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1
INTRODUCTION
Mass and Energy Transport in a
Defonning Earth's Crust
JOHN D. BREDEHOEFT
U.S. Geological Survey, Menlo Park
DENIS L. NORTON
University of Arizona
Groundwater is ubiquitous throughout the crust to depths
of at least 15 to 20 km, perhaps deeper in some places.
Because geologists have traditionally focused on minerals
and lithologic units in their study of Earth processes, the
importance and distribution of the fluid phase have been
overlooked. In fact, fluids are suspected to have a strong
influence over both local state conditions and the
redistribution of mass and energy, and the question arises
as to whether studies that have not taken into account the
presence of water have uncovered the correct mechanisms.
The field and theoretical evidence brought forth by the
contributions to this volume demonstrate that groundwater
plays a major role in crustal processes. The following are
but a few of the phenomena for which fluids clearly play
a central role: migration and entrapment of oil and gas,
formation of ore deposits, metasomatic alteration of large
volumes of rock, changes in the state of stress, failure of
rocks in shear and tension, formation and emplacement of
magmas, formation of geothermal systems, triggering of
earthquakes, cooling of magmas, diagenesis of sedimentary
rocks, movement of landslides, composition of the oceans,
transport of contaminants, and distribution of heat.
Consequently, understanding the geology of the crust
requires that the distribution and behavior of groundwater
be considered.
27
One example of how H2O-rich fluid strongly affects the
evolution of conditions in the crust is the central role it
plays in hydrothermal systems. One can show that a rise
in temperature of 1°C can cause a pressure increase of
several bars in the local fluid pressure. Consequently, the
introduction of a large body of magma into the crust will
inevitably set up a convecting groundwater system around
the magma body, which transports heat away from the
magma at supercritical conditions. Because supercritical
fluid is efficient as a heat-transporting medium, the rates
of groundwater flow will control the cooling rates of the
igneous body. The same convective process also
redistributes chemical components from the host to the
pluton environments, including the ore-forming con-
stituents. Therefore, groundwater flow controls the distri-
bution and grade of many ore deposits.
The magnitude of groundwater convection and
consequently the extent of mineral alteration around a
magma depend on the permeability of rocks. However,
even if the rocks were initially impermeable, the pore
pressure generated by heat dispersed into the host rocks is
high enough to create hydraulic fractures. Thus, the system
generates its own permeability by making fractures. This
fracturing process tends to be episodic because of the
intrinsic strength of the rock. Therefore, repeated sequences
of fracturing followed by mineral deposition in the fractures
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28
{~~;;~)
~ PERTURBATION
CONDUCTIVE
and RATE
CONV£CTI ~ |
COOLING RATE /
i:STATE ~ STRESS:
STRAIN
\ RATE
~ANTI ~
CRYSTALLIZATION
\ TRACTS ~
ABUNDANCE
APERTURE +_~
AD CTION CHEMICAL
RATE REACTION RAT
~CHEW ~Darcy s Law
AFFINITY
rE
FIGURE 1.1 Systematic relationships between
principal transport rates and their products.
Arrows depict directions of energy, mass flow,
and feedback effects of state conditions on rates
(from Norton, 1984~.
are typical of magma environments. As the fractures
gradually fill with minerals, permeability decreases and
fluid flow is mitigated. This cycle of processes forms an
intricately interconnected feedback system (Figure 1.1) in
which the properties of the fluid phase exert primary control
on the evolution of the system.
Because each of the contributions to this volume
discusses ways in which the properties of H2O-rich fluids
determine how processes evolve, this chapter summarizes
some of the general-process dynamics. The symbolic
formalism of mass and energy transport is used in an
attempt to sketch a coherent picture of processes and to
show the usefulness of the equations in revealing the role
of the fluid phase in mass and energy transport within a
deformable medium. The equations presented below are
not original, so the reader is encouraged to explore in
greater depth the transport theories such as those used in
engineering (e.g., Bird et al., 1960; Slattery, 1972) to
solve geologic problems.
CONSERVATION EQUATIONS
The concept of conservation provides a basis for writing
a set of equations that symbolically represent the transport
processes involving water in the Earth's crust. For each of
JOHN D. BREDEHOEFT AND DENIS L. NORTON
the mass, energy, and momentum quantities that participate
in the transport, an equation is formulated that expresses
the conservation of the quantity with respect to the local
system. These equations describe the rate of change in
quantities with respect to time and a representative volume
of the system.
During the 1 800s some carefully conducted experiments
produced a set of empirical laws that express the flux of
mass and energy and in terms of a driving force and a
proportionality constant that incorporates a medium's
properties. These flux laws form the basis for discussing
transport; they include Fick's Law of Diffusion, Fourier's
Law of Heat Conduction, DeDonder's Law of Affinity,
and Darcy's Law. Each has the general form of the flux of
a quantity, where the flux is proportional to the gradient in
a field parameter. The conservation equations for thermal,
mechanical, and chemical energy discussed below all derive
from one or more of these flux laws.
Fundamental to the transport processes in rocks is the
conservation of fluid momentum that derives from the
experiments by Henry Darcy (Darcy, 1856~. In his
experiments Darcy investigated the flow of water through
a sand filter and suggested a general relationship for the
rate of flow versus the drop in hydraulic head. In a classic
paper on the theory of flow in porous media, Hubbert
(1940) showed that Darcy's Law is equivalent to the Navier-
Stokes equation for fluid momentum and can be stated as
follows:
vi = - -- ~ pgh), ~ 1.1
pa Oxj
where vi is the mean velocity in the ith direction, kij is the
permeability, ,u is the fluid viscosity, 0 is the porosity
through which fluid flows, g is the acceleration of gravity,
and h is the hydraulic head. While this is the simplest
form of Darcy's Law, Hubbert went on to show that when
fluid density (p) varies in space, a more general form
stated in terms of fluid pressure (p) is necessary:
vi = _ Kij (ap + pg dz) (1.2)
SO Oxj Ox
Although the hydraulic conductivity, Kij, is commonly
used as the proportionality constant in Darcy's Law, it is a
function of properties of both the rock and the fluid. In
subsurface conditions the medium and fluid properties not
only vary independently but also range over extreme values;
they must be independently expressed in the equations.
Hydraulic conductivity is related to permeability as follows:
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MASS AND ENERGY TRANSPORT IN A DEFORMING EARTH'S CRUST
Kij = P kij.
Permeability is a property of the medium itself that is
nonhomogeneous and aniso~opic and is a symmetric tensor.
Darcy's Law is useful to geologic processes because it is
a macroscopic expression that involves averaging over
some "representative elementary volume" of the porous
medium (see Bear, 1972~.
Flow Equation
The conservation of mass for a single-phase fluid is
a ~Opvi' = ape
Hi at
~ 1 .4)
where p is the fluid density within the interconnected pore
space whose fraction of the system is ¢. The pore pressure
and rock stress are coupled; the coupling is discussed
more fully below under somewhat restrictive assumptions
of coupling. Theis (1935) and Jacob (1940) showed that
by substituting Darcy's Law for the velocity, vi in Eq.
(1.4), and combining the compressibility of both the rock
and the water, we obtain an equation to describe isothermal
flows in compressible media:
axi ( i1 )(axj )
(1.5)
at
where S is a coefficient that expresses the effective vertical
s
compressibility of the coupled rock and fluid, called the
"specific storage" by Jacob (1940~. The flow equation can
also be stated in terms of the fluid pressure:
a pkij rap A az)
. + Pg
Oxi ~ taxi Ox
sS ap
= ~. (1.6)
pg at
As shown below, the flow equation becomes more
complicated as we account for changes in tectonic stress
and nonisothermal conditions. The fluid velocity derived
from these relations forms a basis for the transport of other
. . .
quantities In the system.
D ispersion -D iffusion -Ad section
When we observe the movement of chemical components
through a porous medium, a combination of dispersion,
diffusion, and advection is apparent. This combination of
transport mechanisms can be used to explain variations in
velocities of chemical components. The phenomenon of
dispersion is depicted in Figure 1.2. It occurs both at the
microscopic level, because of the differing velocities of
~-o=,~r`~ - 0~ ~
FIGURE 1.2 Microscopic dispersion (after Freeze and Cherry,
1979~.
flow through the pores, and at the macroscopic level, where
heterogeneities in geologic materials greatly increase the
magnitude of the dispersive process.
Diffusion laterally away from the flow channels into
fracture-controlled matrix blocks was recognized early in
the study of hydrothermal ore deposits as the mechanism
by which alteration haloes and high assays form in the
rock matrix adjacent to fossil flow channels called veins.
Although it is a distinctly different mechanism from the
dispersion of components, its effects are difficult to separate
and are generally treated as a combined process.
The mass flux caused by dispersion and diffusion is
formulated as an expression analogous to Fick's Law of
Diffusion:
jij = Di, aC
Ox
(1.7)
where D*, is the coefficient of dispersion, which is the sum
of the hydrodynamic dispersion coefficient and the diffusion
coefficient
*
Dij = Dij + Dd . (1.8)
Anderson (1984, p. 39) presents a clear discussion of
. · .
Aspersion:
. . . D*~ is the coefficient of mechanical dispersion and Dd is the
coefficient of molecular diffusion. An effective diffusion
coefficient is generally taken to be equal to the diffusion coefficient
of the ion in water (Dd) times a tortuosity factor. The tortuosity
factor has a value less than 1 and is needed to correct for the
obstructing effect of the porous medium. Effective diffusion
coefficients are generally around 10-6 cm2/sec, although a range
of 10-5 to 10-7 cm2/sec is not inconceivable (Grisak and Pickens,
1981~. Except for systems in which groundwater velocities are
very low, the coefficient of mechanical dispersion generally will
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30
be one or more orders of magnitude larger than Dry. Therefore, in
many practical applications the effects of molecular diffusion
may be neglected (D`~ = 0~. The coefficient of mechanical
dispersion is routinely taken to be the product of the magnitude
of the velocity times a parameter known as dispersivity, which is
commonly and somewhat vaguely referred to as a characteristic
mixing length.
Advective transport is caused by fluid flow from one
chemical environment to another. In this situation the fluid
carries along chemical components that alter the local
chemical conditions and shift local equilibrium to far from
equilibrium conditions. The long-distance transport of
components and the imposition of one lithologic
composition onto another over broad regions occur as a
consequence of the carrying-along capacity of the flowing
water.
The general form of the advection-dispersion equation
IS
- (Dl;-) - - (Cvi) + ~ Rk = -, (1.9)
axi axj axi k= ~ at
where C is the concentration of a particular chemical solute
of interest, and Rk is the rate of production of the solute in
the kth reaction. In the case where more than one solute is
of interest, an advection-dispersion equation is written for
each.
Heat Transport
A general expression for heat transport by moving
groundwater can be derived from a statement about the
dispersive as well as the conductive flux of heat, much as
was done for the dispersion-diffusion of mass above. The
dispersive-conductive flux of heat follows directly from
Fourier's Law of Heat Conduction, written here in terms
of temperature:
qi = Lij-, (1.10)
Dxj
where Lij is the coefficient of heat dispersion. The
coefficient of heat dispersion is
L,j = Lij = KT , (1.1 1)
where L is the thermal dispersion coefficient and KT is the
thermal conductivity of the porous medium (fluid and
rock). The usual assumption is that the rock and fluid are
in thermal equilibrium, and consequently the temperature
in Eq. (1.10) is that of the fluid and rock.
The convective movement of heat associated with the
flowing fluid is described by the flux equation:
qconvection = Vi (pcpT)heat consent offluid ~(1.12)
JOHN D. BREDEHOEFT AND DENIS L. NORTON
where cp is the isobaric heat capacity and T is the
temperature of the fluid phase flowing at velocity vi .
Combining the convective, diffusive, and dispersive
fluxes and transforming them into rates of change lead to
the expression for single-phase energy transport in terms
of the temperature and heat capacity of the fluid (Domenico,
1977):
- (Lij-) + - (VipcpT) = pep-, (1.13)
OXi dxj Dxi at
where the fluid velocity is defined by Eq. (1.2) and the
velocity of the rock matrix is presumed to be zero.
Although this set of partial differential equations the
flow equations for pressure and the advection-dispersion
equations for chemical solutes and energy, along with
Darcy's Law form a complete conservation statement,
the effects of stress and strain must also be considered.
Stress and Strain
In the 1920s, in investigating consolidation, Terzaghi
treated soils as a water saturated porous mediums. In a
series of papers Blot (1941, 1955, 1956) introduced a set
of constitutive relationships for stress and strain in a fluid-
filled porous media. Several authors have extended Biot's
work. Following Nur and Byerlee (1971), Palciauskas and
Domenico (1982) extended the theory to nonisothermal
conditions. They suggested that the constitutive
relationship, which includes thermal effects, is
£ij
2 ps ( 3 j ) 9K ~ ~kk bij
--P6ii - aT(T- To),
3H
(1. 14)
where Eij is the strain, Us is the stress, Ok is the sum of the
normal stresses, ~ is the Kroenecker delta, Kb is the bulk
modulus of elasticity, H is a coefficient introduced by
Blot, p is the pore pressure, aT is a coefficient of linear
thermal expansion, and T is the temperature. Pore pressure
changes are usually thought to be the result of volumetric
strain, 8:
= £~ + £22 + £33 = - (~ + o22 + 033)
1
H
~r (P - Sat) (T- To) . (1.15)
For many problems in rock mechanics it is convenient
to think of the total stress as the sum of an effective stress
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Representative terms from entire chapter:
groundwater flow
MASS AND ENERGY TRANSPORT lN A DEFORMING EARTH'S CRUST
plus the pore pressure. Nur and Byerlee (1971) defined a
relationship for effective stress:
32
periods typical for geologic problems, both situations rely
on a similar formulation. Equilibrium is simply a special
case in the more general formulation. The irreversible
formulation requires that we understand the kinetics of the
reactions of interest (Aagaard and Helgeson, 1982~.
The geochemical basis for describing reactions among
minerals and fluids in natural systems derives from the COUPLING
work of Helgeson et al. (1970, 1979~. Once the rigorous
description of mass transfer in systems that are only locally
in equilibrium was placed on a thermodynamic basis, the
rate of the irreversible reactions could be incorporated into
the theory (Aagaard and Helgeson, 1982~. More recently
Helgeson's formulation has been extended to provide for
the diffusion and advection of chemical components
(Lichtner, 1986~.
In the geochemical theory conservation of each of the
basis components,
~i, i = 1, 2, 3 . ..I, (1.25)
where Hi is the concentration of the component in the
respective phase, is described in terms of a transport
equation of the form:
Fluid
a(¢f~f)
+
at
Products Reactants
a`¢p~p' ~ 0(0r~r)
at + ~ at
Advection Diffusion
+ Vf · V~'f - ';¢jpj~f = 0 ~ (1.26)
where the mineral products, p = 1, 2, 3, . . . P. are those
minerals in local equilibrium with the fluid and the mineral
reactants, r = 1, 2, 3, . . . R. are those phases that react
irreversibly with the fluid. Eq. (1.26) is derived from the
general mass transport equation, Eq. (1.9), by assuming
the flow is steady and incompressible, that is,
V Vf = 0 (1.27)
and that the dispersion is negligible. Although this
assumption is questionable, the practical issue of how to
determine the magnitude of the dispersion for either
metamorphism or ore deposits has not been solved. The
equilibrium minerals and the fluid require a set of
thermodynamic relations to describe their activity-
composition relationships, and the minerals out of
equilibrium require a kinetic-rate law consistent with the
thermodynamic standard state to describe their rate of
change. Also, an equation of the form of Eq. (1.26) must
be written for each of the basis components.
In typical problems related to ore deposition, 15 to 20
equations in the form of Eq. (1.26) are necessary. These
equations have nonlinear and stiff features that make them
JOHN D. BREDEHOEFT AND DENIS L. NORTON
difficult to integrate numerically realistic problems
require large computer capacity, often super computers.
For contaminant transport problems as many as five
simultaneous concentration equations have been used to
analyze specific problems.
Chemical reactions within the system that appear as a
source term in Eq. (1.9) are strongly dependent on the
salvation properties of the aqueous phase. Because this
phase controls the dissolution and deposition of minerals,
and hence the porosity distribution, the diffusion and
advection processes are controlled by these properties of
the solvent. The argument can be made that changes in
pressure associated with the flow of fluid through tortuous
pore space can cause substantial changes in the mass of
material deposited in the channels.
If rock strain is considered to be dependent on changes
in total stress as well as temperature [Eq. (1.20~], then
pore pressure is even more strongly coupled to heat transport
[Eq. (1.13~. In some contexts it is possible to ignore the
coupling and thus simplify the appropriate mathematics.
The equations for pressure [Eq. (1.5) and (1.20~],
concentration tEq. (1.9~], temperature [Eq. (1.13~l, and
Darcy's Law [Eqs. (1.1) and (1.2~] are coupled and
nonlinear. The coupling occurs because of the dependence
of both fluid density and viscosity on pressure, temperature,
and concentration. The proportionality constant in Darcy's
Law includes properties of the fluid, both density and
viscosity. Density is influenced by pressure, temperature,
and chemical concentration; that is,
p = flip, T. Xi) . (1.28)
Viscosity is also influenced by all three independent
variables: pressure, temperature, and concentration.
However, for most problems-only the temperature effects
need to be considered:
~ = fop. (1.29)
DISCUSSION: GEOLOGIC PHENOMENA
It is obvious from the discussion above that fluids within
the Earth's crust are intimately involved in many processes
of interest to geologists. Rather than attempt to deal with
all of these processes, we have chosen to focus on one
particular problem, the pore pressure in the active tectonic
areas of the Earth.
Hubbert and Rubey (1959) have pointed out the
mechanical problems associated with large overthrust sheets
such as those observed in the Alps. They argue that one
MASS AND ENERGY TRANSPORT IN A DEFORMING EARTH'S CRUST
possible mechanical solution to large-scale overthrusting
would be for associated pore fluids to have pressures
approaching the lithostatic weight of the overlying rock.
Under such conditions the frictional resistance to sliding
becomes negligible; overthrusting can occur as a
gravitational process (sliding). While the original thought
was to apply this idea to thrust sheets, numerous earth
scientists have extended the idea to the general problem of
frictional failure.
Indeed, the more we observe dynamic processes
operating in the active tectonic areas of the Earth, the more
we see additional evidence suggesting that the pore pressure
in many, if not most, of these regions is high, well above
hydrostatic. Unfortunately, most of the evidence is indirect,
often only inferences from other geophysical observations.
However, the current weight of the circumstantial evidence
is such that a good case can be made for a rather general
condition of high pore pressure.
In stating the case of high pore pressure in the Earth's
active tectonic belts, we could address a number of issues
and, indeed, ramifications of those issues. Yet the purpose
of this volume is not to provide an exhaustive treatise but
rather to indicate the importance of considering the pore
fluids. We believe the chapters that follow illustrate our
case.
HOW PERMEABLE IS THE CRUST?
In considering geologic materials in the upper part of
the crust, everything must be considered permeable.
Geologists have known this since at least the early 1900s,
and more recent quantitative analyses of regional
groundwater systems demonstrate that flow through
lithologies previously considered to be impermeable is
often significant, even through confining beds such as the
Cretaceous Pierre Shale that include layers of bedded
bentonite (Bredehoeft et al., 1983~. Permeability and
resistivity are the two physical parameters associated with
earth materials with the widest range of possible values;
permeability is observed in nature to vary over at least 15
orders of magnitude (see Figure 1.3~.
The depth to which fluids can circulate within the crust
has been a subject of continued investigation by earth
scientists. There are several lines of study. Perhaps the
most direct information comes from earthquakes generated
by fluid injection at the Rocky Mountain Arsenal near
Denver, Colorado. Earthquakes that occurred at the arsenal
to a depth of between 7 and 8 km were rather clearly
triggered by the injection (Hsieh and Bredehoeft, 1981~.
Although other studies of fluid-induced earthquakes show
similar depths, in most instances much less is known about
the nature of the permeability at depth than what is known
at the Rocky Mountain Arsenal.
33
Rocks
~1
Unconsolidated
. deposits
11
0 0
_ ~
._ ~ I
~ ~ 0 s
by E
A :, _
1.~ ~
0 ~
..
c- ~ ~
,,8 O O
0 E ~ O ~
E==
._ c
11
~ O ~
~S=~!uo~i , I
~ . _
1
v,
0
_^
._
cry
1
1 - 1
.O
~ ^~
a, 0 0
~ OO
_
at.'
BE
lo
v
11
O
v,
c
0
1 -'
~1
-
k k K ~ h'
(darey) (an2) (cm/s) (mJs] (gol~doy/tt2)
1 of ,~ , v
-~04 -~0- ~ _'o2 _]o
_lo2 _lo-6 -10-1 -10-3 104
- 10 - 10-7 - lo-2 - too - 1o2
~ - 10-8 - 10-3 - 10-5
- 10-' - 10- - 10- - ,o-6 - 10
_ l o-2 - l o-to - 1 0-5 - l~7 1
- 1 0~3 - 10~~' - 10 - - 10~8 - 10~'
-10-4 _lo-12 -10-7 -10-9
jo-S -lo-13 -)0-8 -10-10 _lo
-,o-6 - 10-14 - 10-9 - 10-~' ~o-S
10~7 lo-'s - lo-'o _10-l2 ,o-6
- 10-. .~-~6 ,^-tl 10 1 _
.~-, .~2
- 10
_ 1
- 1o 7
FIGURE 1.3 Ranges of permeability and hydraulic conductivity
for a variety of rocks.
Deep resistivity measurements of the Earth's crust also
indicate reasonable porosity to rather substantial depths.
Brace (1980) summarized the information and suggested
that some permeability must exist within the crust to depths
of between 13 and 20 km.
There is additional evidence for deep circulation of
fluids within the crust from metamorphic and igneous
petrology. Taylor (Chapter 5, this volume) presents RIO/
~60 isotope data from a number of major intrusive bodies
indicating circulation of groundwater to 10 to 15 km.
Walther (Chapter 4, this volume) estimates that fluid flow
has occurred as deep as 20 km, and Wickham and Taylor
(Chapter 6, this volume) come to a similar conclusion after
studying regional metamorphism in the Pyrenees. Clearly,
the pressure exerted by the pore fluids plays a fundamental
role in the motion of fluid through the crust.
Field evidence and theoretical relations indicate that
permeabilities in excess of 1~4 cm2 are common in magma
environments (Taylor, 1974, 1977; Norton and Taylor,
1979~. The measured mass transfer of chemical components
in ore deposits, layered intrusions, and batholiths requires
that advection was a dominant process. Norton and Knight
(1977) demonstrated that for the driving forces encountered
in the vicinity of magmas the threshold permeability at
which convective exceeds conductive heat transport occurs
at these permeability values.
34
MECHANISMS THAT CREATE HIGH PORE
PRESSURE
The theory for single-phase, nonisothermal, reactive
transport in a porous medium contains as the central
mechanism the flow of fluid in response to a force. The
force fields that drive this flow are caused by the following
mechanisms:
1. topographic relief,
2. tectonic dilation and compression,
3. diagenesis,
4. geothermal systems, and
5. fluid source.
Although there are other potential driving forces, such as
chemical concentrations across membranes (osmosis) and
electrical potential, they are small relative to those listed
above.
The generation of high pore pressure is a rate-controlled
phenomenon; some mechanism, or set of mechanisms,
operates to generate higher-than-hydrostatic pore pressure.
This pore pressure is dissipated by fluid flow outward
from the source. The amount that the pore pressure
increases depends on the ease with which flow can occur,
which in turn depends on the hydraulic conductivity
(permeability) of the host rock. As suggested above,
observed permeabilities range over 15 orders of magnitude
within the crust.
All of the major mechanisms indicated above that drive
fluid flow are also suggested in this volume to create high
pore pressure in active tectonic regimes of the crust. We
discuss examples of each.
Topographic Relief
On the tectonically stable portions of the continents
most if not all groundwater flow is the result of topographic
relief. The driving force or head, h, is composed of the
pressure and elevation terms in Darcy's Law [Eq. (1.2~.
The relation of these quantities can be stated as
JOHN D. BREDEHOEFT AND DENIS L. NORTON
does not fluctuate a great deal. Under most climatic regimes
the water table can be demonstrated as approximately
stationary over time.
Forces associated with the groundwater table surface
are important enough in understanding groundwater flow
that they should be stated another way. If we install a
piezometer into the groundwater flow system within the
Earth, we generally find the water level, the hydraulic
head, to be within 50 m of the land surface, certainly in
most instances within 100 m. Hubbert and Rubey (1959)
referred to this as the "normal" condition. This condition
implies that the land surface is approximately the upper
boundary condition for the groundwater flow system. The
lower boundary for the system is no flow at some depth.
The flow system is driven by topographic relief on the
water table, the upper boundary for the saturated
groundwater system.
Lateral boundaries are formed by drainage divides that
separate the flow system into differing cells. Toth (1963)
showed that the local topography provides perturbations
on the regional flow system (Figure 1.4~. Because of the
obvious scale effects, with small slow systems
superimposed on larger systems we have to determine the
appropriate scale for the problem of interest.
Low-permeability layers retard flow and sometimes
cause large decreases in hydraulic head. Some changes in
permeability are compensated for by the large areas through
which flow, especially vertical flow, can occur in a regional
flow system (Figure 1.5~.
The patterns of groundwater flow produced by normal
conditions are determined by the topography, and changes
in permeability only distort the pattern in flow. A simple
flow net analysis will convince us that the pattern of flow
is established by the boundary conditions. In the normal
situation topography drives groundwater flow.
Groundwater flows downhill, just like surface water. Under
0.2s
on S
p
h = ~ + z, (1.30)
Pg 0
where h is the height above some arbitrary datum to which
fluid would rise in a manometer. The gradient in the head
generates the field of force that moves groundwater.
In this situation the water table replicates the land
surface. Toth (1963) pointed out that the water table is the
upper boundary for saturated groundwater flow and that
this boundary is usually closely approximated by the land
surface. While the water table may fluctuate seasonally
and from year to year because of wet and dry seasons, it
,
, ,, , ,·, , , , , , · ~· ,, ~, ·,
-0 0.1 S 0.25 0.3S 0.4S 0.5S 0.6S 0.7S 0.8S O.9S ' i
(a)
0.2S
O.1S
_~ ~
O.1S 0.2 S 0.3 S 0.4 S 0.5S 0.6S 0.7 S 0.8 S O.9S
( b )
FIGURE 1.4 Cross section illustrating the role of topographic
relief in groundwater flow (after Toth, 1963).
MASS AND ENERGY TRANSPORT IN A DEFORMING EARTH'S CRUST
BLACK ~
HILLS \4 22.7 18.8
\: ~ _ 14.9
\ ~ ~ :~ \~\~3r~ / ;,b.d ~ ::.:.:::.:::. DAKOTA .::::.;.:
\
iC /
:~ - Tom
FIGURE 1.5 Idealized cross section of the Dakota flow system
in South Dakota in which the quantities of flow through venous
normal conditions the maximum topographic elevation
places an upper bound on the hydraulic head, while the
lowest point imposes a lower bound. The total difference
in elevation forms a constraint on the hydraulic gradient.
Under conditions in which elevation differences are the
driving force for the groundwater flow system, the total
drive on the system is limited by the available topographic
relief.
Flow in most large mature sedimentary basins within
the stable continental craton is generally thought to be
driven by topographic relief. Early in the development of
sedimentary basins both compaction and diagenesis may
also be important driving mechanisms. In most sedimentary
basins significant quantities of flow move vertically across
confining layers (cap rocks). Low-permeability layers
retard flow and sometimes cause large decreases in
hydraulic head. Some changes in permeability are
compensated for by the large areas through which flow,
especially vertical flow, can occur in a regional flow system.
In South Dakota, for example, approximately 50 percent
of the recharge and discharge to and from the Dakota
Sandstone aquifer flows through the overlying Pierre Shale,
a shale that is often thought to be "impermeable." This
phenomenon is typical of flow in many if not most
sedimentary basins.
If one were to raise the pore pressure to levels at which
hydraulic fractures of the rock are possible by simple
topographic relief, there must be very high local or regional
relief. Engelder (Chapter 9, this volume) examined fractures
he believes are generated hydraulically in the Devonian
Catskill sands of the northern Appalachians. Engelder
suggests that the most plausible mechanism to create the
35
SIOUX
RIDGE
/
layers are indicated. The numbers are integrated over the entire
state of South Dakota (from Bredehoeft et al., 1983).
necessary pore pressure is high mountainous relief in the
Appalachians to the east during later Paleozoic time. This
mechanism requires approximately 5 km of relief to
generate the appropriate pore pressure but is often dismissed
because the necessary relief is so high.
Tectonic Dilation and Compression
Changes in tectonic stress change the pore pressure.
The effects are included in the more general form of the
flow equation, Eq. (1.20), by the second and third terms on
the right-hand side, which take into account changes in
normal as well as shear stresses. The degree to which the
pore pressure is increased by changes in stress depends on
the rate at which flow can dissipate the pressure.
Tectonic dilation, as we have defined it, takes a number
of forms. Perhaps the most universal is pressure solution,
which gradually over time reduces the porosity of the rock
and compresses the fluids within the pore space. Nur and
Walder (Chapter 7, this volume) argue that this is a universal
phenomenon that can lead to high pore pressure when the
process operates. Pressure solution is most commonly
observed in quartz sandstones, but it is not restricted in
rock type.
Various tectonic strains lead to a volume decrease of
the host rock. Usually the volume strain rate is small.
Currently along the San Andreas Fault in central California
the rate of shear strain is on the order of 10-6 per year. This
rate is accompanied by a volume strain of approximately
10-8 per year, two orders of magnitude smaller but capable
of producing high pore pressure as long as the permeability
of the rocks is small enough. Unfortunately, in active
36
tectonic areas such as along the San Andreas Fault, we
have no direct observations of pore pressure below 2 to 3
km. While it is clear from theoretical considerations that
active rock deformation causes pore pressure changes, the
data to demonstrate these effects are limited.
Oliver (Chapter 8, this volume), in perhaps the most
speculative paper of this volume, suggests that volume
strain associated with tectonic deformation between plates
has caused fluid hydrocarbons to flow laterally into
reservoirs within more stable areas of the continents. Vrolijk
and Myers (Chapter 10, this volume) investigated the
Kodiak accretionary complex and suggest that the high
pore pressure there is the result of active subduction.
Once tectonic strain ceases, any pore pressure change
caused by the strain becomes a transient phenomenon that
will be dissipated by flow over time. However, if the
rocks are of low permeability, such transient effects will
not dissipate rapidly; indeed, in some instances of very
low permeability material, these transient effects may persist
for periods of geologic interest (Figure 1.6) (see Bredehoeft
and Hanshaw, 1968; Hanshaw and Bredehoeft, 1968~.
Diagenesis
To reconcile the competing ideas of high crustal
permeabilities and associated hydrostatic fluid pressures
with evidence for hydraulically sealed low-permeability
rocks with elevated fluid pressures, Nur and Walder
(Chapter 7, this volume) propose a time-dependent process
FIGURE 1.6 Calculated pore pressure
versus depth profiles for a subsiding basin
such as the Gulf Coast. The model used
for these calculations considers only purely
mechanical compaction (from Bredehoeft
and Hanshaw, 1968~.
\\ \ Nt
E _
o
-- 5
UJ
1C
.0~
\
<\~f
\\ \ \\
~` Won Woo
^o~ \23 \3
~ VO :O
\ \
1 1 1 ~ 1\ 1
10
PRESSURE HEAD (103 m OF WATER)
JOHN D. BREDEHOEFT AND DENIS L. NORTON
to relate fluid pressure, flow pathways, and fluid volumes.
In their model crustal porosity, permeability, and hence
fluid pressures are in general time dependent due to the
gradual closure of crustal pore space via heating, sealing,
and inelastic deformation. Under certain circumstances
this process will lead to local drying out of the crust.
Under other circumstances in which the pore fluid cannot
escape fast enough, pore pressure will build up, leading
perhaps to natural hydraulic fracturing, fluid release, pore
pressure drop, and resealing of the system. If there is an
adequate supply of fluid, this process could repeat leading
to the intriguing possibility of cyclic episodes of pore
pressure buildup and natural hydraulic fracturing. This
model involving coupling of hydrological and mechanical
processes needs to be rigorously evaluated with in situ
observations and measurements.
A number of young geologic basins that are actively
receiving sediments have anomalously high pore pressures.
Several such basins-the Gulf Coast Basin in the United
States (Dickinson, 1953) and the Caspian Basin in the
Soviet Union, to name two-have pore pressures that
approach the total weight of the overburden (Figure 1.7~.
The simplest mechanism to rate such high pore pressure
is one in which low-permeability sediments are deposited
in the basin at rates sufficiently high that consolidation
cannot keep pace. The effective stress law, Eq. (1.17),
implies that the total stress can be decomposed into two
components the effective stress (the so-called grain-to-
grain stress) and the pore pressure. If the rate of loading
Rate of sedimentation = 500 m/106yr
Period of sedimentation = 20 x 106yr
Ss = 3 x 10~5/cm
I = 10,000 m
20 25
MASS AND ENERGY TRANSPORT IN A DEFORMING EARTH'S CRUST
go. L
patio CONDUCTIVE: FLOW Tie=
. ~ ,
. · 1.'
U'-l'lr ~11--1~(~,
CONDUCTIVE: NO fLOW
FIGURE 1.7 Convective cells established in the groundwater
around a hot igneous intrusion (from Norton, 1982~.
caused by sedimentation in the basin exceeds the rate at
which the pore pressure can dissipate through the expulsion
of fluid, then pore pressures that approach the lithostatic
load will result tthis process is described mathematically
by the flow equation, Eq. (1.20~. Under these conditions
there is little or no effective stress, and little or no
consolidation takes place.
The simple model of rapid sedimentation can account
for abnormally high pore pressures such as exist in the
Gulf Coast Basin (Bredehoeft and Hanshaw, 1968~. There
are, however, other complicating processes at work here.
Montmorillonite changes to illite with a release of free
water from the clay structure at approximately the same
depth as the first occurrence of the anomalous pore pressure
(Burst, 1969~. In addition, there may be a thermal
generation of high pore pressure caused by somewhat higher
thermal gradients (Sharp and Domenico, 19761.
In contrast to the Gulf Coast Basin, the Caspian Basin
is subsiding at a rate of 1100 m per million years. The
37
maximum thickness of accumulated sediments is 25 km.
The basin has a very low thermal gradient, 16°C/km;
temperatures are only approximately 110°C at 6000 m.
There is no change of montmorillonite to illite to depths of
6 km, but there are substantial overpressures. The south
Caspian Basin is at least one instance where simple
mechanical loading at a high rate can account for high
pore pressures.
The extent to which the simple sedimentation-
consolidation model is the appropriate mechanism for
generating high pore pressure depends on the weight of
the overburden. High pore pressure in a subsiding basin is
a dynamic process that requires active loading to be
maintained. Once subsidence and sedimentation cease,
the high pore pressure becomes a transient within the
system that will dissipate with time. The rate at which the
anomalous pore pressure dissipates depends on the
hydraulic diffusivity of the deposits and the boundary
conditions (Hanshaw and Bredehoeft, 1968~.
Geothermal Systems
Fluid density is a function of temperature, pressure, and
concentration of solute. Perturbations in any of these fields
induce a buoyancy effect on the fluid and consequently
cause the fluid to circulate, but the most common cause is
temperature changes. Increases in temperature also reduce
the fluid viscosity, making it easier for the fluid to flow
[see Darcy's Law, Eqs. (1.1) and (1.2~.
Fluid density changes are incorporated into the flow
equations through the Boussinesq approximation
(Boussinesq, 1903; Rayleigh, 1916) in which the density
is considered to be a function of the ambient, pa, and a
small density perturbation, bp:
P = Po + UP, (1.31)
where the perturbation density is in turn a function of
temperature, pressure, and solute content. This linear
approximation to the buoyancy force permits the highly
nonlinear variations in the density of supercritical fluid to
be incorporated into the driving force field that generates
convective flows.
As Eq. (1.24) indicates, the pore pressure is also strongly
affected by changes in temperature and pore volume caused
by the deposition or dissolution of mineral phases. The
magnitude of the change in pore pressure induced by such
changes is a function of the relative rate of change with
respect to the rate at which groundwater flow dissipates
the pressure increases. The rate of flow, in turn, depends
on the permeability of the material. If the permeability is
sufficiently high, flow occurs away from the density
38
perturbation and mitigates the pore pressure increase. On
the other hand, if the permeability of the material is very
low, pore pressure can increase until it exceeds the effective
strength of the pore wall.
The introduction into the crust of a source of heat such
as an igneous intrusive body creates a heat engine for the
groundwater system, causing flow. Several investigators
have looked at fluid circulation associated with igneous
plutons; more recent attempts to analyze the flow system
have resulted in numerical models that solve the appropriate
coupled partial differential equations (Cathles, 1977; Norton
and Knight, 1977; Faust and Mercer, 1979a,b; Norton,
19841. Norton (Chapter 2, this volume) summarizes results
associated with fluid variation in the near-field region
around magma bodies. Convection cells are set up in the
groundwater flow around the intrusive body. Of interest is
how the convecting groundwater might redistribute metals
and ore-forming fluids emanating from the igneous body.
In other instances the convecting groundwater leaches
metals from the surrounding country rocks and crystallized
portions of the intrusion and redistributes them.
Large pore pressure pulses can be generated by the
introduction of hot, or molten, intrusions. Knapp and
Knight (1977) showed that it is possible to get pore pressure
increases as large as 20 bars/°C even in regions of rather
low temperature and pressure simply as a consequence of
the properties of water. Palciauskas and Domenico (1982)
reexamined the problem taking into account the
compressibility of the rocks; they suggest that these pulses
in pressure can range from 5 to 10 bars/°C in low-
permeability rocks. In either case the change in pore
pressure associated with heating is potentially enormous.
Pore pressure increases are limited by the stress at which
failure is the hydraulic fracture in which the pore pressure
increases to the point where the rock fails. Failure can
occur either as shear or tension. If the failure occurs in
tension, it is an "hydraulic fracture" (Hubbert and Willis,
1957~. At the point at which the rock fails, a fracture is
created. The energy release associated with these fractures
accounts for the microseismic noise noted in active
geothermal areas (Knapp and Knight, 1977; Palciauskas
and Domenico, 1980~. Thermal shock may also account
for the seismic noise.
Fractures increase the permeability of the rock mass
and allow high pore pressure to dissipate rapidly. The
stress conditions at which failure will occur limit the
maximum pore pressure that can be sustained. Many of
the fractures are associated with vein-forming ore deposits.
Titley (Chapter 3, this volume) discusses the sequences of
fracturing associated with ore deposits at Sierrita, Arizona.
Near the critical point, water has a very steep pressure-
density relationship. Whether fractures fill with vein
JOHN D. BREDEHOEFT AND DENIS L. NORTON
forming crystalline deposits depends on this thermodynamic
behavior in the region near the critical point.
Moving groundwater is an efficient mechanism to
transport heat introduced by an igneous body. Convective
heat transport is much more efficient in transporting heat
than pure conduction [see Eq. (1.13~. The rate of cooling
of the igneous body strongly depends on the rate of heat
dissipation by circulating groundwater (Cathles, 1977;
Norton and Knight, 1977~.
Fossil geothermal systems often constitute hydrothermal
ore deposits. In these systems ore has been emplaced by
the moving fluids (Norton, 1984~. It seems possible that
many vein-type ore deposits are formed in hydraulic
fractures held open by the pore pressure during mineral
deposition. Most sills and dikes seem also to be simple
hydraulic fractures into which molten igneous rock was
injected. Hubbert and Willis (1957) discuss the basic
physics of hydraulic fracturing; their explanation is still
the seminal work on the problem.
Geothermal power development exploits a concentrated
heat source in the crust. Whether commercial development
of such a heat source is feasible depends almost entirely
on whether the permeability of the hot rock is high enough
to allow enough groundwater flow to recover the heat
efficiently (Donaldson, 1982~. Heat transport in a
geothermal system depends on flowing groundwater (or
steam). Geothermal reservoir engineering today involves
numerical simulation of the reservoir's performance, which
requires that the coupled partial differential equations be
solved for the boundary conditions of interest.
Source of Fluids
What are the sources of fluids in the crust? Although
fluids within a few thousand meters of the Earth's surface
are likely to be derived from precipitation, the source of
fluids at deeper crustal levels is problematic. Fluids released
from crystallizing magmas-must contribute some fluid to
the crustal fluid reservoir. Rocks that are heated as they
are buried to deeper levels, or that are in the vicinity of
intruding magma bodies, will also contribute fluids as they
recrystallize to new mineral assemblages that are more
compatible with their new thermal and pressure
environment. Finally, fluids may be directly released from
the mantle to the overlying crust during "degassing" events.
The relative proportions of these sources in any particular
region have not been well established, although some
isotopic studies have been undertaken to evaluate the
relative roles of these processes.
Hydration or dehydration of minerals changes the fluid
mass in the pores and will change the pore pressure. Such
an addition or removal of fluid is represented as an
MASS AND ENERGY TRANSPORT IN A DEFORMING EARTH'S CRUST
.
~.
appropriate source or sink terms In the general set or
equations given above. Depending on the volume change
associated with the dehydration, a source of fluids can
increase the pore pressure. Perhaps the most commonly
discussed example is the montmorillonite-illite
transformation that Burst (1969) first suggested as the
explanation for high pore pressure in the Gulf Coast.
Various higher-temperature metamorphic reactions release
free water. Walther (Chapter 4, this volume) discusses
fluid dynamics during progressive metamorphism.
The magnitude of the fluid-pressure change that
accompanies such a reaction is rate dependent. If fluid is
generated at such a rate that it cannot flow readily away
from the source, then fluid pressure will build. If the
permeability of the surrounding host rock is sufficiently
small, the buildup in fluid pressure may be quite large.
One other potential source of fluid flow into the crust is
mantle degassing. Rubey (1951) presented the case that
the volatiles associated with the Earth's surface were not
present early in the life of Me planet and have accumulated
with time. Perhaps the most commonly accepted hypothesis
is that the volatiles have been degassed from the mantle.
There are venous arguments about this topic. Gold (1979)
and Gold and Soter (1985) argue that the basic composition
is methane; numerous other workers believe that the
principal carbon-containing volatile is carbon dioxide.
Barnes et al. (1978) systematically mapped the distribution
of carbon dioxide springs around the Earth and showed
that they are almost exclusively associated with active
tectonic areas. Irwin and Barnes (1975) suggest that carbon
dioxide could give rise to high pore pressures. Bredehoeft
and Ingebntsen (Chapter 11, this volume) examine the
question of whether the current suggested rates of carbon
dioxide outgassing could give rise to high pore pressure
and conclude that it is possible; however, the permeabilities
of the host rocks would have to be quite low.
SUMMARY
The central role of H2O-nch fluids in determining the
dynamic conditions in the Earth's crust is apparent in the
repeated occurrence of fluid properties in all of the transport
equations. This symbolic depiction of the processes shows
not only the influence of processes on one another but also
that this coupling condition is a consequence of the presence
of an often sparse but essential occurrence of water in the
systems. Each chapter that follows demonstrates that water
is an active agent of the mechanical, chemical, and thermal
processes that control the tectonic regimes of the crust.
Recognition of the role of water as the material that
controls the extent of coupling among processes shows
39
promise of reducing Me magnitude of the analytical problem
to one that focuses on the controlling links in the system.
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