle
non-Newtonian.
Tectonophysics,
of extrapolation
work indicates
types of behaviour
rocks, the uncertainties except
as an upper
but the quantitative
to the analysis is to assume a
model, calculate the predictions of the model for chosen values of the physical parameters that are required to specify particular situations, and compare the predictions with field observations. The physical parameters to be chosen will include the deformation properties of the rocks involved, which are normally obtained by some sort of extrapolation from laboratory measurements. The aim of this paper is to discuss the nature of the problems that arise in connection with this extrapolation and to stress the limitations that still exist. At relatively low temperatures and pressures rocks fail under non-hydrostatic stress by brittle fracture. However, they tend to become ductile as temperature and pressure are raised, although the presence of high pore-fluid pressures can greatly extend the brittle field. It is the ductile behaviour on which interest is usually focused in connection 0 1987 Elsevier Science Publishers
B.V.
would
bound;
the fugacity
laws governing
in extrapolating
that, within present and near-Newtonian
rheological with two
uncertainties or grain size
by the present
and it is still premature
of trace to attempt
and the scale of dispersion
their influence
large-scale
133: 33-43.
are then illustrated
be influenced
are even greater
with
A common theoretical approach of tectonic processes in the earth
data.
could be either grain size dependent Both
Introduction
004@1951/87/$03.50
rheological
regimes that need to be taken into account
to the earth are reviewed.
the flow in the upper
and distinctly
of laboratory
tectonic
of the
are not yet clear.
situations
involving
the
lower crust and upper mantle, except where the occurrence of earthquakes points to brittle behaviour. This paper will deal only with creep in rocks in the ductile field; the brittle field calls for separate consideration (Brace and Kohlstedt, 1980; Kirby, 1980, 1983; Paterson, 1978) and cataclastic effects in the ductile field, potentially important in upper crustal processes, will also not be dealt with. In relating the laboratory observations to tectonophysical processes, three types of consideration enter: (a) The form of the flow criterion for general stress states. (b) The environmental and internal variables that need be considered. (c) The circumstance that different flow mechanisms many predominant under different conditions, giving rise to a variety of flow laws for a given rock. These three topics will now be discussed in turn.
34
The
formof
Almost
the flow law all
laboratory
rheological
ments on rocks are at present pression
under superposed
confining
is, with
ur > a2 = u3 where
principal
stresses (compression
difference stress”. strain
rate
confining principal
ur,
pressure, u2, u3 are
positive).
u = u1 - u3 is reported determined
as a function
i = i,
at a given
pressure strains,
measure-
made in axial comthe
The stress
as the “flow of the
temperature
axial T and
u2 = u3(e1 > e2 > e3 are
shortening
that
positive).
the
For a given
rock deforming by a given mechanism under given circumstances the experimental flow behaviour at elevated temperatures is therefore viewed as a function of the three primary variables, and the flow law has the form:
u, i. T
f( u, i, T) = 0
(1) More generally, when account has to be taken of the total strain e and of other environmental and
internal different ered or account expressed F(u,
variables y . . . such as may examples of a given rock type other variables have to be (see next section), the flow in the form:
;, T; c, y...)
=0
enter when are considtaken into law can be
(2)
In either case, an explicit form for (1) and (2) is usually sought in which the strain rate i is expressed as a function of the other variables. In creep tests, in which u, T and environmental variables are held constant, it is generally observed that the strain rate changes as straining progresses but at a diminishing rate so that there is a tendency to approach a steady state. In analysing this behaviour, the observed strain rate is often regarded as being the sum of two components, a constant component equal to the eventual steady state rate and an ephemeral component that decreases as the straining progresses, respectively known as the steady state and transient components. However, it should be noted that such a separation is an empirical one which may not have a physical basis in terms of separate mechanisms of flow. In high temperature tests on metals (Garofolo, 1965, p. 16) the transient component of the creep strain can often be represented satisfactorily by
the expression <,[l - exp( - t/t,)] where t is the elapsed time and e,, 2, are empirical constants; the transient component of the strain rate is then (e/f,)
exp( - t/t,).
It has also been noted (Amin
et al., 1970) that, for a wide variety of metals, the relaxation time t, = 10*/i, where i, is the steady state component years
of the strain rate (thus, f,, - 10’ s-~‘). If the latter empirical
if i, - lo-l4
relationship deforming
could be validly extrapolated at geological
sient effects could of time.
Regarding
to rocks
strain rates, then the tran-
persist
for a geological
the value
period
of cr. it is often
found in laboratory tests that, over the range of strain of the experiment and within some accepted approximation (say, a few percent), a steady state is reached after several percent strain at high temperatures, that is, et - 10p2. If this observation were to extend to geological situations, it would suggest that one should be cautious about attempting to relate steady state creep observations to tectonophysical phenomena such as postglacial uplift where the strains are quite small (a similar view has been expressed recently in the tectonophysics literature; see Peltier, 1985; Sabadini et al., 1985; and Weertman, 1985). There have so far been relatively few experimental studies of transient creep in rocks at high temperature (for example, Post, 1977; Carter and Kirby, 1978; Carter et al., 1981). Provided the maximum principal stress difference is not very high, the steady state component of the strain rate observed in experiments on rocks at high temperature and pressure can commonly be fitted with good approximation to the “ power law”: i = Au” exp( -Q/RT)
(3)
where A, n and Q are empirical parameters, R is the gas constant and T the absolute temperature. As applied to the normal axisymmetric compression test, u is the maximum principal stress difference and i is the maximum principal shortening strain rate. Newtonian flow (linear viscosity) results if n = 1 but values of n = 3 to 5 or more are commonly found. In the remainder of this paper we shall only discuss steady state flow. In tectonophysical situations the stress states will, in general, no longer correspond to those in
35
axisymmetric expe~ments
and so a generalization
for i and u in cases of general stress states), the
of the steady state flow law (3) is needed. In the
flow behaviour can in practice be influenced by a
absence of appropriate studies on rocks subjected
number of additional variables which relate to the
to complex stress states, it seems most reasonable
environmental
to adopt the procedure of Odqvist (1935) often used in engineering creep calculations for gener-
composition or microstructure of the rock itself, as
alizing flow criteria, of expressing the flow law in terms of an “equivalent stress” u* and an “equiv-
indicated by y , . . in (2). We shall now list, and comment on, additional variables that might need
alent strain rate” i*,
to be considered in particular cases.
defined (neglecting
elastic
strains) as:
cJ* =
subjected
conditions
to which the rock is
or which represent
(1) Mean stress or pressure.
i* = (fi/3)[((,
- i,)*
+ (i, - &)2
il)y2
The influence of
stress, often loosely referred to as the influence of
-ul)2]1/2 (4)
+(i3 -
in the
the mean stress or hydrostatic component of the
(l/JT)[(u,- u*)*+ (a* -u3j2 +(u3
variations
(5)
the “pressure”, is relatively small and difficult to measure in the laboratory (in experimental studies the confining pressure us is often referred to as the “pressure” in this connection although it only approximates the mean stress (a, + u2 + us)/3 if a3 is small compared with a,; however, the use of a3 in extrapolating to the deep interior of (Jl -
It will be seen that, provided the volume remains constant, as is implicit, the quantities u = ui - Us and i = i, used in (3) to express the results of axisymmetric experiments correspond to equivalent stress and equivalent strain rate, respectively, as defined by (4) and (5). Thus, if Odqvist’s procedure is followed, the form (3) can be extended to general stress states in terms of i* and (I*. It is to be remembered, however, that the relation (3) is essentially empirical and that other forms might have been used. Also the term “effective” is often used in place of “equivalent” in describing u* and <*, in which case the Von Mises “effective stress” (I* needs to be distinguished
from the Terzhagi
“effective
stress”
used in treating pore pressure in the brittle field (Paterson, 1978, ch. 5). Further, in using (3) in terms of ci* and u* as the constitutive equation of flow, it is implicit that effects associated with variation of the hydrostatic component of the stress or with changes in other environmental variables can be dealt with by suitably varying the parameters A and Q. Other variables Even if we accept relation (3) as the basic form of steady state flow law f(i, u, T) = 0 with the primary variables, i, u and T (using E’* and u*
the earth might be justified insofar as the “pressure” in the earth, as calculated from depth of burial, is also taken to be the least principal stress Us). When extrapolation to higher “pressures” is done, it is usual to assume that the effect of pressure is mainly to increase the quantity Q in (3) putting Q = Q, +pV*, where p is the “pressure”, Q, the apparent activation energy at low pressure and 1/* the apparent activation volume. The parameter I’* is commonly assumed to be more or less equal to the molar volume of the slowest diffusing species (for example, Stocker and Ashby, 1973), but in silicates the identity of this species may not be obvious. The only experimentally determined values of V* for silicate rock are those of Ross et al. (1979) and Green and Hobbs (1984), determined in solid medium apparatus in the confining pressure ranges 500-1500 MPa and 1200-2700 MPa, respectively. They obtained I’* = 11 to 15 X 10e6 and 25 to 40 X 10m6 m3 mol-“, respectively, for olivine-rich rock, taking p to be the confining pressure, from which it is evident that the experimental situation is far from clear. See Poirier (1985, ch. 5) for a more detailed discussion of pressure effects. (2) Chemical environment. The chemical state of a rock may affect its rheological behaviour at high temperature. This state can be specified through equilibration with a thermodynamic en-
36
vironment
described
components
oxide, regarded ment,
and olivine
of condensed
is the intragranular
that has been demonstrated and probably
of di-
species in the environ-
such as silica. One chemical
importance
weakening (Griggs
as volatile
water or carbon
or in terms of the activities
components major
in terms of the fugacities
such as oxygen,
effect of hydrolytic in quartz
occurs in other silicates
and Blacic, 1964, 1965; Blacic, 1972). Ex-
actly how the influence be incorporated
of the water content
should
in the flow law is not well estab-
lished. An illustration
of the overall effect of water
in a rock is given by the measurements
on dunite
by Chopra and Paterson (1981, 1984) who have shown experimentally that the strain rate at a given stress can be changed by two orders of magnitude by drying the rock from a total water content of - lop3 mole fraction (- 0.01 wt.%) to a content of one or two orders of magnitude less; there has been some controversy about the mode of action of the water but it is clear that both intragranular and grain boundary effects can be involved (Karat0 et al., 1986). (3) Presence of jluid phases. The pore pressure effects that are well known in the brittle field are much less important in ductile behaviour at high temperature insofar as the flow stress is concerned, provided the total pressure minus the pore pressure is large compared with the brittle-ductile transition pressure. However, some influence of pore pressure, with associated dilatancy, can be detected well into the ductile field even at elevated temperatures (Fischer and Paterson, 1985) indicating that subsidiary microcracking is a widespread accompaniment of crystal plasticity in rocks, with implications for increased permeability to fluids during deformation. The presence of fluid phases such as metamorphic water-rich phases or igneous partial melts can thus potentially have important effects involving the transport of material through the fluid phase, contributing in turn to the strain or assisting intergranular accommodation (see, for example, Etheridge et al., 1984). Such effects can be expected to introduce some grain size dependence in the parameter A in (3) and possibly also affect the parameters n and Q. (4) Grain size. Under some conditions, when
the
rate-controlling
volves processes
deformation
at grain boundaries
mechanism
in-
rather than in
grain interiors, the creep rate can be strongly dependent on grain size, as in diffusion creep. In this case the pre-exponential ten as Ad-“’ another
where
empirical
the range
d is the grain size and
constant,
1 to 3. Grain
thus more likely deformation
Gueguen, could
found
to be important of
m is
to be usually
size dependent
sizes, as has been pointed the
factor in (3) is rewritin
creep
is
at small
grain
out in connection
with
mylonites
(Boullier
and
1975); but even at a given grain size it
become
predominant
as low stresses
and
strain rates are approached because the stress exponent n is also smaller than for grain size insensitive creep. Thus grain size sensitivity of creep may have to be taken into consideration in tectonophysics, as will be further discussed later. (5) Preferred crystallographic orientation. Because of crystal anisotropy, departures from random orientation of the crystallographic axes of the grains will tend to introduce anisotropy of flow stress in the rock, requiring different choice of parameters in the flow law for different orientations. Relatively little is known about the flow laws for anisotropic rock, although some observations have been made on marble (Heard and Raleigh, 1972). The development or modification of preferred orientation during deformation will also tend to affect the rate of attainment of a steady state. Thus, the creep behaviour of a given type of rock can be expected to vary substantially as changes occur in factors such as those just discussed. Expressed in terms of strain rate this variation may in some cases be of orders of magnitude (for example, in the case of change in water content). Therefore, although very broad generalizations about the flow law for a given type of rock may be possible, any refined consideration will normally have to take into account other variables than just stress, temperature and pressure. Deformation
mechanisms
and rheological regimes
We now return to consider further the nature of the relationships between strain rate (, stress u and temperature T for a particular set of the
37
variables considered in the previous section. We
servations (Heard, 1963; Heard and Raleigh, 1972;
need to take into account that there is a variety of
Rutter,
mech~sms
over, the values of the parameters for particular
whereby the rock can be deformed,
1974;
Schmid et al., 1977, 1980). More-
which will be reflected in a variety of rheological
regimes vary substantially
regimes in terms of the (, u, T relationships.
limestone to another, so that no common “flow
The mechanisms that can be envisaged for the deformation
of a rock can be considered broadly
in three groups,
distinguished
by the scale of
reorganization involved: (1) Atomic transfer mechanisms, in which the change
of shape is achieved
reorganization,
the transfer
by atom-by-atom
process
itself being
most commonly envisaged to be by diffusion; the various types of diffusion creep (Nab~o-He~ng, Coble, solution transfer, etc.) fall in this category. (2) Crystal plasticity mechanisms, in which, by means of dislocation movement, discrete blocks within crystal grains are translated relative to each other while retaining the integrity of the grain; there are several types of dislocation creep, distinguished according to the different rate-controlling factors that predominate (Poirier, 1985, ch. 4). (3) Granular flow mechanisms, in which whole grains or groups of grains are moved relative to each other without the crystallographic constraints in dislocation gliding; the “grain-swapping” mechanisms associated with super-plastic flow (for example, the Ashby-Verrall mechanism), as well as cataclastic or soil-like flow, fall within this group. Since there are distinctions in rheological behaviour within each group of mechanisms as well as between groups according to the various rate controlling factors that may predominate, it can therefore be expected that, as i, CT,T are varied over a wide range, a given rock in a given environment will be able to exhibit a variety of flow laws, this variety being expressed in terms of different values of A, n, Q insofar as the form (3) can be used. The variety is even greater when rocks from different formations of the same compositions type are considered. Thus, without going into details about possible mechanisms involved, it has been found that, for marbles and limestone, there are at least three high-temperature rheological regimes which can be defined expe~mentally by different values of the parameters required to fit three-parameter flow laws such as (3) to the ob-
from one marble or
law for calcite rock” can be given (a summary of values for Solnhofen
limestone and Carrara and
Yule marbles is given in table II of Schmid et al., 1980). The succeeding discussion will be mainly in terms of the phenomenological behaviour rather than of the deformation
mechanisms;
the rheo-
logical measurements provide the primary data for extrapolation,
while the mechanisms may remain
controversial.
However, an understanding of the
mechanisms operating both in the laboratory and geologically will eventually be needed in deciding or confirming which regime of observed behaviour can be validly extrapolated to given geological conditions. The various regimes of flow behaviour can be depicted conveniently in a diagram analogous to the deformation mechanism map of Ashby (1972), in which the fields of applicability of individual flow laws are delineated. The fields may correspond to different mechanism fields but this correlation need not necessarily have been established. Examples of such rheological regimi maps are shown in Fig. 1 for Carrara marble and Solnhofen limestone, based on the experimental work mentioned above and collated in Schmid et al. (1980). While Ashby maps are usually plotted in temperature/flow-stress coordinates with constant strain rate contours, and rheological regime maps can be similarly plotted, str~n-rate/temperature coordinates with constant stress contours have been used in the figures in this paper on the grounds that the boundary conditions for geological and geophysical problems are commonly expressible in terms of displacement rate or strain rate and temperature, as, for example, in collision tectonics driven by plate movement; the conventional Ashby plot may be more appropriate for situations such as gravity sliding constrained by given body forces. It is seen that the patterns of behaviour for the fine-grained Solnhofen limestone and the coarsergrained Carrara marble are distinctly different, there being little correspondence between the
1000
‘.8 -
900
800
700 -I 3 z P 500 _I z R 500 Regime 2 Q=297
I I -1
kJ mot
LOO
I
-14
Fig. 1. Flow regime boundaries experimental
/
I
i
-12
-10
for Solnhofen
field shown and extrapolated
limestone
towards
f
-8 V-~ fQ (Strain Rate) and Carrara
geological
parameters for the regimes 2 and 3 distinguished in each rock in the given experimental field; moreover, a simple extrapolation predicts that the distinction in behaviour will become more exaggerated as geological conditions are approached. Aspects of extrapolation are discussed by Schmid et al. (1980). We now consider two particular cases, those of olivine-rich and of quartz-rich rocks, respectively typifying the problems of extrapolating to upper mantle and to lower crustal conditions. These examples are not to be viewed as being definitive at this stage but rather as illustrating the uncer-
strain
marble
-4
as determined
i-Ai- -2
by Schmid
et al. (1977, 1980) in the
rates
tainties involved in extrapolation and only a few of the aspects mentioned earlier are addressed. Olivine-rich
rocks
The case of olivine-rich rocks can serve to illustrate the potential application of rheological regime maps in tectonophysics. Mention has already been made of the laboratory observation that the flow strength of dun&e is strongly influenced by trace amounts of water (Carter and AvC Lallemant, 1970; Blacic, 1972; Post, 1977; Chopra and Paterson, 1981, 1984). However, at
39
least in the expe~ments of Chopra and Paterson, the grain boundaries contained small amounts of melt due to decomposition of minor accessory hydrated minerals in the rocks. In order to avoid this complication and to study the role of grain size? a study has more recently been made on pure polyc~st~line olivine synthesized by hot-pressing crushed olivine crystals (Karat0 et al., 1986). This work has demonstrated that, as in the case of fine-grained limestone already mentioned, two strongly contrasting regimes of flow can be distinguished: relatively coarse-grained material shows little grain size dependence and a fairly high stress exponent n - 3 or more (regime l), while finer-grained material shows a strong dependence on grain size d, between i a: d-” and dp3, and a low stress exponent, n - 1 (regime 2). The presence of water weakens the polycrystalline olivine in both regimes and increases the transition grain size from about 25 urn to about 40 pm at laboratory strain rates. In order to explore possible implications of such grain size dependence for tectonophysical behaviour we assume a flow law of the form: i = Aand-”
exp( - Q/RT)
with the following parameters
(a in MPa, d in
pm):
Regime 1 Regime 2
1gA
n
m
Q
3.6 6.85
3.5 1.3
0 2.5
450 kJ mol - ’ 300 kJ mol-’
These values are taken as being typical of wet polycrystalline olivine; the values of n and m derive from the observations of Karat0 et al., and the value of Q for regime 1 from observations on wet dunite already mentioned, while the value of Q for regime 2 is taken as being about the same fraction of that in regime 1 as was found for limestone. Finally, the transition from regime 1 to regime 2 is taken from the observations of Karat0 et al. as occurring at a grain size of 50 pm and a stress difference of 70 MPa under the conditions 13OO*C and lO_’ s-r strain rate (the experimental point A in Fig. 2). The boundary between regimes 1 and 2 can then be calculated to be as shown in Fig. 2 by the heavy line marked “50 pm”‘; constant stress contours for 10, 100, and
1000 MPa have also been added near this boundary. Using the K2-s relationship, the boundary between regimes 1 and 2 can then be recalculated for other grain sizes and is shown for 1 mm and 5 mm in Fig. 2, where some of the corresponding stress contours are also shown adjacent to the regime boundaries. If m were taken as 3 or 2 instead of 2.5, the boundary for 1 mm grain size would be shifted about one order of magnitude towards lower or higher strain rates, respectively, and the boundary for 5 mm grain size would be similarly shifted about one and a half orders of magnitude. It follows from Fig. 2 that, in proceeding from laboratory to geological conditions (indicated by the two shaded areas), regime 1 behaviour would still be expected under geological conditions if the grain size were of the order of 10 mm or larger, provided the other environmental variables were not changed significantly (insofar as they exert rheological influence) and provided no other type of behaviour, not yet discovered, were to intervene. However, when the grain size is reduced to the order of 1 mm, grain size sensitive (regime 2) flow would be predicted to predominate, with the some provisoes. In the latter case, extrapolation from measurements made at large stresses in a high-n “dislocation creep” regime would tend to over-estimate the flow stress. In view of the incomplete knowledge of the role of other variables in the rheology of oliviue-rich rocks, the extrapolations of Fig. 2 should not be taken too seriously as #~str~ning possible behaviour in the upper mantle. However, they suggest that, while high-n dislocation creep may well be expected in coarse-grained rocks (consistent with field observations of strong preferred orientations in coarse-grained nodules, for example, Mercier and Nicolas, 19751, grain size sensitive low-n flow may possibly occur in other cases, especially where the grain size is only a few millimeters or less (consistent with the suggestion of Boullier and Gueguen, 1975). Before att~buting more quantitative significance to these extrapolations, it will be necessary to obtain experimental confirmation of the values of Q and to establish the effects of ~co~orating other phases such as pyroxene, raising the pressure, varying the amount
40 LABORATORY
GEOLOGY
-0.6
-0.7
zi 4 If
x iL0.t
!f
*$
l’
I I
t
I
I
-0.s
Of 7
_&II.
I
/
C9
WC
,rs ‘erg ’
I
-1
I
I
I
R
900
/‘Regime I
011422 ./
1
: lgA=3.8&lPa _ ..LI “2.3.3 u=450
units , i
kJ moi
, i 800 I
_,‘Y
/
,
. I
---I I’ I
-1.
0
’
:
I
I
I’
I ,
I
-----I
Regime
2 : IgA= -1.6(MPa, pm Units) n ~1.3 mP2.5 a=300 kJ moi’
.I
I
,I
/I’
I
I
I
I
I
I
-
1
-12
-14
,._.._L
1.. -L
-10
-8 lg (Strain
Fig. 2. Flow regime boundaries temperature/strain-rate
between
stress contours
parameters
I
_,
-8
for three gram sizes in polycrystalline
for 50 pm gram size has been extrapolated shown
i
AGGREGATE (WET) f i~__L1_ -2 -4
700
Rate)
high 11 and low n behaviour
plane. The behaviour
mm grain sizes, using the estimated gram size. Constant
OLIV~NE
t
(see text) which are normalized
are shown in dotted (regime 1) and broken
to lower strain
olivine, projected
on the
rates and to 1 mm and 10
to fit the experimental
(regime 2) lines in the neighbourhood
point
A for 50 nm
of the boundary
for each gram size.
of water, and altering the environmental or chemical conditions in any other way. Quartz-rich
rocks
The case of quartz-rich rocks is more difficult to analyse than that of olivine-rich rocks because the basic modes of behaviour and the factors
affecting these are less well understood. Quartz has proved to be an especially difficult material to study rheolo~~ally in the laboratory (Blacic and Christie, 1984; Paterson, 1985). In the absence of water, the strength of single crystals is enormous, the flow stress (stress difference) at laboratory strain rates being still well in excess of 1000 MPa at 1300°C and 300 MPa confining pressure (mea-
41
surement by J. Bitmead and MS. Paterson, described in Doukhan and Trepied, 1985). The deformation of quartz is, therefore, generally believed to depend essentially on the presence of water, based on the discovery of ““hydrolytic weakening” by Griggs and Blacic (1964, 1965). The weakening has been correlated with a broadband infrared signature of the responsible species (Kekulawala et al., 1978), corresponding to the presence in the quartz of some form of molecular water (Aines et al., 1984). However, the nature of the weakening effect is still not well understood. Some discussion of single crystal studies in relevant at this point to illustrate the difficulties experienced in obtaining a clear picture of the factors governing the role of water in the laboratory deformation of quartz-rich rocks, factors that need to be understood before the validity of extrapolations to geological conditions for quartzrich rocks can be properly assessed. The difficulties experienced in the single crystal studies seem to arise, first, from kinetic sluggishness, especially in the diffusion of the water-related species, making it difficult to assess whether the quartz is at equilibrium in respect of its water content, and, second, from apparent differences in experimental behaviour in solid-medium apparatus at 1500 MPa and in gas-medium apparatus at 300 MPa. Thus, the earlier experiments carried out in solid-medium apparatus have generally indicated that the water-related species responsible for weakening could be readily incorporated into single crystals to a depth of several millimeters during experiments at elevated temperatures (Blacic, 1981; Mackwell and Paterson, 1985), while no deep penetration was found in experiments in gasmedium apparatus (Paterson and Kekulawala, 1979). These differences have given rise to speculation that there is a large influence of pressure on the diffusivity of the water-related species in the quartz (for example, Paterson, 1985). However, it now appears that the differences are related to the different degrees to which microcracking has occurred in the two types of experiment since it has recently been shown by Kirby and Kronenberg (1984) and Kronenberg et al. (1986), and corroborated by Gerretsen et al. (1985) and Rovetta et al. (1986) that, in the absence of obvious cracking in
the solid-medium experiments, there is no penetration of “broad-band” water-related species that is detectable by the methods used so far, consistent with the observations in gas-medium experiments. Whether this lack of penetration is related to low diffusivity or low solubility, or both, has yet to be established, and the equilibrium solubility is still uncertain after several attempts to measure it or to calculate it (Kekulawala et al., 1981; Mackwell and Paterson, 1985; Doukhan and Trepied, 1985; Ord and Hobbs, 1986; Paterson, 1986). However, it is evident that there is still much obscurity surrounding the incorporation of water in quartz and the kinetics of its redistribution. It is a striking observation that, although the presence of water is evidently essential for the deformation of quartz in the laboratory at moderate stresses (say, below 1 GPa), the creep strengths of single crystals such as milky natural quartz and of most quartz&es, all containing mole fractions of water in excess of 10e3, are generally higher than those of synthetic single crystals and clear natural amethyst crystals containing less than lop3 mole fraction of water, as determined from broad-band OH infrared absorption in the 3 pm region (see, for example, Kekulawala et al., 1978, and Mainprice and Paterson, 1984, for observations at 300 MPa confining pressure). Infrared and microscopical evidence indicates that much of the water is present in the crystal or rock in the form of molecular a~regation, often in clusters or bubbles (McLaren et al., 1983; Aines et al,, 1984). It would therefore appear that the scale of dispersion of the water is important in determining its rheological effect in the laboratory (Paterson and Kekulawala, 1979; Kronenberg and Tullis, 1984; J. Gerretsen and M.S. Paterson, in prep.). Alongside this consideration, it can be expected that, if any redistribution of the water is necessary to ensure its optimum effectiveness during the deformation, the value of the diffusivity of the relevant water-related species will determine a certain scale of equilibration within the specimen (Kronenberg and Tullis, 1984; Paterson, 1985). Thus the rheological behaviour can be expected to depend not only on the chemical potential of the water in the sites or reservoirs in which it is
42
present in the rock but also on the scale of dispersion of the water relative to this scale of equilibra-
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133 (1987) 33-43
Science Publishers
B.V., Amsterdam
- Printed
in The Netherlands
Problems in the extrapolation of laboratory rheological data M.S. PATERSON Research School of Earth Sciences, Au~alian (Received
January
National University, Cunherru, A.C.T.
21,1986;
revised version accepted
2601 (Australirr)
June 10,1986)
Abstract Paterson,
M.S., 1987. Problems
The many types of variables
in the extrapolation and deformation
behaviour
from the laboratory
particular
cases. In the case of olivine-rich
of extrapolation, independent amounts
of water. In the case of quartz-rich
any extrapolation water are probably
to geological two important
conditions variables
The problems
rocks, recent experimental
mantle
non-Newtonian.
Tectonophysics,
of extrapolation
work indicates
types of behaviour
rocks, the uncertainties except
as an upper
but the quantitative
to the analysis is to assume a
model, calculate the predictions of the model for chosen values of the physical parameters that are required to specify particular situations, and compare the predictions with field observations. The physical parameters to be chosen will include the deformation properties of the rocks involved, which are normally obtained by some sort of extrapolation from laboratory measurements. The aim of this paper is to discuss the nature of the problems that arise in connection with this extrapolation and to stress the limitations that still exist. At relatively low temperatures and pressures rocks fail under non-hydrostatic stress by brittle fracture. However, they tend to become ductile as temperature and pressure are raised, although the presence of high pore-fluid pressures can greatly extend the brittle field. It is the ductile behaviour on which interest is usually focused in connection 0 1987 Elsevier Science Publishers
B.V.
would
bound;
the fugacity
laws governing
in extrapolating
that, within present and near-Newtonian
rheological with two
uncertainties or grain size
by the present
and it is still premature
of trace to attempt
and the scale of dispersion
their influence
large-scale
133: 33-43.
are then illustrated
be influenced
are even greater
with
A common theoretical approach of tectonic processes in the earth
data.
could be either grain size dependent Both
Introduction
004@1951/87/$03.50
rheological
regimes that need to be taken into account
to the earth are reviewed.
the flow in the upper
and distinctly
of laboratory
tectonic
of the
are not yet clear.
situations
involving
the
lower crust and upper mantle, except where the occurrence of earthquakes points to brittle behaviour. This paper will deal only with creep in rocks in the ductile field; the brittle field calls for separate consideration (Brace and Kohlstedt, 1980; Kirby, 1980, 1983; Paterson, 1978) and cataclastic effects in the ductile field, potentially important in upper crustal processes, will also not be dealt with. In relating the laboratory observations to tectonophysical processes, three types of consideration enter: (a) The form of the flow criterion for general stress states. (b) The environmental and internal variables that need be considered. (c) The circumstance that different flow mechanisms many predominant under different conditions, giving rise to a variety of flow laws for a given rock. These three topics will now be discussed in turn.
34
The
formof
Almost
the flow law all
laboratory
rheological
ments on rocks are at present pression
under superposed
confining
is, with
ur > a2 = u3 where
principal
stresses (compression
difference stress”. strain
rate
confining principal
ur,
pressure, u2, u3 are
positive).
u = u1 - u3 is reported determined
as a function
i = i,
at a given
pressure strains,
measure-
made in axial comthe
The stress
as the “flow of the
temperature
axial T and
u2 = u3(e1 > e2 > e3 are
shortening
that
positive).
the
For a given
rock deforming by a given mechanism under given circumstances the experimental flow behaviour at elevated temperatures is therefore viewed as a function of the three primary variables, and the flow law has the form:
u, i. T
f( u, i, T) = 0
(1) More generally, when account has to be taken of the total strain e and of other environmental and
internal different ered or account expressed F(u,
variables y . . . such as may examples of a given rock type other variables have to be (see next section), the flow in the form:
;, T; c, y...)
=0
enter when are considtaken into law can be
(2)
In either case, an explicit form for (1) and (2) is usually sought in which the strain rate i is expressed as a function of the other variables. In creep tests, in which u, T and environmental variables are held constant, it is generally observed that the strain rate changes as straining progresses but at a diminishing rate so that there is a tendency to approach a steady state. In analysing this behaviour, the observed strain rate is often regarded as being the sum of two components, a constant component equal to the eventual steady state rate and an ephemeral component that decreases as the straining progresses, respectively known as the steady state and transient components. However, it should be noted that such a separation is an empirical one which may not have a physical basis in terms of separate mechanisms of flow. In high temperature tests on metals (Garofolo, 1965, p. 16) the transient component of the creep strain can often be represented satisfactorily by
the expression <,[l - exp( - t/t,)] where t is the elapsed time and e,, 2, are empirical constants; the transient component of the strain rate is then (e/f,)
exp( - t/t,).
It has also been noted (Amin
et al., 1970) that, for a wide variety of metals, the relaxation time t, = 10*/i, where i, is the steady state component years
of the strain rate (thus, f,, - 10’ s-~‘). If the latter empirical
if i, - lo-l4
relationship deforming
could be validly extrapolated at geological
sient effects could of time.
Regarding
to rocks
strain rates, then the tran-
persist
for a geological
the value
period
of cr. it is often
found in laboratory tests that, over the range of strain of the experiment and within some accepted approximation (say, a few percent), a steady state is reached after several percent strain at high temperatures, that is, et - 10p2. If this observation were to extend to geological situations, it would suggest that one should be cautious about attempting to relate steady state creep observations to tectonophysical phenomena such as postglacial uplift where the strains are quite small (a similar view has been expressed recently in the tectonophysics literature; see Peltier, 1985; Sabadini et al., 1985; and Weertman, 1985). There have so far been relatively few experimental studies of transient creep in rocks at high temperature (for example, Post, 1977; Carter and Kirby, 1978; Carter et al., 1981). Provided the maximum principal stress difference is not very high, the steady state component of the strain rate observed in experiments on rocks at high temperature and pressure can commonly be fitted with good approximation to the “ power law”: i = Au” exp( -Q/RT)
(3)
where A, n and Q are empirical parameters, R is the gas constant and T the absolute temperature. As applied to the normal axisymmetric compression test, u is the maximum principal stress difference and i is the maximum principal shortening strain rate. Newtonian flow (linear viscosity) results if n = 1 but values of n = 3 to 5 or more are commonly found. In the remainder of this paper we shall only discuss steady state flow. In tectonophysical situations the stress states will, in general, no longer correspond to those in
35
axisymmetric expe~ments
and so a generalization
for i and u in cases of general stress states), the
of the steady state flow law (3) is needed. In the
flow behaviour can in practice be influenced by a
absence of appropriate studies on rocks subjected
number of additional variables which relate to the
to complex stress states, it seems most reasonable
environmental
to adopt the procedure of Odqvist (1935) often used in engineering creep calculations for gener-
composition or microstructure of the rock itself, as
alizing flow criteria, of expressing the flow law in terms of an “equivalent stress” u* and an “equiv-
indicated by y , . . in (2). We shall now list, and comment on, additional variables that might need
alent strain rate” i*,
to be considered in particular cases.
defined (neglecting
elastic
strains) as:
cJ* =
subjected
conditions
to which the rock is
or which represent
(1) Mean stress or pressure.
i* = (fi/3)[((,
- i,)*
+ (i, - &)2
il)y2
The influence of
stress, often loosely referred to as the influence of
-ul)2]1/2 (4)
+(i3 -
in the
the mean stress or hydrostatic component of the
(l/JT)[(u,- u*)*+ (a* -u3j2 +(u3
variations
(5)
the “pressure”, is relatively small and difficult to measure in the laboratory (in experimental studies the confining pressure us is often referred to as the “pressure” in this connection although it only approximates the mean stress (a, + u2 + us)/3 if a3 is small compared with a,; however, the use of a3 in extrapolating to the deep interior of (Jl -
It will be seen that, provided the volume remains constant, as is implicit, the quantities u = ui - Us and i = i, used in (3) to express the results of axisymmetric experiments correspond to equivalent stress and equivalent strain rate, respectively, as defined by (4) and (5). Thus, if Odqvist’s procedure is followed, the form (3) can be extended to general stress states in terms of i* and (I*. It is to be remembered, however, that the relation (3) is essentially empirical and that other forms might have been used. Also the term “effective” is often used in place of “equivalent” in describing u* and <*, in which case the Von Mises “effective stress” (I* needs to be distinguished
from the Terzhagi
“effective
stress”
used in treating pore pressure in the brittle field (Paterson, 1978, ch. 5). Further, in using (3) in terms of ci* and u* as the constitutive equation of flow, it is implicit that effects associated with variation of the hydrostatic component of the stress or with changes in other environmental variables can be dealt with by suitably varying the parameters A and Q. Other variables Even if we accept relation (3) as the basic form of steady state flow law f(i, u, T) = 0 with the primary variables, i, u and T (using E’* and u*
the earth might be justified insofar as the “pressure” in the earth, as calculated from depth of burial, is also taken to be the least principal stress Us). When extrapolation to higher “pressures” is done, it is usual to assume that the effect of pressure is mainly to increase the quantity Q in (3) putting Q = Q, +pV*, where p is the “pressure”, Q, the apparent activation energy at low pressure and 1/* the apparent activation volume. The parameter I’* is commonly assumed to be more or less equal to the molar volume of the slowest diffusing species (for example, Stocker and Ashby, 1973), but in silicates the identity of this species may not be obvious. The only experimentally determined values of V* for silicate rock are those of Ross et al. (1979) and Green and Hobbs (1984), determined in solid medium apparatus in the confining pressure ranges 500-1500 MPa and 1200-2700 MPa, respectively. They obtained I’* = 11 to 15 X 10e6 and 25 to 40 X 10m6 m3 mol-“, respectively, for olivine-rich rock, taking p to be the confining pressure, from which it is evident that the experimental situation is far from clear. See Poirier (1985, ch. 5) for a more detailed discussion of pressure effects. (2) Chemical environment. The chemical state of a rock may affect its rheological behaviour at high temperature. This state can be specified through equilibration with a thermodynamic en-
36
vironment
described
components
oxide, regarded ment,
and olivine
of condensed
is the intragranular
that has been demonstrated and probably
of di-
species in the environ-
such as silica. One chemical
importance
weakening (Griggs
as volatile
water or carbon
or in terms of the activities
components major
in terms of the fugacities
such as oxygen,
effect of hydrolytic in quartz
occurs in other silicates
and Blacic, 1964, 1965; Blacic, 1972). Ex-
actly how the influence be incorporated
of the water content
should
in the flow law is not well estab-
lished. An illustration
of the overall effect of water
in a rock is given by the measurements
on dunite
by Chopra and Paterson (1981, 1984) who have shown experimentally that the strain rate at a given stress can be changed by two orders of magnitude by drying the rock from a total water content of - lop3 mole fraction (- 0.01 wt.%) to a content of one or two orders of magnitude less; there has been some controversy about the mode of action of the water but it is clear that both intragranular and grain boundary effects can be involved (Karat0 et al., 1986). (3) Presence of jluid phases. The pore pressure effects that are well known in the brittle field are much less important in ductile behaviour at high temperature insofar as the flow stress is concerned, provided the total pressure minus the pore pressure is large compared with the brittle-ductile transition pressure. However, some influence of pore pressure, with associated dilatancy, can be detected well into the ductile field even at elevated temperatures (Fischer and Paterson, 1985) indicating that subsidiary microcracking is a widespread accompaniment of crystal plasticity in rocks, with implications for increased permeability to fluids during deformation. The presence of fluid phases such as metamorphic water-rich phases or igneous partial melts can thus potentially have important effects involving the transport of material through the fluid phase, contributing in turn to the strain or assisting intergranular accommodation (see, for example, Etheridge et al., 1984). Such effects can be expected to introduce some grain size dependence in the parameter A in (3) and possibly also affect the parameters n and Q. (4) Grain size. Under some conditions, when
the
rate-controlling
volves processes
deformation
at grain boundaries
mechanism
in-
rather than in
grain interiors, the creep rate can be strongly dependent on grain size, as in diffusion creep. In this case the pre-exponential ten as Ad-“’ another
where
empirical
the range
d is the grain size and
constant,
1 to 3. Grain
thus more likely deformation
Gueguen, could
found
to be important of
m is
to be usually
size dependent
sizes, as has been pointed the
factor in (3) is rewritin
creep
is
at small
grain
out in connection
with
mylonites
(Boullier
and
1975); but even at a given grain size it
become
predominant
as low stresses
and
strain rates are approached because the stress exponent n is also smaller than for grain size insensitive creep. Thus grain size sensitivity of creep may have to be taken into consideration in tectonophysics, as will be further discussed later. (5) Preferred crystallographic orientation. Because of crystal anisotropy, departures from random orientation of the crystallographic axes of the grains will tend to introduce anisotropy of flow stress in the rock, requiring different choice of parameters in the flow law for different orientations. Relatively little is known about the flow laws for anisotropic rock, although some observations have been made on marble (Heard and Raleigh, 1972). The development or modification of preferred orientation during deformation will also tend to affect the rate of attainment of a steady state. Thus, the creep behaviour of a given type of rock can be expected to vary substantially as changes occur in factors such as those just discussed. Expressed in terms of strain rate this variation may in some cases be of orders of magnitude (for example, in the case of change in water content). Therefore, although very broad generalizations about the flow law for a given type of rock may be possible, any refined consideration will normally have to take into account other variables than just stress, temperature and pressure. Deformation
mechanisms
and rheological regimes
We now return to consider further the nature of the relationships between strain rate (, stress u and temperature T for a particular set of the
37
variables considered in the previous section. We
servations (Heard, 1963; Heard and Raleigh, 1972;
need to take into account that there is a variety of
Rutter,
mech~sms
over, the values of the parameters for particular
whereby the rock can be deformed,
1974;
Schmid et al., 1977, 1980). More-
which will be reflected in a variety of rheological
regimes vary substantially
regimes in terms of the (, u, T relationships.
limestone to another, so that no common “flow
The mechanisms that can be envisaged for the deformation
of a rock can be considered broadly
in three groups,
distinguished
by the scale of
reorganization involved: (1) Atomic transfer mechanisms, in which the change
of shape is achieved
reorganization,
the transfer
by atom-by-atom
process
itself being
most commonly envisaged to be by diffusion; the various types of diffusion creep (Nab~o-He~ng, Coble, solution transfer, etc.) fall in this category. (2) Crystal plasticity mechanisms, in which, by means of dislocation movement, discrete blocks within crystal grains are translated relative to each other while retaining the integrity of the grain; there are several types of dislocation creep, distinguished according to the different rate-controlling factors that predominate (Poirier, 1985, ch. 4). (3) Granular flow mechanisms, in which whole grains or groups of grains are moved relative to each other without the crystallographic constraints in dislocation gliding; the “grain-swapping” mechanisms associated with super-plastic flow (for example, the Ashby-Verrall mechanism), as well as cataclastic or soil-like flow, fall within this group. Since there are distinctions in rheological behaviour within each group of mechanisms as well as between groups according to the various rate controlling factors that may predominate, it can therefore be expected that, as i, CT,T are varied over a wide range, a given rock in a given environment will be able to exhibit a variety of flow laws, this variety being expressed in terms of different values of A, n, Q insofar as the form (3) can be used. The variety is even greater when rocks from different formations of the same compositions type are considered. Thus, without going into details about possible mechanisms involved, it has been found that, for marbles and limestone, there are at least three high-temperature rheological regimes which can be defined expe~mentally by different values of the parameters required to fit three-parameter flow laws such as (3) to the ob-
from one marble or
law for calcite rock” can be given (a summary of values for Solnhofen
limestone and Carrara and
Yule marbles is given in table II of Schmid et al., 1980). The succeeding discussion will be mainly in terms of the phenomenological behaviour rather than of the deformation
mechanisms;
the rheo-
logical measurements provide the primary data for extrapolation,
while the mechanisms may remain
controversial.
However, an understanding of the
mechanisms operating both in the laboratory and geologically will eventually be needed in deciding or confirming which regime of observed behaviour can be validly extrapolated to given geological conditions. The various regimes of flow behaviour can be depicted conveniently in a diagram analogous to the deformation mechanism map of Ashby (1972), in which the fields of applicability of individual flow laws are delineated. The fields may correspond to different mechanism fields but this correlation need not necessarily have been established. Examples of such rheological regimi maps are shown in Fig. 1 for Carrara marble and Solnhofen limestone, based on the experimental work mentioned above and collated in Schmid et al. (1980). While Ashby maps are usually plotted in temperature/flow-stress coordinates with constant strain rate contours, and rheological regime maps can be similarly plotted, str~n-rate/temperature coordinates with constant stress contours have been used in the figures in this paper on the grounds that the boundary conditions for geological and geophysical problems are commonly expressible in terms of displacement rate or strain rate and temperature, as, for example, in collision tectonics driven by plate movement; the conventional Ashby plot may be more appropriate for situations such as gravity sliding constrained by given body forces. It is seen that the patterns of behaviour for the fine-grained Solnhofen limestone and the coarsergrained Carrara marble are distinctly different, there being little correspondence between the
1000
‘.8 -
900
800
700 -I 3 z P 500 _I z R 500 Regime 2 Q=297
I I -1
kJ mot
LOO
I
-14
Fig. 1. Flow regime boundaries experimental
/
I
i
-12
-10
for Solnhofen
field shown and extrapolated
limestone
towards
f
-8 V-~ fQ (Strain Rate) and Carrara
geological
parameters for the regimes 2 and 3 distinguished in each rock in the given experimental field; moreover, a simple extrapolation predicts that the distinction in behaviour will become more exaggerated as geological conditions are approached. Aspects of extrapolation are discussed by Schmid et al. (1980). We now consider two particular cases, those of olivine-rich and of quartz-rich rocks, respectively typifying the problems of extrapolating to upper mantle and to lower crustal conditions. These examples are not to be viewed as being definitive at this stage but rather as illustrating the uncer-
strain
marble
-4
as determined
i-Ai- -2
by Schmid
et al. (1977, 1980) in the
rates
tainties involved in extrapolation and only a few of the aspects mentioned earlier are addressed. Olivine-rich
rocks
The case of olivine-rich rocks can serve to illustrate the potential application of rheological regime maps in tectonophysics. Mention has already been made of the laboratory observation that the flow strength of dun&e is strongly influenced by trace amounts of water (Carter and AvC Lallemant, 1970; Blacic, 1972; Post, 1977; Chopra and Paterson, 1981, 1984). However, at
39
least in the expe~ments of Chopra and Paterson, the grain boundaries contained small amounts of melt due to decomposition of minor accessory hydrated minerals in the rocks. In order to avoid this complication and to study the role of grain size? a study has more recently been made on pure polyc~st~line olivine synthesized by hot-pressing crushed olivine crystals (Karat0 et al., 1986). This work has demonstrated that, as in the case of fine-grained limestone already mentioned, two strongly contrasting regimes of flow can be distinguished: relatively coarse-grained material shows little grain size dependence and a fairly high stress exponent n - 3 or more (regime l), while finer-grained material shows a strong dependence on grain size d, between i a: d-” and dp3, and a low stress exponent, n - 1 (regime 2). The presence of water weakens the polycrystalline olivine in both regimes and increases the transition grain size from about 25 urn to about 40 pm at laboratory strain rates. In order to explore possible implications of such grain size dependence for tectonophysical behaviour we assume a flow law of the form: i = Aand-”
exp( - Q/RT)
with the following parameters
(a in MPa, d in
pm):
Regime 1 Regime 2
1gA
n
m
Q
3.6 6.85
3.5 1.3
0 2.5
450 kJ mol - ’ 300 kJ mol-’
These values are taken as being typical of wet polycrystalline olivine; the values of n and m derive from the observations of Karat0 et al., and the value of Q for regime 1 from observations on wet dunite already mentioned, while the value of Q for regime 2 is taken as being about the same fraction of that in regime 1 as was found for limestone. Finally, the transition from regime 1 to regime 2 is taken from the observations of Karat0 et al. as occurring at a grain size of 50 pm and a stress difference of 70 MPa under the conditions 13OO*C and lO_’ s-r strain rate (the experimental point A in Fig. 2). The boundary between regimes 1 and 2 can then be calculated to be as shown in Fig. 2 by the heavy line marked “50 pm”‘; constant stress contours for 10, 100, and
1000 MPa have also been added near this boundary. Using the K2-s relationship, the boundary between regimes 1 and 2 can then be recalculated for other grain sizes and is shown for 1 mm and 5 mm in Fig. 2, where some of the corresponding stress contours are also shown adjacent to the regime boundaries. If m were taken as 3 or 2 instead of 2.5, the boundary for 1 mm grain size would be shifted about one order of magnitude towards lower or higher strain rates, respectively, and the boundary for 5 mm grain size would be similarly shifted about one and a half orders of magnitude. It follows from Fig. 2 that, in proceeding from laboratory to geological conditions (indicated by the two shaded areas), regime 1 behaviour would still be expected under geological conditions if the grain size were of the order of 10 mm or larger, provided the other environmental variables were not changed significantly (insofar as they exert rheological influence) and provided no other type of behaviour, not yet discovered, were to intervene. However, when the grain size is reduced to the order of 1 mm, grain size sensitive (regime 2) flow would be predicted to predominate, with the some provisoes. In the latter case, extrapolation from measurements made at large stresses in a high-n “dislocation creep” regime would tend to over-estimate the flow stress. In view of the incomplete knowledge of the role of other variables in the rheology of oliviue-rich rocks, the extrapolations of Fig. 2 should not be taken too seriously as #~str~ning possible behaviour in the upper mantle. However, they suggest that, while high-n dislocation creep may well be expected in coarse-grained rocks (consistent with field observations of strong preferred orientations in coarse-grained nodules, for example, Mercier and Nicolas, 19751, grain size sensitive low-n flow may possibly occur in other cases, especially where the grain size is only a few millimeters or less (consistent with the suggestion of Boullier and Gueguen, 1975). Before att~buting more quantitative significance to these extrapolations, it will be necessary to obtain experimental confirmation of the values of Q and to establish the effects of ~co~orating other phases such as pyroxene, raising the pressure, varying the amount
40 LABORATORY
GEOLOGY
-0.6
-0.7
zi 4 If
x iL0.t
!f
*$
l’
I I
t
I
I
-0.s
Of 7
_&II.
I
/
C9
WC
,rs ‘erg ’
I
-1
I
I
I
R
900
/‘Regime I
011422 ./
1
: lgA=3.8&lPa _ ..LI “2.3.3 u=450
units , i
kJ moi
, i 800 I
_,‘Y
/
,
. I
---I I’ I
-1.
0
’
:
I
I
I’
I ,
I
-----I
Regime
2 : IgA= -1.6(MPa, pm Units) n ~1.3 mP2.5 a=300 kJ moi’
.I
I
,I
/I’
I
I
I
I
I
I
-
1
-12
-14
,._.._L
1.. -L
-10
-8 lg (Strain
Fig. 2. Flow regime boundaries temperature/strain-rate
between
stress contours
parameters
I
_,
-8
for three gram sizes in polycrystalline
for 50 pm gram size has been extrapolated shown
i
AGGREGATE (WET) f i~__L1_ -2 -4
700
Rate)
high 11 and low n behaviour
plane. The behaviour
mm grain sizes, using the estimated gram size. Constant
OLIV~NE
t
(see text) which are normalized
are shown in dotted (regime 1) and broken
to lower strain
olivine, projected
on the
rates and to 1 mm and 10
to fit the experimental
(regime 2) lines in the neighbourhood
point
A for 50 nm
of the boundary
for each gram size.
of water, and altering the environmental or chemical conditions in any other way. Quartz-rich
rocks
The case of quartz-rich rocks is more difficult to analyse than that of olivine-rich rocks because the basic modes of behaviour and the factors
affecting these are less well understood. Quartz has proved to be an especially difficult material to study rheolo~~ally in the laboratory (Blacic and Christie, 1984; Paterson, 1985). In the absence of water, the strength of single crystals is enormous, the flow stress (stress difference) at laboratory strain rates being still well in excess of 1000 MPa at 1300°C and 300 MPa confining pressure (mea-
41
surement by J. Bitmead and MS. Paterson, described in Doukhan and Trepied, 1985). The deformation of quartz is, therefore, generally believed to depend essentially on the presence of water, based on the discovery of ““hydrolytic weakening” by Griggs and Blacic (1964, 1965). The weakening has been correlated with a broadband infrared signature of the responsible species (Kekulawala et al., 1978), corresponding to the presence in the quartz of some form of molecular water (Aines et al., 1984). However, the nature of the weakening effect is still not well understood. Some discussion of single crystal studies in relevant at this point to illustrate the difficulties experienced in obtaining a clear picture of the factors governing the role of water in the laboratory deformation of quartz-rich rocks, factors that need to be understood before the validity of extrapolations to geological conditions for quartzrich rocks can be properly assessed. The difficulties experienced in the single crystal studies seem to arise, first, from kinetic sluggishness, especially in the diffusion of the water-related species, making it difficult to assess whether the quartz is at equilibrium in respect of its water content, and, second, from apparent differences in experimental behaviour in solid-medium apparatus at 1500 MPa and in gas-medium apparatus at 300 MPa. Thus, the earlier experiments carried out in solid-medium apparatus have generally indicated that the water-related species responsible for weakening could be readily incorporated into single crystals to a depth of several millimeters during experiments at elevated temperatures (Blacic, 1981; Mackwell and Paterson, 1985), while no deep penetration was found in experiments in gasmedium apparatus (Paterson and Kekulawala, 1979). These differences have given rise to speculation that there is a large influence of pressure on the diffusivity of the water-related species in the quartz (for example, Paterson, 1985). However, it now appears that the differences are related to the different degrees to which microcracking has occurred in the two types of experiment since it has recently been shown by Kirby and Kronenberg (1984) and Kronenberg et al. (1986), and corroborated by Gerretsen et al. (1985) and Rovetta et al. (1986) that, in the absence of obvious cracking in
the solid-medium experiments, there is no penetration of “broad-band” water-related species that is detectable by the methods used so far, consistent with the observations in gas-medium experiments. Whether this lack of penetration is related to low diffusivity or low solubility, or both, has yet to be established, and the equilibrium solubility is still uncertain after several attempts to measure it or to calculate it (Kekulawala et al., 1981; Mackwell and Paterson, 1985; Doukhan and Trepied, 1985; Ord and Hobbs, 1986; Paterson, 1986). However, it is evident that there is still much obscurity surrounding the incorporation of water in quartz and the kinetics of its redistribution. It is a striking observation that, although the presence of water is evidently essential for the deformation of quartz in the laboratory at moderate stresses (say, below 1 GPa), the creep strengths of single crystals such as milky natural quartz and of most quartz&es, all containing mole fractions of water in excess of 10e3, are generally higher than those of synthetic single crystals and clear natural amethyst crystals containing less than lop3 mole fraction of water, as determined from broad-band OH infrared absorption in the 3 pm region (see, for example, Kekulawala et al., 1978, and Mainprice and Paterson, 1984, for observations at 300 MPa confining pressure). Infrared and microscopical evidence indicates that much of the water is present in the crystal or rock in the form of molecular a~regation, often in clusters or bubbles (McLaren et al., 1983; Aines et al,, 1984). It would therefore appear that the scale of dispersion of the water is important in determining its rheological effect in the laboratory (Paterson and Kekulawala, 1979; Kronenberg and Tullis, 1984; J. Gerretsen and M.S. Paterson, in prep.). Alongside this consideration, it can be expected that, if any redistribution of the water is necessary to ensure its optimum effectiveness during the deformation, the value of the diffusivity of the relevant water-related species will determine a certain scale of equilibration within the specimen (Kronenberg and Tullis, 1984; Paterson, 1985). Thus the rheological behaviour can be expected to depend not only on the chemical potential of the water in the sites or reservoirs in which it is
42
present in the rock but also on the scale of dispersion of the water relative to this scale of equilibra-
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