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Pore Fluid Pressure Near Magma Chambers DENIS L. NORTON University of Arizona INTRODUCTION Fluid pressure in the Earth's crust is a function of variations in the regional stress, temperature, and compo- sition of H2O-rich fluids. Whereas in relatively quiescent geologic environments the effects of fluid pressure are subtle, its effects in magma environments can be easily identified (Knapp and Norton, 1981; Burton and Helgeson, 1983; Lantz, 1984; Norton, 19841. In the near-field re- gions of magmas the occurrence of high fluid pressures is evident from the functional form of the equation of state for the fluids; the distribution and geometry of fluid-filled pores; and relationships among the thermal, chemical, and mechanical processes. This chapter discusses processes in magma-hydrothermal systems to demonstrate the mecha- nisms involved and to show that in these environments fluid pressure plays an essential role in the generation and maintenance of percolation networks for hydrothermal fluid flow. Extreme variations in fluid pressure in the near-field region of magmas are caused by sparse but significant amounts of H2O-rich fluids that are ubiquitous in the host rocks and common in the magmas. Pore fluids typically found in the host rocks have large positive values of the isochoric coefficient of thermal pressure, whereas those in the magma have large negative values. Therefore, fluid pressure increases during the dissipation of thermal en 42 orgy from magmas are a natural consequence of the cool- ing process, where temperature increases in the host and concurrently decreases in the magma. The resultant pres- sure increase in both domains generates large local stresses. Once the rock fails, fracture networks form that are con- tinuous between the domains on either side of the pluton wall. These networks allow fluid flow and convective transport of thermal energy, thereby increasing the rate of pressure change in both environments. Because the evidence for these conditions derives from both field observation and theory, the following discus- sion first reviews the conditions of ambient pressure in crustal environments and then examines the consequences of perturbing these conditions in the context of the distri- bution and geometry of pores. PRESSURE AT DEPTH Pressure conditions that exist prior to the imposition of a thermal perturbation on the crust are critical to predict- ing the magnitude and consequences of the subsequent fluid pressure increases. Pressure at the base of a column of rock that is composed of minerals and fluid-filled pore . . spaces IS glVeI1 By Total = Po + g | pr (T. P. X) dz, (2.1)
PORE FLUID PRESSURE NEAR MAGMA CHAMBERS where g is the magnitude of the gravitational vector and pr(T, P. X) is the volume-averaged density of the rock in the column. The term rock refers to minerals plus pore space, and all pore space is assumed to be filled with fluid. Because density is a function of the material composition and of temperature and pressure, an explicit function is needed to integrate Eq. (2.1~. It expanded into a statement that independently accounts for the density of solid and fluid phases: Liz Total = Po + g J [(>mpm + 4)fpf (T. P. X)]dz, (2.2) zo where (m iS the total volume fraction of mineral phases, Pm is the average density of the mineral assemblage, Of is the volume fraction of the fluid phase, and pf(T, P. X) is the temperature-, pressure-, and composition-dependent den- sity of the phase. Mineral densities are nearly constant relative to fluid density over the state conditions com- monly encountered in the crust. Consequently, the pres- sure contribution from the minerals can be removed from the integral: ~ ta} = Po + g [em pm + rZ aim ~ ~ ] Total pressure can therefore be determined by expressing the density function as an equation of state using relations like those reported by Helgeson and Kirkham (1977) and Johnson and Norton (1989~. However, the crust is generally in a state of heterogene- ous stress that can be represented as a stress tensor, T: .. Oxx Oxy Oxz T = oyx oyy cyz , (2.4) O.zx Cozy Razz where aij refers to the stress exerted along the ith axis normal to the jth direction. The resultant pressure caused by these stresses is equivalent to the trace of the stress tensor and is called the mean pressure, Pm: oxx + thy + ZZ (2.5) 3 The mean (Pm) and total (Prowl) pressures from Eq. (2.3) are equal only in conditions where ~ = ~ = ~ . To ox yy zz demonstrate the role of fluid pressure in deformation, we will assume that the stress field is homogeneous. Pressure conditions actually encountered in the subsur- face are unlikely to correlate closely with those values derived from the integrations using extreme values of (>m in Eq. (2.3~. Neither (dim > 0 for O < Z < Zmax) nor hydro 43 static (him > 0; of = 1 for O < Z < ZmaX) pressure conditions are likely because pressure generally depends on the geo- logic history of the lithologic units. As an example of the effect of dynamic loading history that can alter the pressure from the conditions just men- tioned, computations that describe the burial of a fluid- filled pore are reviewed in this chapter; the details of this experiment can be found in Knapp and Knight (1977~. The loading process can be thought of as diagenesis of a fluid packet, first isolated from the other pores in the rock by near-surface compaction and then subjected to increases in temperature and confining pressure as it subsides to greater depths in the basin. Even though the fluid in this situation may constitute a small fraction of the rock, typically less than 10 wt.%, its density variation in response to changes in state conditions can have a large effect on local pressure conditions. Because temperature and pressure are functions of depth, the total differential of fluid density with respect to depth is dpf (3pf ~ do (0pf ) dP dz gaT )PX dz Pap )TX dz , (0Xi )TPxi dz (2.6) where the partial derivatives are derived from the equation of state for the fluid, f(P, T. p, X) = 0, and the derivatives of pressure and temperature with depth are independent quantities that must be defined from solutions to the heat transfer equations. The partial derivative of composition, on the far right of Eq. (2.6), is significant in all natural environments. However, an important aspect of fluid pressure variations can be demonstrated with a single- component fluid. Therefore, the following discussion focuses on the single-component H2O system and omits the compositional variation in Eq. (2.6~. Density variation with depth for a fluid in the pure system is given by dPf (3pf ) dT (0pf ) dP (2.7, dz OT P dz aP T dz where the dependent partial derivatives can be replaced with the intrinsic properties of the fluid the isobaric coefficient of thermal expansion, Of: pf (aT)p (2.8) and the isothermal coefficient of compressibility, pf: Do (ap] (2.9) pf vaP )T
44 Substitution of these quantities into Eq. (2.7) gives Pf = -pfaf- + puff -. (2.10) dz dz dz If the fluid-filled pore is assumed to remain at a con- stant volume, the variation of pressure within the pore is only a function of the expansivity-to-compressibility ratio and the thermal gradient: dP _ al dT dz Of dz , (2.11) where af/,Bf is the isochoric coefficient of thermal pres- sure. It varies from 5 to 20 bars/°C for ranges in state conditions commonly found in subsiding sedimentary basins (Figure 2.1~. Knapp and Knight (1977) found pressure increases within the pore to be large at shallow depths even for modest thermal gradients. They also found that only for thermal gradients of less than 10°C/km will the pressure within the constant-volume pore remain less than the mean confining pressure, Pm, as subsidence occurs; for larger thermal gradients the fluid pressure increases to values much greater than the confining stress. This overpressure occurs at depths that depend primarily on the magnitude of the thermal gradient. Knapp and Knight's computation showed clearly that fluid pressure can vary with depth if the pore geometry is poorly interconnected and that the variation is a function of the thermal pressure coefficient TEMPERA r URE (·C) 0 200 400 600 800 ~ 000 200 lo: ~ 600 In son 1 000 B ~ R S / TIC ) 0.5 FIGURE 2.1 The isochoric coefficient of thermal pressure, a /pf, computed with the equation of state for the H2O-system (itrom Johnson and Norton, 1989~. DENIS L. NORTON of the fluid. The upper limit of the fluid pressure is a function of the pore wall strength and failure process. Prior to examining the consequences of failure on fluid pressure, the effect of a thermal perturbation imposed on the same pore space as described above is examined. CHANGE IN FLUID PRESSURE Thermal perturbations imposed on a porous rock gener- ate differential stresses because of large differences in the constitutive properties between minerals and fluids. The physical significance of this difference in response to thermal changes is that small thermal changes in the crust can cause large differences between values of mean pres- sure and fluid pressure. For a particular thermal state in the crust the hydrostatic and mean or total confining pres- sure can be defined from Eqs. (2.3) and (2.5~. Perturba- tions from this state caused by changes in the thermal flux are now examined. Consider the pressure changes that occur within an isolated fluid-filled pore. Assume that the fluid pressure exerted on the pore wall is initially in equilibrium with the ^. . mean confining pressure: Pf = Pm . (2.12) The time derivative of fluid pressure increase as a result of temperature and pore volume changes is dPf = (0Pf ) dT + (3Pf) dV ~ (2 13) dt aT v dt av T dt where V is the pore volume. Because the pores are as- sumed to be filled with a single-phase fluid, this volume is equivalent to 1/pf. Again, consider a constant volume process, dV/dt = 0, consistent with regional strain rates that are much smaller than those caused by the local change in temperature and with the situation in which a local increase in fluid pressure does not dilate the pore at a significant rate. Under this condition the change in inter- nal pressure as a function of time is given by dBf (3Pf ) dT, (2.14) dt aT v dt where faPf /aT~v is the ratio of the isobaric coefficient of thermal expansion to the isothermal compressibility, af /0f (Figure 2.1~. Notice that the time derivative of the pres- sure is analogous to its spatial derivative in Eq. (2.12~. As a consequence of Of /,Bf, pressure increases of several bars per degree centigrade can occur over relatively short times
PORE FLUID PRESSURE NEAR MAGMA CHAMBERS . --I ~ Edge - Agerturc in the near-field region of magma bodies, where the change in temperature with time is rapid. The magnitude of pressure increases associated with this situation and the burial computation in the previous section are likely to exceed the strength of the walls of intergranular pore space or of the mineral grains them- selves. The conditions for failure depend on the mode of occurrence of the fluid in the rocks, particularly on the shape of the pore space. FLUIDS IN ROCKS Fluids in rocks are seldom directly observed, but their presence is inferred from the presence of pore space. Although the mode of occurrence of the fluid phase is traditionally expressed in terms of the volume or porosity, the geometry of the fluid and its distribution with respect to the mineral phases are of equal importance to consid- erations of transport processes, particularly the conse- quences of fluid pressure changes. Studies of the mode of occurrence of fluid have revealed that the shape of the pore space is determined by properties of the fluid that fills them and by the local conditions of stress. The total porosity of crystalline igneous rocks ranges from a few percent to less than a fraction of a percent, and of this total pore space the ever-present fractures in these rocks contribute only a very small fraction of pore space to the total, circa 10-3 (Snow, 1970; Norton and Knapp, 19771. Igneous rocks commonly have a relatively large permea- bility, >10-~4 cm2, during their active thermal history, but only a small porosity, or total fluid fraction, is associated with the flow channels. This is because the flow channels 45 FIGURE 2.2 (A) Intersect of fracture with topographic surface. Slit-like form is typical of fracture topology at all scales of observation (Norton, 1987~. (B) Idealized fracture form based on dislocation theory (from Mavko and Nur, 1978; Norton, unpublished field data). are elongate fractures with apertures on the order of a few hundred microns (Norton and Villas, 1977~. The redistri- bution of only a small fraction of the fluid contained in the total pore space of such rocks into fractures can signifi- cantly affect the rock permeability and the mechanism of transport through it. The slit-like nature of fluid-filled pores (Figure 2.2) is caused by the active deformation of their host rock within a heterogeneous stress field and the redistribution of a portion of the fluid phase into fractures whose orientation is a function of the stress trajectories. Once formed, these fractures are extremely sensitive to the differential be- tween the mean confining pressure and the fluid pressure within the fracture. They are therefore delicate indicators of fluid pressure conditions because the mean confining pressure is relatively constant over long times, whereas the fluid within the fractures is sensitive to local changes in temperature. FAILURE CRITERIA The small finite strength of the pore wall implies that for the initial condition of Pf = Pm only a small pressure increase is necessary to reach the yield strength of the wall. The strain rate associated with this deformation is strictly a function of the local time derivative of the tem- perature. The duration of the elastic deformation may range from a few months to years. But the failure of the wall and consequent fracture propagation is an instantane- ous process, in which fluid expands irreversibly against a fixed pressure. Extensive commentary on failure criteria exists in the
46 literature; the findings by Berbabe (1987) indicate that a simple failure law matches experimental data on fracture formation in crystalline rocks. This law describes the failure of the pore walls when the sum of the confining pressure and wall strength is exceeded by the internal fluid pressure: PC +T = Pf (2.15) where ~ is the tensile strength of the rock. This failure criterion oversimplifies the mechanical problem consid erably but highlights the forces involved and accurately describes the transitions from one pore configuration to and another. The generation of pressures in excess of the mean confining pressure requires material strengths that will withstand the excess pressure. Because on a regional scale a rock has no effective strength, pressures in excess of Pm are not likely to develop over large portions of the crust. However, if the initial condition were one in which fluid in equilibrium with the local value of hydrostatic pressure was sealed into an isolated pore and then heated, the fluid pressure might increase several tens of bars up to the local confining pressure before failure would occur. Therefore, the following two failure conditions should be considered: ~ - m~ 1. The condition in which the initial condition was P ~ P . A fluid-filled pore fails and the fluid expands against a pressure only slightly lower than the condition of failure. 2. The condition in which the initial condition was Pf ~ Phylum. Failure may not occur until pressure increases to Pm or even slightly greater depending on the rock strength. Expansion may then occur against PhydIo if the newly formed fracture intersects a regime of hydrostatic pressure. The main difference between the two situations is the amount of energy required to attain failure. The second situation requires more energy simply because the pres- sure increase required to attain failure is greater. In either of the above situations the volume increases associated with failure can be examined in terms of irreversible ex- pansion against a constant pressure value, Pb. VOLUME CHANGE CAUSED BY FAILURE The volume of the newly formed fracture is a function of the expansivity of the fluid and the energy lost by viscose flow. Because fluid flows away from the breached pore wall a relatively short distance and for a limited time, viscose dissipation of energy is likely to be small. The rate of volume change in a fluid subjected to differential DENTS L. NORTON changes in temperature and pressure is given by the total differential of volume with respect to time: dV av dT av dP dV bT P dt LIP dt ' (2.16) where the dependent partial derivatives can be abbreviated as 1 Pave a- - V aT P V (aP )T (2.17) (2.18) Substituting these quantities into Eq. (2.16) gives an ex- pression for volume change in terms of the fluid properties and the change in temperature and pressure: - = Va- - Vp-. (2.19) dt dt dt Both the expansivity and the compressibility of an H2O- rich fluid phase are much greater than those of minerals. Therefore, Eq. (2.19) can be written only in terms of the fluid properties without introducing significant errors in the analysis (Moskowitz and Norton, 1977): dV V dT V ~ dP (2.20) dt dt dt The pressure differential is relatively small where the fail- ure occurs at pressure slightly greater than the confining pressure, but where expansion is against hydrostatic pres- sure the change is substantial. Integration of Eq. (2.20) for a given pressure over fixed temperature limits gives If dVf rT rP Jvf° Vf JTO al (T)dT + J ,Bf(P)dP (2.21) and Vf = Vf exp :[ af(T) dT + ,[ pf(P)dP]. (2.22) The concomitant propagation of the fracture and the thermal expansion of the fluid as it is released by the pore wall failure result in a net increase in porosity that poten- tially augments the flow porosity. The porosity increase that results from the failure of pressurized pores can be
PORE FLUID PRESSURE NEAR MAGMA CHAMBERS computed from the equation of state of the fluid and the ° 2° ~ change in fluid volume defined by Eq. (2.181. The porosity, 0, contributed by the isolated pores in a region is the initial volume fraction of fluid in those pores: Vf Vf . (2.23) ~ Vtotal Vm + Vf c: The initial fluid volume is a function of the initial pores- 2 0.1 0 ity, _ 0.15- - Vf = <) Vtotal , (2.24) and the total volume of minerals in the rock is Vf = ( 1 - ¢) VtOtal . (2.25) Therefore, the fluid porosity at a time following a thermal change can be expressed as a combination of Eqs. (2.23) and (2.17~: of = ofF(T) (2.26) (1 - (f) + ~fF(T) where the function, F(TJ, in Eq. (2.26) is F(T) = exp [iT af(T) dT ip0 of ] Porosity increases in a super-exponential manner be- cause of the exponential term and because At increases exponentially in the critical region of the H2O system. Under near-critical conditions the porosity doubles in re- sponse to only a few degree increase in temperature (Fig- ure 2.3~. Because a large portion of the total porosity in a crys- talline rock is in the form of isolated porosity, circa 0.01 to 0.1, the thermal expansion of pore fluid could be quite effective in increasing the interconnected pore space, which constitutes the flow network in a rock and therefore di- rectly affects the value of permeability. Furthermore, the flow porosity required for moderately large values of per- meability is on the order of 104 to 10-s. Conversion of even a small percentage of the isolated pores into oriented fractures can lead to enormous increases in permeability. ENERGY REQUIRED FOR FRACTURE The total energy available from a volatile-rich mag- matic body is ~200 cal/g. In regions where the alteration process produces hydrous phases, the exothermic heat can add an additional amount of up to 40 cal/g. Mafic and 4~ '~/4,~, 400 `800 _, ~-~ _~~ 47 005- 1 ~ ~ 1 ~ ~ ~ ~ 1 ~ ~ ~ ~ 1 ~ ~ ~ ~ I 0 100 200 300 400 500 TEMPERATURE (°C) FIGURE 2.3 Maximum change in porosity as a function of temperature change for an isolated fluid-filled pore that propa- gates through the rock matrix on failure (Moskowitz and Norton, 1977). volatile-poor magmas contain more thermal energy than felsic magmas because of the large 100 cal/g heat of crys- tallization. The thermal energy converted to mechanical energy during the failure process described is only a few calories per gram (Knapp and Knight, 1977~. Although the energy expended in a typical situation amounts to only a few calories per gram, the increase in permeability that can result from the failure is significant. This increase allows a small fraction of the pressure to be dissipated by viscose flow. However, the generation of an interconnected network over a broad region also disperses the fluid force field generated by the buoyancy forces. Therefore, the increase in permeability results in an in- crease in fluid velocity and consequently an increase in the advective transport of chemical components and heat. ADVECTION AND PORE PRESSURE Increases in rock permeability cause proportionate in- creases in fluid velocity. The fluid flow is driven by the pervasive buoyancy force field, which is an inevitable consequence of changes in heat flux, particularly those changes caused by the infiltration of magma. Therefore, when the synchronous propagation of fractures produces an interconnected network of flow channels, fluid motion ensues and augments the flux of heat and chemical compo- nents. Localized flow associated with the individual fail- ure events also augments the flux but is less effective than the buoyancy force driven flow. If local equilibrium between the fluid in an isolated pore and the minerals that form the pore wall is assumed,
48 then at the time of pore failure and propagation of a frac- ture a gradient in fluid composition will be established along the interconnected fracture network because of gra- dients in state conditions. This compositional gradient in the fluid composition and the coinciding fluid velocity advect chemical components from one environment to another. As an example of this process, consider a system in which the lithologic units can be represented by the sys- tem SiO2-H2O; they contain only quartz and an aqueous fluid locally in equilibrium with quartz. Chemical changes caused by advective flow in a system, where temperature, pressure, and composition gradients occur and where equilibrium prevails locally between quartz and fluid, are symbolically depicted by Fluid Quartz Advecuon a(~)fiO2 + a(~)Sti02 at a' - + vf ~Yfi°2 = 0 (228) where the relationship between the rate of change in quartz and the concentration of aqueous silica can be defined by the equilibrium equation: Si02 = Aqueous (2.29) The partial derivatives of 9ifi02 in the fluid phase with respect to time, t, and distance, 1, can then be defined in terms of the standard state partial molal enthalpy, AHr°, and partial molal volume, AVr°, of the equilibrium reaction between quartz and aqueous silica, Eq. (2.29): If 2 ~SiO2 0 026 (^H~° AT AV,° ~: Oaf = ~fit'2 0 026 (SHE AT _ ~Vr i) ~ (2 31) Ol RT2 dl RT dl where the assumption has been made that the activity of aqueous silica is equal to its morality, and the concentra- tion of silica in Eqs. (2.30) and (2.31) is in grams per cubic centimeter. The distance operator in the advection term in Eq. (2.31) is equated to the distance along the flow path, 1. Therefore, the rate of change in the amount of quartz within a fracture is a function of both the thermodynamic properties of the reaction and the temporal and spatial derivatives of temperature and fluid pressure: a S i02 AdvechOn O dT vrO -=-Vf·~fiO20.026(- - - - -~ OtRT2 dl RT dl ) Rate of charge in fluid _~fi02 0~026 (~2- - --) . (2.32) RT dt RT dt DENTS L. NORTON This relation demonstrates the coupling among the thermal, pressure, and fluid flow fields in which the depo- sition or dissolution of a mineral phase in the flow net- work can effectively change its continuity. The manner in which the mineral actually blocks the flow channel is a consequence of detailed interaction be- tween the local flow field within the fracture (Brown, 1987) and the local gradients in temperature and pressure. These interactions are unfortunately not predictable from the transport theory. However, there is ample evidence in the textures of veins to indicate that sealing of the flow channel actually occurs numerous times during the history of the thermal event. SUMMARY Field relations indicate that the processes that lead to the formation of fracture-controlled percolation networks and that fill these networks with vein minerals are related through an episodic series of events. The fractures that constitute the networks are slit-like features of limited extent; they occur in interconnected sets, and in any one environment there may be many such sets of different orientation and distinct chronology (see Titley, Chapter 3, this volume). The vein material within a fracture is per- vaded with discordant contacts that suggest many discrete events of fracture and vein fill. Transport equations demonstrate that the functional relations among the field variables form a coupled set of differential equations in which the principal transport mechanism has a hyperbolic form. In all but the simplest of systems in which hyperbolic functions depict the mecha- nism of transport, the system evolves in a chaotic manner, particularly if other nonlinearities are present. In the situ- ation discussed in this chapter, highly nonlinear properties of H2O-rich fluids have long been recognized as exerting primary control on the evolution of temperature, pressure, and fluid velocity (Norton and Knight, 1977~. Independent lines of evidence-one from transport theory and the other from the geometric properties of veins and fracture systems point to anomalous fluid pressure as the force that not only generates systematic fractures and causes them to interconnect and form perco- lation networks but that also retards flow through mineral deposition. REFERENCES Berbabe, Y. (1987~. The effective pressure law for permeability during pore pressure and confining pressure cycling of several crystalline rocks, Journal of Geophysical Research 92, 649- 657. Brown, S. R. (1987~. Fluid flow through rock joints: The effect of surface roughness, Journal of Geophysical Re search 92, 1337-1347.
PORE FLUID PRESSURE NEAR MAGMA CHAMBERS Burton, C. J., and H. C. Helgeson (1983~. Calculation of the chemical and thermodynamic consequences of differences between fluid and geostatic pressure in hydrothermal systems, American Journal of Science 283-A, 540-548. Helgeson, H. C., and D. H. Kirkham (1974~. Theoretical predic- tion of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures. I. Summary of the thermody- namic/electrostatic properties of the solvent, American Jour- nal of Science 274, 1089-1198. Hubbert, M. K., and D. G. Willis (1957~. Mechanics of hydrau- lic fracturing, Transactions, Society of Petroleum Engineer- ing, AIME 210, 153-166. Johnson, J. W., and D. Norton (1989~. Critical phenomena in magma-hydrothermal systems: I. State, thermodynamic, trans- port, and electrostatic properties of H2O in the critical region, American Journal of Science, in press. Knapp, R. B., and J. E. Knight (1977~. Differential thermal expansion of pore fluids: Fracture propagation and microearth- quake production in hot pluton environments, Journal of Geophysical Research 82, 2515-2522. Knapp, R. B., and D. Norton (1981~. Preliminary numerical analysis of processes related to magma crystallization and stress evolution in cooling pluton environments, American Journal of Science 281, 35-68. 49 Lantz, R. (1984~. The influence of the geometry of the pluton- host rock interface on the orientations of thermally induced hydrofractures at the Cochise Stronghold pluton, Cochise County, Arizona, M.S. thesis, University of Arizona, Tucson. Mavko, G., and A. Nur (1978~. The effect of nonelliptical cracks on the compressibility of rocks, Journal of Geophysical Re- search 83, 4459-4468. Moskowitz, B. M., and D. Norton (1977~. A preliminary analy- sis of intrinsic fluid and rock resistivity in active hydrothermal systems, Journal of Geophysical Research 82, 5787-5795. Norton, D. (1984~. A theory of hydrothermal systems, Annual Reviews of Earth and Planetary Sciences 12, 155-177. Norton, D., and R. Knapp (1977~. Transport phenomena in hydrothermal systems: The nature of porosity, American Jour- nal of Science 277, 913-936. Norton, D., and J. E. Knight (19771. Transport phenomena in hydrothermal systems: Cooling plutons, American Journal of Science 277, 937-981. Norton, D., and R. N. Villas (1977~. Irreversible mass transfer between circulating hydrothermal fluids and the Mayflower stock, Economic Geology 72, 1471 - 1504. Snow, D. T. (1970~. The frequency and aperture of fractures in rocks, Journal of Rock Mechanics 7, 23-40.
