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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 1C 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 CONVCTI ~ | 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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MASS AND ENERGY TRANSPORT lN A DEFORMING EARTH'S CRUST plus the pore pressure. Nur and Byerlee (1971) defined a relationship for effective stress: OCR for page 27
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

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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.

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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).

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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

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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

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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, 16C/km; temperatures are only approximately 110C 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

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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

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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. REFERENCES Aagaard, P., and H. C. Helgeson (1982~. Thermodynamic and kinetic constraints on reaction rates among minerals and aqueous solutions. I. Theoretical considerations, American Journal of Science 282, 237-285. Anderson, M. P. ( 1984~. Movement of contaminants in groundwater: Groundwater transport-advection and dispersion, in Groundwater Contamination, Studies in Geophysics, National Research Council, National Academy Press, Washington, D.C., pp. 37-45. Barnes, I., W. P. Irwin, and D. E. White (1978~. Global distribution of carbon dioxide discharges and major zones of seismicity, U.S. Geological Survey Water Resources Investigation 78-39, Open File Report, 12 pp. Bear, J. (1972~. Dynamics of Fluids in Porous Media, Elsevier, New York, 764 pp. Blot, M. A. (1955~. Theory of elasticity and consolidation for a porous anisotropic solid, Journal of Applied Physics 26, 182- 185. Blot, M. A. ( 1956~. Thermoelasticity and irreversible thermodynamics, Journal of Applied Physics 27, 240-253. Bird, R. B., W. E. Stewart, and E. N. Lightfoot (1960~. Transport Phenomena, John Wiley & Son, Inc., New York, 780 pp. Boussinesq (1903~. Theorie Analytique de la Chaleur, I. ii. Brace, W. F. (1971~. Resistivity of saturated crustal rocks to 40 km based on laboratory measurements, in The Structure and Physical Properties of the Earth's Crust, J. G. Heacock, ea., American Geophysical Union Monograph 14, Washington, D.C., pp. 243-255. Brace, W. F. (1980~. Permeability of crystalline and argillaceous rocks: Status and problems, International Journal of Rock Mechanics in Mineral Science and Geomechanical Abstracts 17, 876-893. Bredehoeft, J. D., and B. B. Hanshaw (1968~. On the maintenance of anomalous fluid pressure. I. Thick sedimentary sequences, Geological Society of America Bulletin 79, 1097-1106. Bredehoeft, J. D., C. E. Neuzil, and P. C. D. Milly (1983~. Regional flow in the Dakota aquifer, U.S. Geological Survey Water Supply Paper 2237, pp. 1-45. Burst, J. F. (1969~. Diagenesis of Gulf Coast clayey sediments and its possible relation to petroleum migration, American Association of Petroleum Geologists Bulletin 53, 73-79. Cathles, L. M. (1977~. An analysis of the cooling intrusives by ground-water convection which includes boiling, Economic Geology 72, 804-826. Cherry, J. A., R. W. Gillham, and J. F. Barker (1984~. Contaminants in groundwater: Chemical processes, in Groundwater Contamination, Studies in Geophysics, National Research Council, National Academy Press, Washington, D.C., pp. 46-64. Darcy, H. (1856~. Determination of the laws of the flow of water through sand (translated from the French, 1983), in Physical

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40 Hydrogeology. R. A. Freeze and W. Back, eds., Hutchinson Ross, Stroudsburg, Pa., pp. 14-19. Dickinson, G. (1953~. Geological aspects of abnormal reservoir pressures in Gulf Coast Louisiana, American Association of Petroleum Geologists Bulletin 37, 410-432. Domenico, P. A. (19771. Transport phenomena in chemical rate processes in sediments, Annual Reviews of Earth and Planetary Sciences 5, 287-317. Donaldson, I. G. (1982~. Heat and mass circulation in geothermal processes in sediments, Annual Reviews of Earth and Planetary Sciences 10, 155-164. Faust, C. R., and J. W. Mercer (1979a). Geothermal reservoir simulation: I. Mathematical models for liquid- and vapor- dominated hydrothermal systems, Water Resources Research 15, 23-30. Faust, C. R., and J. W. Mercer (1979b). Geothermal reservoir simulation: II. Numerical solution techniques for liquid- and vapor-dominated hydrothermal systems, Water Resources Research 15, 31-46. Freeze, R. A., and J. A. Cherry (1979~. Groundwater, Prentice- Hall, Inc., Englewood Cliffs, N.J., 604 pp. Freeze, R. A., and P. A. Witherspoon (1967~. Theoretical analysis of regional groundwater flow: Effect of water-table configuration and subsurface permeability variation, Water Resources Research 3, 623-634. Gold, T. (1979~. Terrestrial sources of carbon and earthquake outgassing, Journal of Petroleum Geology 1, 3-19. Gold, T., and S. Soter (1985~. Fluid ascent through the lithosphere and its relation to earthquakes, PAGEOPH, 492-530. Grisak, G. E., and J. F. Pickens (1981~. An analytic solution for solute transport through fractured media with matrix diffusion, Journal of Hydrology 52, 47-57. Hanshaw, B. B., and J. D. Bredehoeft (1968~. On the maintenance of anomalous fluid pressures: II. Source layer at depth, Geological Society of America Bulletin 77, 1107- 1122. Helgeson, H. C. (19791. Mass transfer among minerals and hydrothermal solutions, in Geochemistry of Hydrothermal Ore Deposits, 2nd ea., H. L. Barnes, ea., John Wiley & Sons, New York, pp. 568-610. Helgeson, H. C., and D. H. Kirkham (1974a). Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: I. Summary of the thermodynamic/electrostatic properties of the solvent, American Journal of Science 274, 1089-1198. Helgeson, H. C., and D. H. Kirkham (1974b). Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: II. Debye- Huckel parameters for activity coefficients and relative partial molar properties, American Journal of Science 274, 1198- 1261. Helgeson, H. C., T. H. Brown, A. Nigrini, and T. A. Jones (1970~. Calculation of mass transfer in geochemical processes involving aqueous solutions, Geochimica et Cosmochimica Acta 34, 569-592. Hsieh, P. A., and J. D. Bredehoeft (1981~. A reservoir analysis of the Denver earthquakes: A case of induced seismicity, Journal of Geophysical Research 86, 903-920. JOHN D. BREDEHOEFT AND DENIS L. NORTON Hubbert, M. K. (1940~. The theory of ground-water motion, Journal of Geology 48, 785-944. Hubbert, M. K., and W. W. Rubey (1959~. Role of fluid pressure in mechanics of overthrust faulting, Geological Society of America Bulletin 70, 115-166. Hubbert, M. K., and D. G. Willis (19574. Mechanics of hydraulic fracturing, Transactions of the AIME 210, 153-166. Irwin, W. P., and I. Barnes (1975~. Effect of geologic structure and metamorphic fluids on seismic behavior of the San Andreas Fault system in central and northern California, Geology 1, 713-716. Jacob, C. E. (19401. On the flow of water in an elastic artesian aquifer, EOS Transactions American Geophysical Union 21, 574-586. Knapp, R. B., and J. E. Knight (1977~. Differential thermal expansion of pore fluids: Fracture propagation and microearthquake production in hot pluton environments, Journal of Geophysical Research 82, 2515-2522. Lichtner, P. C. (1986~. Continuum model for the simultaneous chemical reactions and mass transport in hydrothermal systems, Geochimica et Cosmochimica Acta 49, 779-800. Norton, D. (1982~. Fluid and heat transport phenomena typical of copper-bearing pluton environments, southeastern Arizona, in Advances in Geology of Porphryr Copper Deposits, Southwestern North America, S. R. Titley, ea., University of Arizona Press, Tucson, pp. 59-72. Norton, D. (1984~. A theory of hydrothermal systems, Annual Reviews of Earth and Planetary Sciences 12, 155-177. Norton, D., and J. Knight (1977~. Transport phenomena in hydrothermal systems cooling plutons, American Journal of Science 277, 937-981. Norton, D., and H. P. Taylor, Jr. (1979~. Quantitative simulation of the hydrothermal systems of crystallizing magmas on the basis of transport theory and oxygen isotope data: An analysis of the Skaergaard Intrusion, Journal of Petrology 20, 421 -486. Nur, A., and J. D. Byerlee (19711. An exact effective stress law for elastic deformation of rocks with fluids, Journal of Geophysical Research 76, 6414-6419. Palciauskas, V. V., and P. A. Domenico (1980~. Microfracture development in compacting sediments: Relations to hydrocarbon-maturation kinetics, American Association of Petroleum Geologists Bulle-tin 64, 927-937. Palciauskas, V. V., and P. A. Domenico (1982~. Characterization of drained and undrained response of thermally loaded repository rocks, Water Resources Research 18, 281-290. Rayleigh, Lord (1916~. On convection currents in a horizontal layer of fluid, when the higher temperature is on the underside, Physical Magazine Series 6~32), 529-545. Rubey, W. W. (19511. Geologic history of the sea, Geological Society of America Bulletin 87, 1111-1148. Sharp, J. M., Jr., and P. A. Domenico (1976~. Energy transport in thick sequences of compacting sediment, Geological Society of America Bulletin 87, 390-400. Skempton, A. W. (1954~. The pore pressure coefficients A and B, Geotechnique 4, 143- 147. Slattery, J. C. (1972~. Momentum, Energy, and Mass Transfer in Continua, McGraw-Hill, New York, 679 pp.

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MASS AND ENERGY TRANSPORT IN A DEFORMING EARTH'S CRUST Taylor, H. P., Jr. (1974). The application of oxygen and hydrogen isotope studies to problems of hydrothermal alteration and ore deposition, Economic Geology 69, 843-883. Taylor, H. P., Jr. (1977~. Water/rock interactions and the origin of H2O in granitic batholiths, Journal of the Geological Society of London 133, 509-558. Terzaghi, K. (1943~. Theoretical Soil Mechanics, Wiley, New York, 510 pp. Theis, C. V. (1935~. The relation between the lowering of the 41 piezometric surface and the rate and duration of discharge of a well using groundwater storage, EOS Transactions American Geophysical Union 16, 519-524. Toth, J. (1963~. A theoretical analysis of ground-water flow in small drainage basins, Journal of Geophysical Research 68, 4795-4812. Van der Kamp, G., and J. E. Gale (19834. Theory of Earth tide and barometric effects in porous formations with compressible grains, Water Resources Research 19, 538-544.