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OCR for page 42
Pore Fluid Pressure Near
Magma Chambers
DENIS L. NORTON
University of Arizona
INTRODUCTION
Fluid pressure in the Earth's crust is a function of
variations in the regional stress, temperature, and compo
sition of H2Orich fluids. Whereas in relatively quiescent
geologic environments the effects of fluid pressure are
subtle, its effects in magma environments can be easily
identified (Knapp and Norton, 1981; Burton and Helgeson,
1983; Lantz, 1984; Norton, 19841. In the nearfield re
gions of magmas the occurrence of high fluid pressures is
evident from the functional form of the equation of state
for the fluids; the distribution and geometry of fluidfilled
pores; and relationships among the thermal, chemical, and
mechanical processes. This chapter discusses processes in
magmahydrothermal systems to demonstrate the mecha
nisms involved and to show that in these environments
fluid pressure plays an essential role in the generation and
maintenance of percolation networks for hydrothermal fluid
flow.
Extreme variations in fluid pressure in the nearfield
region of magmas are caused by sparse but significant
amounts of H2Orich fluids that are ubiquitous in the host
rocks and common in the magmas. Pore fluids typically
found in the host rocks have large positive values of the
isochoric coefficient of thermal pressure, whereas those in
the magma have large negative values. Therefore, fluid
pressure increases during the dissipation of thermal en
42
orgy from magmas are a natural consequence of the cool
ing process, where temperature increases in the host and
concurrently decreases in the magma. The resultant pres
sure increase in both domains generates large local stresses.
Once the rock fails, fracture networks form that are con
tinuous between the domains on either side of the pluton
wall. These networks allow fluid flow and convective
transport of thermal energy, thereby increasing the rate of
pressure change in both environments.
Because the evidence for these conditions derives from
both field observation and theory, the following discus
sion first reviews the conditions of ambient pressure in
crustal environments and then examines the consequences
of perturbing these conditions in the context of the distri
bution and geometry of pores.
PRESSURE AT DEPTH
Pressure conditions that exist prior to the imposition of
a thermal perturbation on the crust are critical to predict
ing the magnitude and consequences of the subsequent
fluid pressure increases. Pressure at the base of a column
of rock that is composed of minerals and fluidfilled pore
. .
spaces IS glVeI1 By
Total = Po + g  pr (T. P. X) dz, (2.1)
OCR for page 42
PORE FLUID PRESSURE NEAR MAGMA CHAMBERS
where g is the magnitude of the gravitational vector and
pr(T, P. X) is the volumeaveraged density of the rock in
the column. The term rock refers to minerals plus pore
space, and all pore space is assumed to be filled with fluid.
Because density is a function of the material composition
and of temperature and pressure, an explicit function is
needed to integrate Eq. (2.1~. It expanded into a statement
that independently accounts for the density of solid and
fluid phases:
Liz
Total = Po + g J [(>mpm + 4)fpf (T. P. X)]dz, (2.2)
zo
where (m iS the total volume fraction of mineral phases, Pm
is the average density of the mineral assemblage, Of is the
volume fraction of the fluid phase, and pf(T, P. X) is the
temperature, pressure, and compositiondependent den
sity of the phase. Mineral densities are nearly constant
relative to fluid density over the state conditions com
monly encountered in the crust. Consequently, the pres
sure contribution from the minerals can be removed from
the integral:
~ ta} = Po + g [em pm + rZ aim ~ ~ ]
Total pressure can therefore be determined by expressing
the density function as an equation of state using relations
like those reported by Helgeson and Kirkham (1977) and
Johnson and Norton (1989~.
However, the crust is generally in a state of heterogene
ous stress that can be represented as a stress tensor, T:
.. Oxx Oxy Oxz
T = oyx oyy cyz , (2.4)
O.zx Cozy Razz
where aij refers to the stress exerted along the ith axis
normal to the jth direction. The resultant pressure caused
by these stresses is equivalent to the trace of the stress
tensor and is called the mean pressure, Pm:
oxx + thy + ZZ (2.5)
3
The mean (Pm) and total (Prowl) pressures from Eq. (2.3)
are equal only in conditions where ~ = ~ = ~ . To
ox yy zz
demonstrate the role of fluid pressure in deformation, we
will assume that the stress field is homogeneous.
Pressure conditions actually encountered in the subsur
face are unlikely to correlate closely with those values
derived from the integrations using extreme values of (>m
in Eq. (2.3~. Neither (dim > 0 for O < Z < Zmax) nor hydro
43
static (him > 0; of = 1 for O < Z < ZmaX) pressure conditions
are likely because pressure generally depends on the geo
logic history of the lithologic units.
As an example of the effect of dynamic loading history
that can alter the pressure from the conditions just men
tioned, computations that describe the burial of a fluid
filled pore are reviewed in this chapter; the details of this
experiment can be found in Knapp and Knight (1977~.
The loading process can be thought of as diagenesis of a
fluid packet, first isolated from the other pores in the rock
by nearsurface compaction and then subjected to increases
in temperature and confining pressure as it subsides to
greater depths in the basin.
Even though the fluid in this situation may constitute a
small fraction of the rock, typically less than 10 wt.%, its
density variation in response to changes in state conditions
can have a large effect on local pressure conditions. Because
temperature and pressure are functions of depth, the total
differential of fluid density with respect to depth is
dpf (3pf ~ do (0pf ) dP
dz gaT )PX dz Pap )TX dz
, (0Xi )TPxi dz (2.6)
where the partial derivatives are derived from the equation
of state for the fluid, f(P, T. p, X) = 0, and the derivatives
of pressure and temperature with depth are independent
quantities that must be defined from solutions to the heat
transfer equations. The partial derivative of composition,
on the far right of Eq. (2.6), is significant in all natural
environments. However, an important aspect of fluid
pressure variations can be demonstrated with a single
component fluid. Therefore, the following discussion
focuses on the singlecomponent H2O system and omits
the compositional variation in Eq. (2.6~.
Density variation with depth for a fluid in the pure
system is given by
dPf (3pf ) dT (0pf ) dP (2.7,
dz OT P dz aP T dz
where the dependent partial derivatives can be replaced
with the intrinsic properties of the fluid the isobaric
coefficient of thermal expansion, Of:
pf (aT)p (2.8)
and the isothermal coefficient of compressibility, pf:
Do
(ap] (2.9)
pf vaP )T
OCR for page 42
44
Substitution of these quantities into Eq. (2.7) gives
Pf = pfaf + puff . (2.10)
dz dz dz
If the fluidfilled pore is assumed to remain at a con
stant volume, the variation of pressure within the pore is
only a function of the expansivitytocompressibility ratio
and the thermal gradient:
dP _ al dT
dz Of dz
, (2.11)
where af/,Bf is the isochoric coefficient of thermal pres
sure. It varies from 5 to 20 bars/°C for ranges in state
conditions commonly found in subsiding sedimentary basins
(Figure 2.1~.
Knapp and Knight (1977) found pressure increases
within the pore to be large at shallow depths even for
modest thermal gradients. They also found that only for
thermal gradients of less than 10°C/km will the pressure
within the constantvolume pore remain less than the mean
confining pressure, Pm, as subsidence occurs; for larger
thermal gradients the fluid pressure increases to values
much greater than the confining stress. This overpressure
occurs at depths that depend primarily on the magnitude of
the thermal gradient. Knapp and Knight's computation
showed clearly that fluid pressure can vary with depth if
the pore geometry is poorly interconnected and that the
variation is a function of the thermal pressure coefficient
TEMPERA r URE (·C)
0 200 400 600 800 ~ 000
200
lo:
~ 600
In
son
1 000
B ~ R S / TIC )
0.5
FIGURE 2.1 The isochoric coefficient of thermal pressure,
a /pf, computed with the equation of state for the H2Osystem
(itrom Johnson and Norton, 1989~.
DENIS L. NORTON
of the fluid. The upper limit of the fluid pressure is a
function of the pore wall strength and failure process.
Prior to examining the consequences of failure on fluid
pressure, the effect of a thermal perturbation imposed on
the same pore space as described above is examined.
CHANGE IN FLUID PRESSURE
Thermal perturbations imposed on a porous rock gener
ate differential stresses because of large differences in the
constitutive properties between minerals and fluids. The
physical significance of this difference in response to
thermal changes is that small thermal changes in the crust
can cause large differences between values of mean pres
sure and fluid pressure. For a particular thermal state in
the crust the hydrostatic and mean or total confining pres
sure can be defined from Eqs. (2.3) and (2.5~. Perturba
tions from this state caused by changes in the thermal flux
are now examined.
Consider the pressure changes that occur within an
isolated fluidfilled pore. Assume that the fluid pressure
exerted on the pore wall is initially in equilibrium with the
^. .
mean confining pressure:
Pf = Pm . (2.12)
The time derivative of fluid pressure increase as a result of
temperature and pore volume changes is
dPf = (0Pf ) dT + (3Pf) dV ~ (2 13)
dt aT v dt av T dt
where V is the pore volume. Because the pores are as
sumed to be filled with a singlephase fluid, this volume is
equivalent to 1/pf. Again, consider a constant volume
process, dV/dt = 0, consistent with regional strain rates
that are much smaller than those caused by the local change
in temperature and with the situation in which a local
increase in fluid pressure does not dilate the pore at a
significant rate. Under this condition the change in inter
nal pressure as a function of time is given by
dBf (3Pf ) dT, (2.14)
dt aT v dt
where faPf /aT~v is the ratio of the isobaric coefficient of
thermal expansion to the isothermal compressibility, af /0f
(Figure 2.1~. Notice that the time derivative of the pres
sure is analogous to its spatial derivative in Eq. (2.12~. As
a consequence of Of /,Bf, pressure increases of several bars
per degree centigrade can occur over relatively short times
OCR for page 42
PORE FLUID PRESSURE NEAR MAGMA CHAMBERS
.
I ~
Edge

Agerturc
in the nearfield region of magma bodies, where the change
in temperature with time is rapid.
The magnitude of pressure increases associated with
this situation and the burial computation in the previous
section are likely to exceed the strength of the walls of
intergranular pore space or of the mineral grains them
selves. The conditions for failure depend on the mode of
occurrence of the fluid in the rocks, particularly on the
shape of the pore space.
FLUIDS IN ROCKS
Fluids in rocks are seldom directly observed, but their
presence is inferred from the presence of pore space.
Although the mode of occurrence of the fluid phase is
traditionally expressed in terms of the volume or porosity,
the geometry of the fluid and its distribution with respect
to the mineral phases are of equal importance to consid
erations of transport processes, particularly the conse
quences of fluid pressure changes. Studies of the mode of
occurrence of fluid have revealed that the shape of the
pore space is determined by properties of the fluid that
fills them and by the local conditions of stress.
The total porosity of crystalline igneous rocks ranges
from a few percent to less than a fraction of a percent, and
of this total pore space the everpresent fractures in these
rocks contribute only a very small fraction of pore space to
the total, circa 103 (Snow, 1970; Norton and Knapp, 19771.
Igneous rocks commonly have a relatively large permea
bility, >10~4 cm2, during their active thermal history, but
only a small porosity, or total fluid fraction, is associated
with the flow channels. This is because the flow channels
45
FIGURE 2.2 (A) Intersect of fracture with
topographic surface. Slitlike form is
typical of fracture topology at all scales of
observation (Norton, 1987~. (B) Idealized
fracture form based on dislocation theory
(from Mavko and Nur, 1978; Norton,
unpublished field data).
are elongate fractures with apertures on the order of a few
hundred microns (Norton and Villas, 1977~. The redistri
bution of only a small fraction of the fluid contained in the
total pore space of such rocks into fractures can signifi
cantly affect the rock permeability and the mechanism of
transport through it.
The slitlike nature of fluidfilled pores (Figure 2.2) is
caused by the active deformation of their host rock within
a heterogeneous stress field and the redistribution of a
portion of the fluid phase into fractures whose orientation
is a function of the stress trajectories. Once formed, these
fractures are extremely sensitive to the differential be
tween the mean confining pressure and the fluid pressure
within the fracture. They are therefore delicate indicators
of fluid pressure conditions because the mean confining
pressure is relatively constant over long times, whereas
the fluid within the fractures is sensitive to local changes
in temperature.
FAILURE CRITERIA
The small finite strength of the pore wall implies that
for the initial condition of Pf = Pm only a small pressure
increase is necessary to reach the yield strength of the
wall. The strain rate associated with this deformation is
strictly a function of the local time derivative of the tem
perature. The duration of the elastic deformation may
range from a few months to years. But the failure of the
wall and consequent fracture propagation is an instantane
ous process, in which fluid expands irreversibly against a
fixed pressure.
Extensive commentary on failure criteria exists in the
OCR for page 42
46
literature; the findings by Berbabe (1987) indicate that a
simple failure law matches experimental data on fracture
formation in crystalline rocks. This law describes the
failure of the pore walls when the sum of the confining
pressure and wall strength is exceeded by the internal fluid
pressure:
PC +T = Pf (2.15)
where ~ is the tensile strength of the rock. This failure
criterion oversimplifies the mechanical problem consid
erably but highlights the forces involved and accurately
describes the transitions from one pore configuration to and
another.
The generation of pressures in excess of the mean
confining pressure requires material strengths that will
withstand the excess pressure. Because on a regional scale
a rock has no effective strength, pressures in excess of Pm
are not likely to develop over large portions of the crust.
However, if the initial condition were one in which fluid in
equilibrium with the local value of hydrostatic pressure
was sealed into an isolated pore and then heated, the fluid
pressure might increase several tens of bars up to the local
confining pressure before failure would occur. Therefore,
the following two failure conditions should be considered:
~  m~
1. The condition in which the initial condition was
P ~ P . A fluidfilled pore fails and the fluid expands
against a pressure only slightly lower than the condition of
failure.
2. The condition in which the initial condition was
Pf ~ Phylum. Failure may not occur until pressure increases
to Pm or even slightly greater depending on the rock strength.
Expansion may then occur against PhydIo if the newly formed
fracture intersects a regime of hydrostatic pressure.
The main difference between the two situations is the
amount of energy required to attain failure. The second
situation requires more energy simply because the pres
sure increase required to attain failure is greater. In either
of the above situations the volume increases associated
with failure can be examined in terms of irreversible ex
pansion against a constant pressure value, Pb.
VOLUME CHANGE CAUSED BY FAILURE
The volume of the newly formed fracture is a function
of the expansivity of the fluid and the energy lost by
viscose flow. Because fluid flows away from the breached
pore wall a relatively short distance and for a limited time,
viscose dissipation of energy is likely to be small. The
rate of volume change in a fluid subjected to differential
DENTS L. NORTON
changes in temperature and pressure is given by the total
differential of volume with respect to time:
dV av dT av dP
dV bT P dt LIP dt ' (2.16)
where the dependent partial derivatives can be abbreviated
as
1 Pave
a 
V aT P
V (aP )T
(2.17)
(2.18)
Substituting these quantities into Eq. (2.16) gives an ex
pression for volume change in terms of the fluid properties
and the change in temperature and pressure:
 = Va  Vp. (2.19)
dt dt dt
Both the expansivity and the compressibility of an H2O
rich fluid phase are much greater than those of minerals.
Therefore, Eq. (2.19) can be written only in terms of the
fluid properties without introducing significant errors in
the analysis (Moskowitz and Norton, 1977):
dV V dT V ~ dP (2.20)
dt dt dt
The pressure differential is relatively small where the fail
ure occurs at pressure slightly greater than the confining
pressure, but where expansion is against hydrostatic pres
sure the change is substantial. Integration of Eq. (2.20) for
a given pressure over fixed temperature limits gives
If dVf rT rP
Jvf° Vf JTO al (T)dT + J ,Bf(P)dP (2.21)
and
Vf = Vf exp :[ af(T) dT + ,[ pf(P)dP]. (2.22)
The concomitant propagation of the fracture and the
thermal expansion of the fluid as it is released by the pore
wall failure result in a net increase in porosity that poten
tially augments the flow porosity. The porosity increase
that results from the failure of pressurized pores can be
OCR for page 42
PORE FLUID PRESSURE NEAR MAGMA CHAMBERS
computed from the equation of state of the fluid and the ° 2° ~
change in fluid volume defined by Eq. (2.181. The porosity,
0, contributed by the isolated pores in a region is the initial
volume fraction of fluid in those pores:
Vf Vf . (2.23) ~
Vtotal Vm + Vf c:
The initial fluid volume is a function of the initial pores 2 0.1 0
ity,
_ 0.15

Vf = <) Vtotal , (2.24)
and the total volume of minerals in the rock is
Vf = ( 1  ¢) VtOtal . (2.25)
Therefore, the fluid porosity at a time following a thermal
change can be expressed as a combination of Eqs. (2.23)
and (2.17~:
of =
ofF(T) (2.26)
(1  (f) + ~fF(T)
where the function, F(TJ, in Eq. (2.26) is
F(T) = exp [iT af(T) dT ip0 of ]
Porosity increases in a superexponential manner be
cause of the exponential term and because At increases
exponentially in the critical region of the H2O system.
Under nearcritical conditions the porosity doubles in re
sponse to only a few degree increase in temperature (Fig
ure 2.3~.
Because a large portion of the total porosity in a crys
talline rock is in the form of isolated porosity, circa 0.01 to
0.1, the thermal expansion of pore fluid could be quite
effective in increasing the interconnected pore space, which
constitutes the flow network in a rock and therefore di
rectly affects the value of permeability. Furthermore, the
flow porosity required for moderately large values of per
meability is on the order of 104 to 10s. Conversion of
even a small percentage of the isolated pores into oriented
fractures can lead to enormous increases in permeability.
ENERGY REQUIRED FOR FRACTURE
The total energy available from a volatilerich mag
matic body is ~200 cal/g. In regions where the alteration
process produces hydrous phases, the exothermic heat can
add an additional amount of up to 40 cal/g. Mafic and
4~ '~/4,~,
400 `800 _, ~~
_~~
47
005 1 ~ ~ 1 ~ ~ ~ ~ 1 ~ ~ ~ ~ 1 ~ ~ ~ ~ I
0 100 200 300 400 500
TEMPERATURE (°C)
FIGURE 2.3 Maximum change in porosity as a function of
temperature change for an isolated fluidfilled pore that propa
gates through the rock matrix on failure (Moskowitz and Norton,
1977).
volatilepoor magmas contain more thermal energy than
felsic magmas because of the large 100 cal/g heat of crys
tallization. The thermal energy converted to mechanical
energy during the failure process described is only a few
calories per gram (Knapp and Knight, 1977~.
Although the energy expended in a typical situation
amounts to only a few calories per gram, the increase in
permeability that can result from the failure is significant.
This increase allows a small fraction of the pressure to be
dissipated by viscose flow. However, the generation of an
interconnected network over a broad region also disperses
the fluid force field generated by the buoyancy forces.
Therefore, the increase in permeability results in an in
crease in fluid velocity and consequently an increase in the
advective transport of chemical components and heat.
ADVECTION AND PORE PRESSURE
Increases in rock permeability cause proportionate in
creases in fluid velocity. The fluid flow is driven by the
pervasive buoyancy force field, which is an inevitable
consequence of changes in heat flux, particularly those
changes caused by the infiltration of magma. Therefore,
when the synchronous propagation of fractures produces
an interconnected network of flow channels, fluid motion
ensues and augments the flux of heat and chemical compo
nents. Localized flow associated with the individual fail
ure events also augments the flux but is less effective than
the buoyancy force driven flow.
If local equilibrium between the fluid in an isolated
pore and the minerals that form the pore wall is assumed,
OCR for page 42
48
then at the time of pore failure and propagation of a frac
ture a gradient in fluid composition will be established
along the interconnected fracture network because of gra
dients in state conditions. This compositional gradient in
the fluid composition and the coinciding fluid velocity
advect chemical components from one environment to
another.
As an example of this process, consider a system in
which the lithologic units can be represented by the sys
tem SiO2H2O; they contain only quartz and an aqueous
fluid locally in equilibrium with quartz. Chemical changes
caused by advective flow in a system, where temperature,
pressure, and composition gradients occur and where
equilibrium prevails locally between quartz and fluid, are
symbolically depicted by
Fluid Quartz Advecuon
a(~)fiO2 + a(~)Sti02
at a'
 + vf ~Yfi°2 = 0 (228)
where the relationship between the rate of change in quartz
and the concentration of aqueous silica can be defined by
the equilibrium equation:
Si02 = Aqueous (2.29)
The partial derivatives of 9ifi02 in the fluid phase with
respect to time, t, and distance, 1, can then be defined in
terms of the standard state partial molal enthalpy, AHr°,
and partial molal volume, AVr°, of the equilibrium reaction
between quartz and aqueous silica, Eq. (2.29):
If 2 ~SiO2 0 026 (^H~° AT AV,° ~:
Oaf = ~fit'2 0 026 (SHE AT _ ~Vr i) ~ (2 31)
Ol RT2 dl RT dl
where the assumption has been made that the activity of
aqueous silica is equal to its morality, and the concentra
tion of silica in Eqs. (2.30) and (2.31) is in grams per cubic
centimeter. The distance operator in the advection term in
Eq. (2.31) is equated to the distance along the flow path, 1.
Therefore, the rate of change in the amount of quartz
within a fracture is a function of both the thermodynamic
properties of the reaction and the temporal and spatial
derivatives of temperature and fluid pressure:
a S i02 AdvechOn O dT vrO
=Vf·~fiO20.026(    ~
OtRT2 dl RT dl )
Rate of charge in fluid
_~fi02 0~026 (~2  ) . (2.32)
RT dt RT dt
DENTS L. NORTON
This relation demonstrates the coupling among the
thermal, pressure, and fluid flow fields in which the depo
sition or dissolution of a mineral phase in the flow net
work can effectively change its continuity.
The manner in which the mineral actually blocks the
flow channel is a consequence of detailed interaction be
tween the local flow field within the fracture (Brown,
1987) and the local gradients in temperature and pressure.
These interactions are unfortunately not predictable from
the transport theory. However, there is ample evidence in
the textures of veins to indicate that sealing of the flow
channel actually occurs numerous times during the history
of the thermal event.
SUMMARY
Field relations indicate that the processes that lead to
the formation of fracturecontrolled percolation networks
and that fill these networks with vein minerals are related
through an episodic series of events. The fractures that
constitute the networks are slitlike features of limited
extent; they occur in interconnected sets, and in any one
environment there may be many such sets of different
orientation and distinct chronology (see Titley, Chapter 3,
this volume). The vein material within a fracture is per
vaded with discordant contacts that suggest many discrete
events of fracture and vein fill.
Transport equations demonstrate that the functional
relations among the field variables form a coupled set of
differential equations in which the principal transport
mechanism has a hyperbolic form. In all but the simplest
of systems in which hyperbolic functions depict the mecha
nism of transport, the system evolves in a chaotic manner,
particularly if other nonlinearities are present. In the situ
ation discussed in this chapter, highly nonlinear properties
of H2Orich fluids have long been recognized as exerting
primary control on the evolution of temperature, pressure,
and fluid velocity (Norton and Knight, 1977~.
Independent lines of evidenceone from transport
theory and the other from the geometric properties of
veins and fracture systems point to anomalous fluid
pressure as the force that not only generates systematic
fractures and causes them to interconnect and form perco
lation networks but that also retards flow through mineral
deposition.
REFERENCES
Berbabe, Y. (1987~. The effective pressure law for permeability
during pore pressure and confining pressure cycling of several
crystalline rocks, Journal of Geophysical Research 92, 649
657.
Brown, S. R. (1987~. Fluid flow through rock joints: The
effect of surface roughness, Journal of Geophysical Re
search 92, 13371347.
OCR for page 42
PORE FLUID PRESSURE NEAR MAGMA CHAMBERS
Burton, C. J., and H. C. Helgeson (1983~. Calculation of the
chemical and thermodynamic consequences of differences
between fluid and geostatic pressure in hydrothermal systems,
American Journal of Science 283A, 540548.
Helgeson, H. C., and D. H. Kirkham (1974~. Theoretical predic
tion of the thermodynamic behavior of aqueous electrolytes at
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nal of Science 274, 10891198.
Hubbert, M. K., and D. G. Willis (1957~. Mechanics of hydrau
lic fracturing, Transactions, Society of Petroleum Engineer
ing, AIME 210, 153166.
Johnson, J. W., and D. Norton (1989~. Critical phenomena in
magmahydrothermal systems: I. State, thermodynamic, trans
port, and electrostatic properties of H2O in the critical region,
American Journal of Science, in press.
Knapp, R. B., and J. E. Knight (1977~. Differential thermal
expansion of pore fluids: Fracture propagation and microearth
quake production in hot pluton environments, Journal of
Geophysical Research 82, 25152522.
Knapp, R. B., and D. Norton (1981~. Preliminary numerical
analysis of processes related to magma crystallization and
stress evolution in cooling pluton environments, American
Journal of Science 281, 3568.
49
Lantz, R. (1984~. The influence of the geometry of the pluton
host rock interface on the orientations of thermally induced
hydrofractures at the Cochise Stronghold pluton, Cochise
County, Arizona, M.S. thesis, University of Arizona, Tucson.
Mavko, G., and A. Nur (1978~. The effect of nonelliptical cracks
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Moskowitz, B. M., and D. Norton (1977~. A preliminary analy
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Norton, D. (1984~. A theory of hydrothermal systems, Annual
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Norton, D., and R. Knapp (1977~. Transport phenomena in
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Norton, D., and J. E. Knight (19771. Transport phenomena in
hydrothermal systems: Cooling plutons, American Journal of
Science 277, 937981.
Norton, D., and R. N. Villas (1977~. Irreversible mass transfer
between circulating hydrothermal fluids and the Mayflower
stock, Economic Geology 72, 1471  1504.
Snow, D. T. (1970~. The frequency and aperture of fractures in
rocks, Journal of Rock Mechanics 7, 2340.