Field studies of Cordilleran metamorphic core complexes indicate that meteoric fluids permeated the upper crust down to the detachment shear zone and interacted with highly deformed and recrystallized (mylonitic) rocks. The presence of fluids in the brittle/ductile transition zone is recorded in the oxygen and hydrogen stable isotope compositions of the mylonites and may play an important role in the thermomechanical evolution of the detachment shear zone. Geochemical data show that fluid flow in the brittle upper crust is primarily controlled by the large-scale fault-zone architecture. We conduct continuum-scale (i.e., large-scale, partial bounce-back) lattice-Boltzman fluid, heat, and oxygen isotope transport simulations of an idealized cross section of a metamorphic core complex. The simulations investigate the effects of crust and fault permeability fields as well as buoyancy-driven flow on two-way coupled fluid and heat transfer and resultant exchange of oxygen isotopes between meteoric fluid and rock. Results show that fluid migration to middle to lower crustal levels is fault controlled and depends primarily on the permeability contrast between the fault zone and the crustal rocks. High fault/crust permeability ratios lead to channelized flow in the fault and shear zones, while lower ratios allow leakage of the fluids from the fault into the crust. Buoyancy affects mainly flow patterns (more upward directed) and, to a lesser extent, temperature distributions (disturbance of the geothermal field by ~25°C). Channelized fluid flow in the shear zone leads to strong vertical and horizontal thermal gradients, comparable to field observations. The oxygen isotope results show δ18O depletion concentrated along the fault and shear zones, similar to field data.
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 A growing body of studies suggests that water plays a major role in the thermal, mechanical, and geochemical evolution of the Earth, from crust to mantle [Austrheim, 1987; Nesbitt and Muehlenbachs, 1989; Marquer and Burkhard, 1992; Hirth and Kohlstedt, 1996; Oliver, 1996; Koons et al., 1998; Manning and Ingebritsen, 1999; Iio et al., 2002; Saar and Manga, 2003; Hirschmann et al., 2005; Bürgmann and Dresen, 2008; Ingebritsen and Manning, 2010]. In active tectonic areas, faults, cross-cutting lower to upper crust, are heterogeneous and anisotropic structures that can serve as conduits or barriers or combined conduit-barrier systems for groundwater flow, with ramifications for heat transport and ore formation [Smith et al., 1990; Antonellini and Aydin, 1994; Forster et al., 1994; Newman and Mitra, 1994; Lopez and Smith, 1995; Caine et al., 1996; Sibson and Scott, 1998; Saar and Manga, 2004; Saar et al., 2005; Bense and Person, 2006; Person et al., 2007; Saar, 2011]. Therefore, understanding the geometry and permeability structure of such flow zones has important implications for the thermomechanics of lithospheric deformation.
 Faults in crustal rocks are complex structures that consist of a fault core of gouge, cataclasite, and mylonite, where most of the displacement is accommodated [Caine et al., 1996]. This core is mechanically associated with an outer damage zone consisting of small faults, fractures, veins, and folds. Beyond these first-order observations, fault geometry and permeability are not well understood, although there appears to be consensus that fault zones represent hydrologic units [Haneberg, 1995; Bense and Person, 2006; Sutherland et al., 2012] as they typically direct fluid flow either within their high-permeability zones or outside, along their low-permeability zones. In this paper, we address crustal-scale fluid flow in regions of crustal extension, where normal and detachment fault zones provide a relatively simple flow system. Following Person et al. , we are particularly interested in the thermal implications of fluid flow in this tectonic regime and in the resulting oxygen isotope interaction between fluid and rock at elevated temperatures. This two-dimensional model offers first-order insights into the relationship between fluid flow, the thermal structure of the crust, and associated oxygen isotope exchange, so that the sensitivity of fluid-rock interaction to crustal and fault permeability fields can be evaluated.
 At elevated temperature, fluid-rock interaction is quantified most effectively by stable isotope exchange [e.g., Valley, Taylor and O'Neil, 1986]. The distribution of oxygen and hydrogen isotopes in fluids and minerals has become an important fingerprint to elucidate the origin and fluxes of fluids over a wide range of timescales and geological settings such as intrusions and fault zones, including thrust faults [Burkhard and Kerrich, 1988; Rye and Bradbury, 1988; Banks et al., 1991; Moore and Vrolijk, 1992; Sibson, 1994; Sibson and Scott, 1998; Crespo-Blanc et al., 1995; Goddard and Evans, 1995; McCaig et al., 1995; Ghisetti et al., 2001; Kirschner and Kennedy, 2001], strike-slip faults [O'Neil and Hanks, 1980; Unruh et al., 1992; Chester et al., 1993; Barton et al., 1995; Sibson and Scott, 1998; Kharaka et al., 1999; Pili et al., 2002], and extensional detachment systems [Bowman and Willett, 1991; Bowman et al., 1994; Gerdes et al., 1998; Cook et al., 1997; Mulch et al., 2007; Gébelin et al., 2011]. Although the details of fluid transport and exchange with rock in the transition between brittle and ductile deformation regimes are still poorly understood, several stable isotope and other studies have shown that meteoric water can circulate to significant depths (down to the ductile crust) in normal fault systems [e.g., Wickham and Taylor, 1987; Fricke et al., 1992; Morrison, 1994; Morrison and Anderson, 1998; Bebout et al., 2001; Mulch et al., 2004, 2006, 2007; Person et al., 2007; Ingebritsen and Manning, 2010; Gébelin et al., 2011; Gottardi et al., 2011].
 The focus of this paper is to study the effects of permeability fields on groundwater flow, heat distribution, and isotope composition in a fault-controlled system associated with uplift and exhumation of metamorphic rocks at a detachment zone. The model is intended to idealize the geometry of an extension system affecting hot and thick crust, such as found in Cordilleran metamorphic core complexes [Vanderhaeghe and Teyssier, 2001]. The geometry of the extensional domain is inspired by a model proposed by Person et al. , but the method to compute the fluid, heat, and isotope transport follows a macroscale lattice-Boltzmann approach. In the first part of the paper, we summarize the field evidence for fluid circulation and fluid-rock isotope exchange that has been reported from North American Cordilleran metamorphic core complexes. We then set up the model domain based on generalized field observations and introduce the numerical method. Model results are presented in terms of the thermal and isotope records for various permeability fields and fault zone connectivity cases. Finally, modeling results are compared to temperature and isotope ratios measured in rocks pegged to the field setting. This provides insight into the permeability conditions that must be fulfilled to explain observed temperature and isotope values.
2 Geologic Evidence for Crustal-Scale Fluid Flow
 Our study aims to model crustal-scale fluid circulation in the context of continental extension that accompanies the collapse of thickened, orogenic crust. On first order, the continental crust causes extension in two layers (brittle and ductile) with contrasting thermal and mechanical behaviors [Kusznir and Park, 1987]. The brittle upper crust thins by normal faulting and is separated from the ductilely flowing lower crust by a detachment zone. In this framework, localized extension leads to the formation of metamorphic core complexes [Brun et al., 1994; Rey et al., 2001; Teyssier et al., 2005]. Stable isotope studies have shown that at the time of extension, surface-derived fluids permeate the brittle upper crust down to the detachment footwall (Figure 1), where meteoric fluid signatures are preserved in metamorphic minerals [Fricke et al., 1992; Morrison, 1994; Holk, 1997, 2000; Mulch et al., 2004, 2006; Gébelin et al., 2011; Gottardi et al., 2011]. In extensional systems, arrays of normal faults crosscut the upper crust, are rooted in the detachment surface, and provide natural pathways for fluid flow. The high geothermal gradient that characterizes extension zones enhances buoyancy-driven convective circulation of surface fluids from the surface to the detachment at the base of the brittle crust (Figure 1) [Mulch et al., 2004, 2007]. Although salinity-related convection effects may be important in extensional environments, they are not considered in this study [Duffy and Al-Hassan, 1988; Leising et al., 1995].
 Following a protracted Mesozoic tectonic history of compression, accretion, and crustal thickening that built the North American Cordillera, extensional detachments evolved rapidly during Paleocene-Eocene time along crustal-scale, low-angle shear zones [Armstrong, 1982; Coney et al., 1984; Brown et al., 1986; Price, 1986; Parrish et al., 1988; Carr, 1992; Vanderhaeghe and Teyssier, 1997; Crowley et al., 2001; Bird, 2002]. Thermochronological and geochemical studies of core complexes show two groups (Figure 2): North of the Snake River plain, from British Columbia to Washington State, the Omineca belt comprises the largest region of Cordilleran metamorphic core complexes [Armstrong, 1982; Coney, 1987; Brown et al., 1986]. In this belt, 40Ar/39Ar ages of fabric-forming, synkinematic white mica concentrate narrowly between 49 and 46 Ma (Table 1 and Figure 3). Below the detachment zone, metamorphic core complexes were characterized by significant partial melting, granite intrusion, and flow of lower crust at the time of extensional activity (Table 1 and Figures 3 and 4) [Parrish et al., 1988; Vanderhaeghe and Teyssier, 1997, 2001; Vanderhaeghe et al., 1999, 2003; Foster et al., 2001; Foster and Raza, 2002; Teyssier et al., 2005]. In contrast, south of the Snake River Plain (Figure 2), the southern core complexes of the Raft River-Grouse Creek-Albion Mountains, Ruby-East Humboldt Range, Snake Range, and Whipple Mountain show a more prolonged extension history from Eocene to recent Basin-and-Range extension, with a pulse of mylonite development in Oligo-Miocene time (Table 1 and Figure 3).
Table 1. Available Data on Metamorphic Core Complexes
 Detailed stable isotope work in the Shuswap metamorphic complex, including the Thor-Odin dome [Holk, 1997, 2000; Mulch et al., 2004, 2006] and Kettle dome [Mulch et al., 2007] documents fluid-rock exchange (Table 1 and Figure 3). Along the Thor-Odin detachment (portion of the Columbia River detachment that bounds the Thor-Odin dome to the east), the δD values of syntectonic muscovite fish are extremely low (between −140 and −160‰) and relatively constant in the top few hundred meters of the detachment. The quartzite protolith in the detachment footwall has muscovite with −80 < δD < −40‰, consistent with metamorphic muscovite (Table 1 and Figure 3) [Mulch et al., 2004, 2006; Quilichini, 2012]. Quartz-muscovite oxygen stable isotope thermometry estimates the temperature of mylonitization at 420 ± 40°C for the Thor-Odin detachment mylonite; the isotopic composition of the water that interacted with the mica at those temperatures was therefore −140 < δD < −120‰ (Table 1 and Figure 3). Surface water is the only reasonable candidate for such very low deuterium values [Taylor, 1977]. Similarly, synkinematic muscovite displays δD values as low as −135‰ for the Kettle dome detachment [Mulch et al., 2007; Quilichini, 2012] (Table 1 and Figure 3). The Kettle dome quartzite mylonite shows little variation in δ18O and displays oxygen isotope equilibrium with deformation temperatures of 400 ± 40°C [Mulch et al., 2007] and 365 ± 45°C [Quilichini, 2012] (Table 1 and Figure 3). Based on the hydrogen isotope composition of muscovite, muscovite at this temperature exchanged isotopically with meteoric water with maximum δDwater = −115 ± 5‰ in the Kettle dome. In the Bitterroot detachment shear zone, Kerrich and Hyndman  estimated the temperature of mylonitization at 550°C based on quartz-muscovite oxygen stable isotope thermometry (Table 1 and Figure 3), but Quilichini  proposes a temperature of mylonitization with a range from 500 to 300°C. Similarly, synkinematic muscovite has a composition ranging from −140‰ to −120‰, indicating that isotopic exchange occurred with meteoric water with maximum δDwater = −110 ± 5‰ in the Bitterroot detachment.
 South of the Snake River Plain (Figure 2), 40Ar/39Ar ages of mylonitization are more complicated, showing a protracted history. However, all of the core complexes were finally exhumed in Oligo −Miocene time: 22–10 Ma for the Raft River [Wells et al., 2000], 27–21 Ma for the Ruby-East Humboldt Range [Mueller and Snoke, 1993; McGrew and Snee, 1994], 27–17 Ma for the Snake Range [Lee, 1995; Miller et al., 1999; Gébelin et al., 2011], and 26–16 Ma for the Whipple Mountains [Davis, 1988] (Table 1 and Figure 3). The peak temperatures for this latest metamorphism, based on oxygen stable isotope thermometry, are estimated at 345–485°C for the Raft River Mountains [Wells et al., 2000; Gottardi et al., 2011], 400 ± 50°C for the Snake Range: [Gébelin et al., 2011], and 460°C–535°C for the Whipple Mountains [Anderson, 1985; Davis et al., 1982] (Table 1 and Figure 3). In the Ruby Range, petrological constraints put the temperature of Miocene metamorphism around 550°C [Mueller and Snoke, 1993; McGrew and Snee, 1994] (Table 1 and Figure 3), and mylonite development likely occurred down to 480°C [Fricke et al., 1992].
 Stable isotope data are also available for the metamorphic core complexes south of the Snake River Plain. In the Raft River detachment, Gottardi et al.  revealed a steep thermal gradient (~140°C / 100 m) based on quartz/muscovite stable isotope thermometry. δ18O values of quartz and muscovite are variable, with values ranging from 9.8‰ to 12.7‰ for quartz and 7.1 to 9.4‰ for muscovite (Table 1 and Figure 3). Hydrogen isotope composition of muscovite varies from −120‰ to −90‰, in agreement with a fluid composition of −100‰ to −70‰ [Suzuoki and Epstein, 1976] (Table 1 and Figure 3). In addition, quartz fluid inclusions analyzed in five samples over the entire section have a narrow range between −94 and −82‰; these values are in agreement with those calculated from the hydrogen isotope composition of muscovite and are also consistent with meteoric fluids.
 In the Ruby Mountains, Fricke et al.  report very low quartz δ18O values (as low as 2.8‰) from the mylonite zone compared to the unmylonitized quartz from adjacent basins (values around 10.0 to 12.0‰), and fluid inclusion δD yields a composition of −119‰ (Table 1 and Figure 3). Mylonitized leucogranite also shows large oxygen isotope disequilibrium. Fricke et al.  concluded that their measurements can only be explained by a fluid composition depleted in 18O and D from interaction with a meteoric fluid. Very low muscovite hydrogen isotope compositions are also reported in the uppermost part of the detachment in the Snake Range by Gébelin et al.  (−150 to −145‰) (Table 1 and Figure 3). In the Whipple Mountains, Morrison and Anderson  report an extreme thermal gradient (82°C over 38 m) below the detachment fault. Oxygen isotope compositions of epidote and quartz from chloritic breccias that underlie the detachment fault yield quartz-epidote fractionations that range from 4.1 to 6.4‰ (Table 1 and Figure 3) and increase systematically toward the fault, giving mean temperatures that decrease from ~432°C at 50 m below the fault to ~350°C at 12 m below the fault (82°C over 38 m) (Table 1 and Figure 3). Morrison and Anderson  pioneered the concept of fluid-induced fault-zone refrigeration. In general, it has long been recognized that groundwater recharge typically leads to subsurface cooling [e.g., Saar, 2011 for a review].
3 Idealized Geometry of the Model
 The preceding summary of isotope studies on metamorphic core complexes shows that significant oxygen isotope depletion in detachment zones is caused by the infiltration and interaction of meteoric fluids with deforming recrystallizing rocks. During orogenic collapse, brittle faulting and subsequent domino-style evolution of the upper crust opens numerous fault pathways for surface fluids to percolate deep into the crust (Figures 1 and 4). The geometry of the model represents such a generalized Cordilleran metamorphic core complex during the onset of extension, prior to exhumation of the detachment zone (Figure 5).
 The modeled upper crust is divided into ~10 km blocks (6, 6.75, and 7 km thick), separated by two 750 m thick high-angle fault zones that root into the shear zone. Several metamorphic core complexes have preserved syndeformational basins in the hanging wall of normal faults that developed by domino-style faulting during the evolution of the detachment (e.g., Trinity Hills and Enderby basins in the Okanagan detachment to the west of Thor-Odin, British Columbia, and Sacramento Pass basin in the Snake Range). Therefore, the model geometry also includes two 2 km thick near-surface basins in close association with the faults. There is a 1500 m elevation difference over 22.5 km between the lowest and highest block, which corresponds to a ~6.6% topography gradient, a relatively common slope in mountainous areas [Woodbury and Smith, 1988]. This slope induces some topography-driven flow and ensures that recharge will occur in the higher elevation fault; flow toward the lower fault will maintain a fairly stable fluid circulation during modeling. The base of the model includes a 4.5 km thick crust below the detachment zone that is represented by a 1500 m thick shear zone.
 We are particularly interested in fluid flow in the shear zone that connects the two high-angle faults, because this is the zone for which we have structural, thermal, and isotopic data from natural detachment systems. Although the shear zone can reach ~1 km in thickness, field observations and isotopic data from the Shuswap detachment in particular (Thor-Odin dome) [Mulch et al., 2006] suggest that a thinner deformation zone was active at any given time (order of 100 m thick). This model setup was also used in Person et al. .
4 Numerical Modeling Method
 In the last two decades, lattice-Boltzmann (LB) methods have been developed as promising numerical techniques for modeling fluid flow, solute or heat transport, reactive flow, and multicomponent and multiphase flow in heterogeneous porous media over both microscopic and macroscopic spatial scales [Chen and Doolen, 1998; Wolf-Gladrow, 2000; Succi, 2001; Guo and Zhao, 2005; Walsh and Saar, 2010; Myre et al., 2011]. These methods are different from conventional computational fluid dynamics (CFD) approaches that are based on discretized forms of continuum-based macroscopic equations. In contrast, LB methods employ a mesoscopic version of the classic Boltzmann equations to solve complex flow and transport behavior. LB methods include two steps: (1) a streaming step in which the physical properties (represented by fluid, heat, or solute “packages”) are propagated between two neighboring nodes and (2) a collision step which redistributes the fluid momentum, thermal energy, or solute concentration based on kinetic theory. These features of LB methods provide the advantage of modeling non-equilibrium dynamics with relatively simple algorithms, and simulating porous media flows with interfacial dynamics and complex boundaries [Chen and Doolen, 1998]. In the present study, we develop a simple, reliable, and highly efficient computational tool for simulating large-scale fluid flow and heat transport with fluid-rock isotope exchange, based on our previous LB algorithms [Myre et al., 2011; Walsh et al. 2009; Kao and Saar, Lattice Boltzmann models for local transports equilibrium (LTE) and non-equilibrium (LTNE) advective-diffusive equations, submitted to International Journal of Heat and Mass Transfer, 2013; Kao and Saar, in preparation, 2013].
4.1 Groundwater Flow
 In standard LB algorithms, streaming and collision steps are implemented as described above, to simulate fluid flow through explicitly described small-scale pore-space geometries, as opposed to providing permeability tensor fields of discretized, large-scale continuum space—the approach used in standard macroscopic groundwater flow modeling.
 A relative simple way to introduce the capability of simulating macroscale porous medium fluid flow in LB methods is to add a partial-bounce-back step into the standard two-step LB scheme [Dardis and McCloskey, 1998a, 1998b; Sukop and Thorne, 2006; Walsh et al., 2009]. Here, the degree of bounce-back is analytically determined according to local permeability via a solid fraction, as described in Walsh et al. . This partial-bounce-back scheme exactly satisfies Darcy's Law in the incompressible limit [Walsh et al., 2009].
 A standard two-dimensional (2-D) version of the groundwater flow equation under incompressible and isotropic conditions can be given as [Garven and Freeze, 1984]:
where k is the local, here isotropic, permeability of the heterogeneous medium, ν is the kinematic fluid viscosity, ρf is the local fluid density, which is a function of local temperature, g is Earth's (vertical) gravitational acceleration, P is the fluid pressure given as P = ρf g h, where h is the hydraulic head, i.e., the sum of pressure head and elevation head), is a reference fluid density at a reference temperature of 25°C and specified at the ground surface (i.e., upper boundary), and is the vertical characteristic length along the gravitational direction, y, which is evaluated at a depth of interest, measured from the surface elevation (with ) to the local crust elevation (with ρf).
 The left-hand side in equation (1a) is the flow momentum term, which can be solved employing the partial-bounce-back LB scheme as it reaches steady state. The buoyant momentum on the right-hand side of equation (1a) is treated as a body force term in the LB algorithm, as proposed by Kao and Yang . Note that a porous medium buoyant characteristic velocity (qBuoy) is defined as
and thus, the buoyant momentum, , satisfies the dimensions of ∇q as applied in the flow momentum equation as shown in Garven and Freeze . This buoyant source term was also used in Person et al. , but in that study, the buoyant momentum term does not satisfy the dimensions of ∇q which causes an overevaluation of the buoyant momentum that results from the basal heat flux. For this reason, we test the actual significance of the geothermal buoyancy effect on fluid flow and resultant temperature and oxygen isotope fields in our sensitivity study of crust permeabilities (section 6.1).
 The LB formulation uses a dimensionless unit system with some numerical limitations. In order to implement geofluids simulations, we perform a scaling analysis to establish the relationship between numerical and physical unit systems (section A).
4.2 Coupled Groundwater and Heat Transfer
 As shown in Table 2, the thermal diffusivities of water (1.4 × 10−7 m2/s) and rock (1.3 × 10−6 m2/s) result in a linearly weighted [e.g., Saar and Manga, 2004; Walsh and Saar, 2010] local mixed fluid-solid diffusivity range of 1.2 × 10−6 m2/s < Dm < 1.3 × 10−6 m2/s for the relatively low porosities of 0.08 to 0.03. These thermal diffusivity values are much larger than diffusivities of a solute (isotope) in water (10−10 m2/s). The solid-phase solute diffusion can typically be ignored for computing isotope transport and exchange [Bowman et al., 1994]. As a result, the timescale differences between diffusive heat and isotope transport are significant, i.e., diffusive isotope transport is much slower than diffusive (i.e., conductive) heat transport. Similar to Garven and Freeze  and Person et al. , we employ quasi steady state conditions in which larger-scale solid deformation occurs over larger time scales than fluid flow or thermal as well as isotope transport and fluid-mineral exchange. Such condition allows the coupled fluid flow and heat transport equations to be modeled until steady state conditions are reached.
Table 2. Physical Properties Used in the Numerical Simulation
Thermal diffusivity of water (Df)
1.4 × 10−7 m2/s
Thermal diffusivity of rock (Ds)
1.3 × 10−6 m2/s
Solute (isotope) diffusivity (Dh)
Fluid Kinematic viscosity (ν)
9.8 × 10−7 m2/s
 In the present study, the macroscopic hydrothermal behavior is formulated by an advective-diffusive thermal transport equation under so-called “local-transport-equilibrium” (LTE) conditions, given as
where Tm is the local mixed fluid-solid temperature that is based on the LTE condition, qx and qy are Darcy velocities for the horizontal, x, and vertical, y, dimensions, respectively, ρfCpf is the volumetric heat capacity of the fluid, ρmCpm is the local mixed fluid-solid volumetric heat capacity inside the medium, which is determined by a linear weighting, ρmCpm = ϕ ⋅ ρfCpf + (1 − ϕ) ⋅ ρsCps, where ϕ is the (relatively small) local porosity, and ρsCps is the rock volumetric heat capacity, and Km = ϕ ⋅ Kf + (1 − ϕ) ⋅ Ks is the mixed thermal conductivity, where Kf and Ks are the thermal conductivities in the fluid and in the solid, respectively.
 Note that in order to evaluate the buoyant momentum properly, the time-accurate term in equation (2) is preserved when coupling the thermal computation with the fluid flow algorithm to evaluate the buoyancy effect at each numerical time step. To solve equation (2), we apply a macroscopic local-transport-equilibrium lattice-Boltzmann (LTE-LB) model. The numerical-physical scaling analysis is described in section A.
4.3 Isotope Transport and Fluid-Solid Exchange
 As stated above, the solute (isotope) diffusivity is approximately four orders of magnitude smaller than the thermal diffusivities in the solid phase, in the fluid phase, or in the mixed fluid-solid complex phase. This implies that the diffusive isotope transport and fluid-solid isotope exchange is much slower than the diffusive (i.e., conductive) heat transfer. Hence, a time-dependent advective-diffusive solute transport equation with a kinetic reaction term for fluid-solid isotope exchange must be considered [Bowman et al., 1994], such as
where Rf is the oxygen isotope in the fluid phase, Dh is the medium solute diffusivity evaluated by the linear weighting of Dh = ϕ ⋅ Df, where Df is the solute diffusivity in the fluid phase, and ux = qx/ϕ and uy = qy/ϕ are the seepage velocities in the x and y directions, respectively.
 The reaction term in Equation (3a) employs a kinetic interfacial reaction and satisfies the conservation principle of solute exchange,
where Rs is the oxygen isotope in the solid phase, αr is the equilibrium solute exchange factor, is the local Damköhler number, where is the bulk solid surface area, rk is the local reaction rate, is the characteristic length of interfacial reaction, and the characteristic seepage speed of the fluid is given as , where ux and uy are seepage velocity components, obtained from dividing Darcy velocity components by the local porosity. In equation (3b), both αr and rk are determined as a function of the local temperature and the mineral components of the crustal unit [Person et al., 2007].
 Based on the formulations of equations (3a) and (3b), we implement a macroscopic lattice-Boltzmann fluid, heat, and isotope reactive transport model, the latter employing an implicit reaction algorithm as described in Kao and Saar (in preparation, 2013). The numerical-physical scaling analysis is described in section A.
4.4 Computational Domain and Simulating Conditions
 The spatial dimensions of the computational domain, illustrated in Figure 5, are 22.5 km horizontally and 12.5 km vertically. The upper plate is up to 7.5 km thick, the horizontal shear zone that simulates the brittle-ductile transition is 1.3 km thick, and the underlying lower plate is 4.2 km thick. We also study two different horizontal fault configurations within the horizontal shear zone. In “connected configuration,” there exists a 150 m thick horizontal subfault, or channel, within the shear zone (with the same permeability as the subvertical fault zones). This channel connects the two subvertical fault zones in the upper plate crust. Results for this Configuration 1 are shown in Figures 6, 8, and 10. In the “disconnected configuration,” the subvertical upper plate fault zones are not connected. Results for this Configuration 2 are shown in Figures 7, 9, and 11.
Table 3. Various Cases Presented in This Study, With Associated Permeabilities
Log of Permeability (m2)
Lower Crust: −19, Basins: −14
 The lattice-Boltzmann (LB) numerical simulations employ the D2Q9 model, a squared two-dimensional (D2), nine-velocity (Q9) lattice system. The computational domain is divided into 150 × 90 = 13,500 grid blocks for the horizontal, x, and vertical, y, dimensions, respectively. Applying our scaling analysis (section A), this grid number of 13,500 is sufficient to obtain accurate results using our LB models. Hence, special grid treatments, employing complex computational techniques, are not required. In addition, and as mentioned in section 4.1, the buoyancy effect employed by Person et al.  has been corrected here.
 Therefore, in this paper, we investigate (1) cases in which the buoyancy effect due to heat transfer is turned on or off, (2) the sensitivity of fluid and heat transport and isotope exchange dependence to various combinations of crust and fault permeability structures, and (3) cases in which the two steep fault zones in the upper plate are and are not connected by a high-permeability, horizontal channel within the shear zone.
 In our simulations, we apply no fluid flow boundary conditions on the two sides and at the bottom of the computational domain. A constant pressure hydraulic head boundary condition is implemented at the upper land-surface boundary shown in Figure 5 determine the hydraulic head. For heat transfer computations, the two side walls are specified as no heat flux boundary conditions . The upper boundary is set by a constant temperature Tupper = 25°C at the ground surface and the model employs a constant geothermal gradient of 45°C/km at the bottom boundary. For simulating fluid-rock oxygen isotope exchange, the equilibrium initial condition, , is implemented throughout the computational domain, where denotes ‰. The boundary value of Rf. LB = 0 (for δ18Of. phy = − 18‰) is used at upper boundaries to represent the oxygen isotope groundwater, and an extrapolating scheme is applied for the condition of
 In our numerical studies, computations of groundwater flow and advective-diffusive heat transfer are coupled by the buoyancy source term during each numerical time step until steady state conditions are reached. The steady state fluid and heat transfer solution is then used to simulate oxygen isotope transport and its fluid-rock exchange. Parameters used are listed in Table 2.
5 Numerical Modeling Results
 We conduct a numerical modeling sensitivity study to investigate the effects of permeability fields on groundwater flow patterns, heat transfer, and oxygen stable isotope transport and fluid-rock exchange. Different combinations of permeabilities are used for the upper crust, the steep fault zones, and the horizontal shear zone; six representative cases are detailed in this paper (Table 3). For simplicity, and to keep the number of variable parameters to a minimum, all rock units in the model are isotropic with respect to permeability; the second-rank permeability tensor is reduced to a single, local permeability value that is applicable to both the horizontal and vertical directions. The basin and the lower crust (Figure 5) have constant permeabilities of k = 10−14 m2 and 10−19 m2, respectively. The permeability of the fault and shear zones are varied from 10−19 m2 to 10−14 m2 and from 10−18 m2 to 10−16 m2, respectively. However, as mentioned previously, the shear zone may contain a 150 m thick horizontal subfault, or channel, with the same permeability as the subvertical fault zones. We refer to this as the “connected configuration.” When no such horizontal, high-permeability channel exists within the shear zone, we refer to a “disconnected configuration.” Upper crust permeability ranges from 10−19 m2 to 10−17 m2 (Table 3). Our numerical modeling technique also allows us to turn buoyancy-driven convection on and off to isolate the effects of buoyancy on flow patterns, temperature redistributions, and resultant oxygen isotope distributions.
5.1 Fluid Flow Patterns
 In our simulations, Case 1 is the base model (Table 3). In this case, the upper and lower crusts have the same low permeability (10−19 m2), and the permeability contrast between the fault zones (10−15 m2) and the upper crust (10−19 m2) is high (four orders of magnitude), thus forcing the fluid flow into the fault zones (Figure 6). In the connected configuration (Figure 6), when the faults are connected by a high-permeability, horizontal channel within the shear zone, the Darcy velocity, or fluid flux, increases along the path from the recharge fault zone (~1.5 m/yr) to the shear zone channel (~1.8 m/yr) and up the discharge fault zone (~3.2 m/yr). In contrast, in the disconnected configuration (Figure 7), because of the high-permeability contrast between the fault and the shear zones (3 orders of magnitude), fluid flow is inhibited between the recharge and the discharge fault zones, leading to lower fluid flow values of 0.6 m/yr in the recharge fault zone and 1.2 m/yr in the discharge fault zone.
 In Case 2, the permeabilities of the upper crust and the fault zones remain the same as in Case 1, but the shear zone permeability is now 10−16 m2, only 1 order of magnitude lower than that of the fault zones (Table 3). The higher-permeability contrast between the upper crust and shear zone enhances fluid flow through the shear zone, leading to higher fluid fluxes in the discharging fault zone (Figure 6). In the connected configuration, the flow pattern is similar to Case 1: fluid flux increases along the fault zone from lower values (~2 m/yr) in the recharge fault zone to ~3 m/yr in the shear zone channel, to the highest values as the discharge fault zone connects to the near-surface basin (~3.8 m/yr). In the disconnected configuration, when the connection between the fault zones is not enhanced by a high-permeability channel within the shear zone (Figure 7), the still elevated permeability of the shear zone still concentrates fluid flow from the recharge to the discharge fault zone albeit, at slightly lower fluid flux values than in the connected configuration (~1.8 m/yr in the recharge fault zone, ~3.4 m/yr in the discharge fault zone).
 Case 3 has higher-permeability fault zones (Table 3) than the base model (Case 1). The permeability contrast between the fault zones and the upper crust is 5 orders of magnitudes (10−14 m2 and 10−19 m2, respectively), 1 order of magnitude higher than the permeability contrast in the base model. Consequently, in the connected configuration, fluid flow is even more restricted to the fault zones, and fluid flux values increase along the flow path from ~3.6 m/yr in the recharging fault zone to ~ 4.0 m/yr in the shear zone channel, and up to ~ 4.6 m/yr in the discharge fault zone close to the near-surface basin (Figure 6). In the disconnected configuration, corresponding values are ~1.2 m/yr in the recharge fault zone and ~2.8 m/yr in the discharge fault zone (Figure 7). Thus, increasing the permeability contrast by one order of magnitude between Cases 1 and 3 leads to a significant increase of fluid fluxes in the fault zone (about 2 to 4 times higher in the disconnected and connected configurations, respectively).
 In Cases 4–6, permeability of the upper crust is increased from 10−19 m2 to 10−17 m2. This results in overall significantly higher fluid flow rates.
 In Case 4, the upper crust and the shear zone have the same permeability of 10−17 m2, whereas the fault zones remain at a permeability of 10−15 m2. Intermediate permeabilities of the upper crust and shear zone lead to a higher fluid flux (~2 m/yr). Because the fault zone/upper crust permeability contrast is reduced to 2 orders of magnitudes, fluid flow is less restricted to the fault zones and fluid “leaks” from the fault zones into the upper crust and shear zone (Figure 6). In the disconnected configuration, the fluid flux reaches high values in the recharge and discharge regions (top of the fault zones) but decreases rapidly with depth (Figure 7).
 Case 5 has the same configuration as Case 4, except that the permeability of the fault zone is one order of magnitude higher (10−14 m2). In the connected configuration, a higher fault/upper crust permeability contrast again enhances fluid flow in the fault zones and the shear zone (Figure 6), leading to a twofold increase in fluid fluxes between Case 4 (~12 m/yr) and Case 5 (~24 m/yr). The disconnected configuration is similar to Case 4 (Figure 7).
 In Case 6, the permeability of the fault zones is very low (10−19 m2) so that the fault zones serve largely as barriers to across-fault fluid flow. The permeability of the fault zones is two orders of magnitude smaller than that of the upper crust (10−17 m2), leading to very small fluid fluxes across and within the fault zones in both the connected (Figure 6) and disconnected (Figure 7) configurations. The permeability of the upper crust is identical to Cases 4 and 5, and the shear zone has a higher permeability (10−16 m2), leading to similar fluid flow rates (~2 m/yr).
5.2 Temperature Distribution
 In the connected configuration of the base model (Case 1), flow along the recharge fault zone carries cool fluids downwards, deflecting the isotherms (Figure 8), the temperature at the bottom of the recharge fault zone is only about 175°C. Within the horizontal, high-permeability shear-zone channel, the fluids are heated by ~100°C, resulting in a temperature of ~275°C at the bottom of the discharge fault zone. The rapid flow of relatively cool fluid in the shear-zone channel produces a concave deflection and tightening of the isotherms in the lower crust. Flow in the discharge fault zone displaces the isotherms upward and heats the basin in the vicinity of the discharge fault zone by ~25°C (Figure 8). In the disconnected configuration (Figure 9), the temperature field around the recharge fault zone is only slightly reduced by ~25°C. However, the isotherms are also compressed at the bottom of the discharge fault zone, leading to a ~100°C vertical gradient over ~2 km across the shear zone.
 In Case 2, the permeability of the shear zone (outside the high-permeability channel) is 10−16 m2, higher than in the base case, so that the permeability contrast between the shear zone and the crust is also higher (by 3 orders of magnitude). This leads to a strong channeling of the fluids within the shear zone, with minor leakage into the lower crust (Figure 8). The temperature distribution is similar to the base model (Case 1). In the disconnected configuration (Figure 9), the temperature pattern is also similar to that in Case 1, with a more pronounced compression of the isotherms effect at the bottom of the discharge fault zone (~125°C of vertical gradient over the 2 km thick across the shear zone), and a reduced thermal gradient along the discharge fault zone.
 In Case 3, the fault zones have a high permeability (10−14 m2) so that the permeability contrast with both the shear zone and upper crust is significant (4 and 5 orders of magnitude, respectively) and is responsible for concentrating fluid flow into the shear zone channel. Consequently, in the connected configuration, the upper crust experiences significant cooling with temperatures as low as ~125°C at the bottom of the recharge fault zone at a depth of ~6 km (Figure 8). The isotherms in the upper crust to the right of the recharge fault zone become bent downwards significantly in the vicinity of the fault zone, creating temperature perturbations of ~ −50°C. The entire block between the two fault zones is significantly cooled. Beneath the shear zone, the cool fluids produce a strong downward deflection and a tightening of the isotherms in the lower crust. As the fluids find their way upward in the discharge fault zone, they heat the discharge area, with temperature perturbations greater than +50°C in the basin. In contrast, the disconnected configuration temperature pattern is similar to that in Case 2 (Figure 9).
 In Case 4, because the permeability contrast between the fault zones and the upper crust is smaller, fluid flow is more distributed within the crust, and cooling is less localized than in Case 3 (Figure 8). However, in the connected configuration, the same concave tightening of isotherms under the shear zone and sharper vertical thermal gradients across the shear zone channel (150°C/2 km) are observed, as in Case 3. The more distributed flow in the crust leads to strong upward flow in the discharge area and heating of the discharge basin. Again, the disconnected configuration pattern is similar to Case 3 (Figure 9).
 In Case 5, the permeability of the fault zones is decreased by one order of magnitude (10−14 m2) and the patterns observed are similar to those in Case 4 but are more pronounced (Figure 8). In the connected configuration, the high permeability of the recharge fault zone concentrates fluid flow down into the shear zone, leading to a dramatic cooling of the recharge fault zone (approximately −65°C temperature perturbations), with temperatures < 100°C as the flow reaches the shear zone at ~6 km depth. The block between the two fault zones is significantly cooled, and the same strong bending of the isotherms to the right of the recharge fault zone is observed. Fluids are heated along the shear zone, producing a strong thermal gradient from the recharge fault zone (~100°C) to the discharge fault zone (~250°C). Hot fluids are brought up to the surface along the discharge fault zone, heating the discharge basin and resulting in temperature perturbations of greater than +25°C. The same downward deflection and tightening of isotherms is observed in the lower crust. In contrast, the disconnected configuration is similar to Case 3 (Figure 9).
 In Case 6, the permeability of the fault zone (fluid barrier) is low (10−19 m2) compared to that of the shear zone (10−16 m2), and upper crust (10−17 m2). Hence, in the connected configuration, fluid flow is mostly vertical and confined to each permeability field region or block. Horizontal flow is impeded by the subvertical, low-permeability fault zones that act as barriers, leading to subhorizontal isotherms. Only when reaching the shear zone can the fluids connect from the high-elevation block to the low-elevation block, leading to a perturbation of the isotherms within the shear zone. Hot fluids descending from the high-elevation block are channelized by the shear zone, and discharged into the low-elevation block, resulting in a general heating of the low-elevation block. In Case 6, the difference between the connected versus unconnected configuration is not relevant, as the fault zone permeabilities are lower than those of the shear zone. Similar temperature distributions are observed in the connected and disconnected configurations (Figures 8 and 9, respectively).
5.3 Oxygen Isotope Distribution
 A review of oxygen and hydrogen stable isotope data available for metamorphic core complexes allows us to constrain the oxygen isotopic composition of the fluid and the starting value of the rock. Typical δD values estimated for fluids percolating a detachment system during mylonitization range from −80 to −120‰ (Table 1 and Figure 3). Using the calibration of Suzuoki and Epstein  and assuming a temperature of mylonitization of 450°C, a fluid with a δD of −120‰ yields a δ18O value of approximately −18‰. Therefore, the fluid composition is set to δ18O = −18‰ in our model. Typical δ18O of quartz from the protolith of mylonitic zones is about 13‰. Therefore, the starting oxygen isotope composition of the rock in our model is set to δ18O = +13‰.
 Given these starting conditions, the spatial distribution of the rock oxygen isotope composition is directly related to the geometry of the system, the temperature field, and the time-integrated fluid flux. The rock stable isotope results presented in Figures 10 and 11 are the final steady state equilibrium values and cannot be directly compared to transient distribution at a particular time interval [as presented in Person et al., 2007]. Therefore, the stable isotope distribution shown here should be interpreted in light of the temperature distribution, in that equilibrium will be reached more rapidly at high temperature than at low temperature. Modeling results are therefore most comparable to the natural system in the lower half of the model, including the modeled shear zone that connects the high-angle faults and for which we have isotopic measurements.
 In the connected configuration of the base model Case 1, the infiltration of δ18O depleted water is recorded at shallow depth (upper 1 km) and the oxygen isotope composition of the rock is gradually less depleted with depth, from ~3‰ at the surface to normal values of 13‰ at ~4 km below the surface (Figure 10). The oxygen depletion is more pronounced on the right side of the model where the hydraulic head gradient is the highest. The rock oxygen isotope composition is only slightly modified (~12‰) along the recharge fault zone and the shear zone channel. However, the combination of compressed isotherms (Figure 8) and high fluid fluxes (Figure 6) on the discharging limb of the fault system yields more depleted (~10‰) rocks along the discharging fault zone. In the disconnected configuration (Figure 11), the pinning effect of compressed isotherms under the discharging fault zone leads to strong oxygen isotope depletion down to ~7‰. The only major depletion occurs along the discharging path, with values ~10‰ in the fault zone.
 In Case 2, the higher-permeability shear zone concentrates flow, leading to stronger depletion of the rock oxygen isotope composition. In the connected configuration, relatively depleted values (~8‰) are reached at the base of the recharge fault zone, where cool fluids reach the shear zone (Figure 10). Fluids are progressively heated along the shear zone (Figure 8), where composition is less affected (~12‰). In the discharging fault zone, the same pattern as in Case 1 is observed, but more pronounced, with values down to ~7‰ in the discharging fault zone. In the disconnected configuration (Figure 11), the isotopic pattern is similar to Case 1, but depletion at the bottom of the discharging fault zone is more pronounced (~4‰). The depletion spreads outward from the discharging fault zone.
 In Case 3, isotope results for both the connected and the disconnected configurations show a pattern intermediate to Cases 1 and 2, except that the block between the two fault zones is more depleted because of the more pronounced channelized fluid flow induced by the faults (Figures 10 and 11).
 Case 4 has a more permeable crust (10−17 m2), and fluid flow through the system is faster (Figure 6). A lower permeability contrast between the crust and the fault zones (here only 2 orders of magnitude) allows “leakage” of fluids through the crust. Therefore, in the connected configuration (Figure 10), a strong depletion in oxygen isotope values is observed in areas of high hydraulic head gradients. At the top of the recharge fault zone, the fluid flux is too high, compared to the rate of fluid-rock oxygen isotope exchange, to affect the rock composition, which maintains a value of ~12‰. However, the rock in the upper crust near the fault zone shows oxygen isotope values as low as 6‰, indicating strong fluid-rock oxygen-isotope interaction. On the right side of the model, the isocomposition curves (Figure 10) show the same deflection as the isotherms (Figure 8). Along the discharging fault zone, high fluid flux strongly depletes the oxygen isotope values (down to ~4‰). In contrast, discharge fluid flux on the left side of the model is sufficiently vigorous to maintain the starting oxygen isotope composition. A similar depletion pattern is observed for the disconnected configuration (Figure 11), with oxygen isotope values as low as ~4‰ along the fault zones. The shear zone in the disconnected configuration, however, almost preserves its original oxygen isotope composition of ~12‰. In both the disconnected and the connected configurations, Case 5 is very similar to Case 4, except that the block between the two fault zones is more depleted in oxygen isotopes.
 The oxygen isotope distribution in Case 6 is similar to Cases 4 and 5, with a more pronounced depletion along the fault zones. Since the low-permeability (10−19 m2) fault zones largely act as barriers to fluid flow, intense fluid-rock interaction occurs on the right side of both fault zones, yielding very depleted oxygen isotope compositions (down to ~3‰). The high-permeability (10−16 m2) shear zone, which provides a connection between the high-elevation block and the low-elevation block, channels fluid flow and therefore enhances fluid-rock interaction, resulting in low oxygen isotope values (~3‰) in the connected configuration. In the disconnected configuration, the fault zones still show significant oxygen isotope depletion (down to ~3‰), but the shear zone is much less affected (~11‰).
 Here we discuss the effects of buoyancy-driven flow on fluid flow, temperature, and oxygen isotope fields (section 6.1), fluid flow patterns and rates as well as resultant temperature and oxygen isotope distributions (section 6.2) and comparisons with field observations (section 6.3).
6.1 Effect of Buoyancy
 Fluid buoyancy owing to temperature-induced density inversions affects fluid flow patterns, temperature, oxygen isotope transfer, and fluid-rock isotope exchange. When buoyancy is turned off (Figure 12), fluid flow is driven only by hydraulic head gradients owing to water table (topography) gradients, here from the high-elevation block on the right to the low-elevation block on the left (Figure 5). When buoyancy is turned on (Figure 12), a significant upward component is added to the fluid flow, which in turn causes significant downward flow due to conservation of fluid mass. Buoyancy-driven upward flow in the lower crust and topography-driven flow in the upper crust meet in the model, such that flow lines converge on the left side of the model. Even though there is a significant difference in the flow direction, the magnitude of fluid flow, which is largely controlled by permeability [Manning and Ingebritsen, 1999], shows little to no variation. When buoyancy is on, upward flow in the lower crust transfers significant heat, which leads to a different temperature distribution compared to the buoyancy-off case (Figure 12), and the temperature field shows positive deviations of about 25°C compared to the case when buoyancy is off, although the model isotherms have essentially the same shape and distribution. This temperature increase is not sufficient to affect oxygen isotope fluid-rock exchange, and isotope values remain the same whether buoyancy is turned on or off.
6.2 Effects of Permeability Fields and a High-Permeability Connection Between the Fault Zones on Temperature and Oxygen Isotope Distributions
 Our motivation to connect the subvertical fault zones by a horizontal, high-permeability channel within the shear zone is to strongly enhance fluid flow through the system and, in particular, along the detachment shear zone. Detachment zones usually develop at the brittle-ductile transition, where deforming rocks develop a strong foliation [Lister and Williams, 1979; Davis and Coney, 1979; Bell, 1981]. This foliation is usually subhorizontal and thus constitutes a major permeability anisotropy that promotes fluid flow in the horizontal direction [Sibson, 1996; Zharikov et al., 2003]. Connecting the fault zones is a way to take into account such anisotropy without introducing anisotropic permeability tensors. Without this enhanced connection, fluid flow is controlled by the permeability contrast between the shear zone, the fault zones, and the upper/lower crust. If this permeability contrast is small (one order of magnitude or less), then flow is not sufficient to produce major thermal and/or isotopic perturbations.
 Field data also show that the most depleted oxygen isotope values are measured within the detachment shear zone [Mulch et al., 2006]. δ18O of quartz are typically several per mil lower in the shear zone than in the protolith (section 2, Table 1, and Figure 3). Such degree of oxygen isotope depletion can only be achieved by enhanced fluid flow through the detachment shear zone, hence our motivation to connect the two fault zones.
 In the disconnected configuration, isotherms are relatively flat around the shear zone, with a temperature of about 250°C. In contrast, in the connected configuration, cool fluids from the high-elevation block are focused into the shear zone, deflecting the isotherms downward by more than 50°C (Figure 12). At the same time, the isotherms become subvertical in the shear zone, leading to a temperature increase along the flow path within the shear zone of about +75°C from the right to the left of the shear zone. This effect becomes stronger as the permeability contrast between the crust and the shear zone increases and fluids are more channeled into the shear zone.
 The flow and temperature fields are reflected in the oxygen isotope distribution. In the connected configuration, enhanced flow in the upper part of the recharge fault zone produces strong isotope depletion of the rock (down to δ18O = ~5‰). However, as fluid migrates toward the bottom of the recharge fault zone, flow slows and isotope exchange is limited, resulting in δ18O values of ~7‰ in the crustal rocks. In the disconnected configuration, we observe progressively lower δ18O values along the fluid flow path through the recharge fault and shear zone, where δ18O values reach ~10‰. In the connected configuration, the rocks that form the core of the fault zones usually remain at a higher oxygen isotope value than the surrounding rocks outside the fault zones. This could be explained by the “leakage” of fluids away from the fault zones and into rocks that are at higher temperatures. Transport within the fault zones is too fast, compared to fluid-rock oxygen isotope exchange rates, to cause significant oxygen isotope depletion in the fault zones themselves. This effect is more pronounced as the fault zone/crust permeability contrast increases and is thus very strong in Case 5.
 The case that best documents these relations is Case 5, where the fault zone/crust permeability contrast is 2 orders of magnitude. The low-permeability recharge fault zone brings cool fluids down to the shear zone. In the disconnected configuration, little deflection effect of the isotherms is observed, because the fluids do not flow efficiently past the lower end of the recharge fault zone. In the connected configuration, however, the fluids have an “outlet” from the recharge fault zone into the shear zone, and thus the recharge fault zone experiences significant cooling (T = 100°C at 5 km depth). In this case, the fluid flow continues into the shear zone, where it is heated along its path, yielding a strong temperature gradient along the shear zone of ~ 150°C from the recharge to the discharge fault zone. In this case, the temperature at the base of the shear zone is ~200°C. In contrast, in the disconnected configuration, the temperature at the base of the shear zone is ~300°C. Hence, fault zone connection has a very strong influence on both temperature and oxygen isotope distribution around the fault and shear zones.
 Another important observation is the shape of the isotherms in the vicinity of the discharge fault zone. In the disconnected configuration, isotherms are deflected downward at the bottom of the discharge fault zone. In the connected configuration, no such feature is observed and isotherms are always deflected upward in the discharging fault zone. Along the discharging fault zone, the isotopes tend to show a gradient from higher δ18O at the bottom end of the discharge fault zone to lower values near the surface. The magnitude of this δ18O gradient depends on the fault zone/crust permeability contrast. When the contrast is higher, the gradient is stronger.
 Finally, significant heat perturbations are typically observed in the near-surface basin. The high permeability of the basin “attracts” the fluid coming up the fault, leading to heating of ~ 50°C in the vicinity of the discharge fault zone. In the connected configuration, substantial discharge of hot fluids produces an even stronger temperature perturbation in the basin near the discharge fault zone, nearly 100°C. Although this strong thermal perturbation is observed, flow of the hot fluid is too fast, compared to the isotopic exchange rate, to produce major perturbations of the δ18O composition.
6.3 Comparison with Field Observations
 Extensional tectonics, driven by orogenic collapse and leading to the development of metamorphic core complexes, generates high-amplitude relief, for example through domino-style tilting of upper crustal blocks as observed in the present topography of the Basin-and-Range province [e.g., Lister and Davis, 1989]. Hot ductile lower crust is brought in contact with the brittle upper crust that thins by normal faulting and locally opens major crustal-scale conduits for upward or downward fluid flow, which enhances heat transfer [e.g., Sibson et al., 1988; Lopez and Smith, 1995; Sibson and Scott, 1998]. In such a system, fluid circulation is driven by the high-amplitude topography and the fluid buoyancy that is generated by the high heat flux in the lower crust [Woodbury and Smith, 1988; Raffensperger and Garven, 1995a, 1995b].
 Our numerical lattice-Boltzmann (LB) fluid flow model attempts to reproduce this scenario and shows that significant heat and oxygen isotope perturbations result from such fluid flow of originally meteoric waters and that the magnitude of these changes depends on the permeability contrast between the different hydrologic units. Results show that significant heat transfer and isotope depletion of crustal rocks occurs when the permeability contrast between the fault zones and the crust is at least two orders of magnitudes (10−15 m2 to 10−14 m2 for the fault zones, 10−19 m2 to 10−17 m2 for the crustal units). Heat transfer and oxygen isotope depletion is more pronounced when the steep fault zones are connected by a horizontal fault zone or “channel” with a permeability similar to the subvertical fault zones that is located within the shear zone. In our model, this connecting channel is 150 m thick (like the fault zones) and is comparable in nature to the development of strongly foliated mylonitic or cataclastic shear zones of similar thicknesses (Figure 5).
 In nature, very high preserved transient thermal gradients have been reported based on oxygen stable isotope thermometry studies, ranging from 82°C/38 m to 140°C/100 m [Whipple Mountains, Morrison, 1994; Raft River Mountains, Gottardi et al., 2011, respectively]. Such sharp transient thermal gradients are the result of crust thinning and heat advection. Most detachment shear zones that develop during the exhumation of metamorphic core complexes probably evolved at a shallow dip angle (for example in the Raft River Mountains, Wells et al., 2000; in the Snake Range, Miller et al., 1999), a process that can lead to a high metamorphic gradient only if the shear zone accommodates large displacement. Thinning of the lower crust also leads to heat advection. Even though our model does not include deformation, it produces instantaneous thermal gradients across the shear zone of up to ~75°C/km. Coupling extensional deformation to the model could lead to major compression of the isotherms and even higher thermal gradients, as observed in large-scale deformation models [Rey et al., 2009a, 2009b]. Alternatively, transient variations in permeability could result in significant lateral heat transport [Weis et al., 2012] in a highly channelized fluid flow system.
 Another outcome of the numerical models is the static horizontal temperature gradient along the shear zone between recharge and discharge fault zones that is significant (~25°C/km) and could affect rocks sampled along the horizontal fluid transport direction. However, our 2-D models do not allow fluid flow out of the plane of the model and instead rely on the presence of recharge and discharge faults that can be variably connected to drive fluid flow. In nature, it is likely that convective fluid flow occurs along a single fault surface as a function of its evolving permeability structure, for example, during and between seismic cycles (seismic pumping) [Person et al., 2007]. Therefore, the horizontal temperature gradients obtained in our model are not necessarily developed in nature along the motion direction.
 Our numerical thermal results also show that depending on the permeability of the upper crust, significant cooling of the recharge fault zone and the recharge basin can be produced as well as heating of the discharge fault zone and the discharge basin. The magnitude of these perturbations (up to ± 50°C) is sufficiently large to affect low-temperature thermochronometers.
 The numerical model constructed in this study yields temperatures in the upper crust and the shear zone that are lower than temperature of mylonitization estimated from field studies: the highest temperatures are produced in Cases 1 and 2, where the temperature at the top of the shear zone (~7 km deep) is ~200°C (160°C, 140°C, and 110°C for Cases 3, 4, and 5, respectively). Field data (Table 1 and Figure 3) shows that the mid-crustal shear zone of a metamorphic core complex develops at ~5 to 8 km depth, at temperatures ranging from 300°C to 500°C, substantially higher than our modeling results. This discrepancy between our modeling results and the measurements has several implications. First, if the modeling results are taken at face value, mid-crustal shear zones develop much deeper in the crust, probably around 10–15 km. Our model setup is designed to simulate the early extension stage, when the crust is still thick and the detachment is still deep. At this stage, the brittle-ductile transition is likely at ~10 to 15 km depth, given the depth limit of seismicity that is observed in zones of hot crust, like Tibet [e.g., McKenzie and Fairhead, 1997; Maggi et al., 2000; Watts, 2001; Watts and Burov, 2003; Handya and Brun, 2004]. These observations are consistent with our pressure estimates, although temperature is usually much better constrained than pressure.
 Second, we used a 45°C/km geothermal gradient in our calculation, so if we want to reproduce the natural measurement (~400°C shear zone at ~7 km depth), we would need to use a much higher geothermal gradient. This implies that natural detachment system may develop under higher geothermal gradients than usually estimated. Unfortunately, the LB treatment employed here restricts the geothermal gradient that can be used to be less than 55°C/km, so that the effects of larger geothermal gradients could not be tested. Finally, if the temperatures in natural systems are higher, then more fluid-rock isotope exchange is expected since elevated temperature enhances fluid-rock isotope exchange.
 As mentioned in section 5.3, the stable isotope distributions presented here are final steady-state values, and should be interpreted carefully when compared to field data. Metamorphic core complexes are short-lived structures, exhumed on a million year timescale (Table 1 and Figure 2). Hence, the oxygen stable isotope distribution of the upper 3 km of the model, where the temperature is low, is unlikely to be observed in nature, because the time needed to achieve isotopic equilibrium at such low temperatures is unrealistically long. However, the modeled isotopic shifts at the depth of the shear zone connecting the high-angle faults correlate well with natural observations. This validates the common assumption that rock and fluid reach isotopic equilibrium in natural systems.
 Typical quartz δ18O compositions from the unmylonitized core of Cordilleran metamorphic core complexes is ~ 13‰ [Kettle Dome, Mulch et al., 2004, 2006; Bitterroot, Foster et al., 1997, 2002] and the lowest quartz δ18O values range from 7.90‰ [Morrison, 1994] to 9.8‰ [Bitterroot, Foster et al., 1997, 2002; Gottardi et al., 2011], 10‰ [Thor-Odin, Mulch et al., 2004, 2006], and 11.4‰ [Snake Range, Gébelin et al., 2011]. However, the extreme depletion observed in the Ruby Mountains [δ18Oqz = 2.8‰, Fricke et al., 1992, and 5.0‰ in the Kettle dome, Mulch et al., 2007] are not predicted. In addition, several studies show that in general, surface fluids do not penetrate into the lower crust below the detachment fault zone [Fricke et al., 1992; Wickham, Taylor, Snoke and O'Neil, 1991] but do penetrate to the detachment shear zones and the ductile layer in the footwall of the detachment fault [Mulch et al., 2004, 2006, 2007]. This is observed in our model as well: the oxygen isotope depletion is restricted to the fault zones and along the detachment zone at the interface between upper and lower crust, with oxygen isotope ratios lowered from 13‰ (starting value) down to ~ 6‰. There are few studies of stable isotopes from supra-detachment basins available in the metamorphic core complex literature. A detailed study [Wickham et al., 1993; Bickle et al., 1995] of the Lizzies Basin, which developed during the exhumation of the Ruby Mountains, shows preservation of high δ18O values in rocks of the basin (12 to 13‰.). Similar oxygen isotope distribution patterns are produced in our model. Fluids approaching the ground surface within a discharging fault zone spread into the high-permeability basin, where the temperature is not high enough and the flow rate is too large, compared to fluid-rock oxygen isotope exchange rates, to alter oxygen isotope composition, thereby preserving the original oxygen isotope composition of the rocks in that region.
 Several oxygen and hydrogen isotope studies of North American core complexes show that circulation of meteoric fluids during the development of the detachment shear zone is ubiquitous. The fluid-rock oxygen isotope exchange is a result of the interplay between rock type, temperature, fluid flow, duration of isotope exchange, permeability, and fluid pathway configurations.
 This study presents the results of continuum-scale (i.e., large-scale, partial bounce-back) lattice-Boltzman fluid, heat, and oxygen isotope transport simulations of an idealized cross section of a metamorphic core complex. The simulations investigate the effects of crust and fault zone permeability fields as well as buoyancy-driven flow on two-way coupled fluid and heat transfer and resultant exchange of oxygen isotopes between meteoric fluid and rock. The model is composed from bottom to top of a lower crustal block, a shear zone and three upper crustal blocks that are separated from one another by subvertical fault zones. The model has the possibility to connect or disconnect the two fault zones, allowing significant fluid flow through the subvertical fault zones in the connected configuration.
 The role of buoyancy was investigated and the results show that, although buoyancy affects fluid flow direction, the magnitude of fluid flow (controlled by permeability) shows little to no variation. Furthermore, when buoyancy is on, upward flow in the lower crust transfers significant heat, leading to temperature perturbations of about 25°C compared to the buoyancy-off case, although the model isotherms keep the same shape and distribution. This temperature perturbation is not sufficient to affect oxygen isotope fluid-rock exchange, and, therefore, isotope values remain the same whether buoyancy is turned on or off.
 The results of our numerical model suggest that the characteristics of the permeability field primarily control the flow of meteoric fluids and the resultant distribution of temperature and oxygen isotopes. Results show that fluid migration to middle to lower crustal levels is fault controlled and depends primarily on the permeability contrast between the fault zones and the crustal rocks. High fault/crust permeability ratios (two orders of magnitude) lead to channelized flow in the fault and shear zones while lower ratios allow leakage of the fluids from the fault zones into the crust.
 Strong thermal gradients (vertical and lateral) are produced by our model when the subvertical fault zones are connected, i.e., when fluid flow is channelized into the horizontal shear zone between the bottom ends of the fault zones, corresponding, in nature, to the development of planar anisotropy in the shear zone due to foliation-enhancing horizontal flow. The disconnected configuration shows relatively flat isotherms in the shear zone, with a temperature of about 250°C at the top of the shear zone. In contrast, in the connected configuration, draining of cool fluids from the high-elevation block is focused into the shear zone, strongly deflecting the isotherms downward by 50°C up to 150°C. At the same time, horizontal gradients within the shear zone are also observed, with gradients from the right to the left of the shear zone ranging from 75°C to 150°C. This effect becomes stronger as the permeability contrast between the crust and the shear zone increases and fluids are channeled more into the shear zone causing more advective heat transfer.
 The oxygen isotope results show profound oxygen depletion (starting value of δ18O = +13‰ down to 4‰) concentrated along the fault and shear zones, similar to field data. The connected configuration produces the strongest isotopic depletion of the rock, with values down to δ18O = ~5‰. However, the isotope depletion is limited by the ratio between transport rate and fluid-rock oxygen isotope exchange rate. In some cases, the transport within the fault zones is too fast, compared to fluid-rock oxygen isotope exchange rates, to cause significant oxygen isotope depletion in the fault zones themselves. This effect is more pronounced as the fault zone/crust permeability contrast increases.
 Our two-dimensional simulation show that the results are non-unique, different permeability combinations can produce similar temperature and oxygen stable isotope distribution. However, the results of this study show that profound thermal and isotopic perturbations, both horizontal and vertical, can be achieved by deep fluid circulation, which would be worth testing in areas with sufficient field exposure. Our results also show that the evolution of the rock fabric, in the fault zones, but especially in the shear zone, with the development of planar anisotropy-like foliation, profoundly impacts fluid flow and, hence, also temperature and isotopic depletion patterns. Adding crustal deformation models to our or similar numerical, large-scale fluid, heat, and oxygen isotope transfer model would be the next step in investigating further the role of fluids in active tectonic settings.
Appendix A: Scaling Analyses for Groundwater Flow, Heat Transfer, and Isotope Transport and Fluid-Solid Exchange
 In order to scale the mesoscopic dimensionless lattice-Boltzmann algorithms to physical fluid flow, and heat transport with fluid-solid isotope exchange over large spatial scales, we apply scaling analyses to establish the relationship between the numerical and physical unit systems.
A1 Length Scaling
 Dimensional (physical) spatial length scale for unit grid size is chosen by Δxphy = 150 (m) for current simulations. Thus, the physical length scale (Lphy) can be computed by
where NLB is grid numbers used in the computational domain.
A2 Pressure Scaling
 In LB algorithm, numerical pressure is determined by the equation of state, i.e., , where cs is the speed of sound, and ρLB is numerical fluid density. The numerical-physical pressure relationship can be normalized as
where and Pphy is physical pressure, which is determined by ρfg h (h is the hydraulic head).
A3 Temperature Scaling
 Normalization is also applied for the numerical-physical temperature relationship with the maximum TLB. max = 1.0 and minimum TLB. min = 0.0 values, so that
A4 Crust Permeability Scaling
 For simulating geofluid cases here, we implement the second-order partial-bounce-back LB scheme, in which, the available value range of numerical permeability is restricted by kLB = 0.5833 ~ 0.0072. However, the practical (physical) permeability range in Earth's crust is usually within the range: kphy = 10− 14 ~ 10− 19 (m2). In order to overcome this numerical limitation and difficulty, a logarithm-normalization algorithm is proposed to model the numerical-physical permeability relationship, such that
 Once physical and numerical maximum/minimum permeabilities are defined by kLB. max = 0.583333333, kLB. min = 0.007201646, kphy. max = 10− 14 (m2), and kphy. min = 10− 19 (m2) in equation (A4a). Permeability scaling equations can then be obtained:
 So that , where
and , where
A5 Velocity Scaling
 A format for numerical-physical velocity relationship is chosen by
where q is Darcy's velocity in porous media and ΔtFlow/Thermal. phy is the physical time scale (seconds) per unit numerical time step (ΔtLB = 1) for fluid flow and thermal transfer, as modeled in section A6.
 However, the grid number mapped in faults zones (Figure 5) shall be NLB. Faults > 4 to provide the sufficient resolution in computing accurate numerical results, where the computational fault width is larger than the real (physical) width in fault layers. Hence, we apply a velocity correcting factor (qwt), based on the flow-rate conservation principle, to obtain the reality velocity inside faults zones:
 In current simulations, the characteristic fault width is chosen by 100 (m) and mapped by NLB. Faults = 5, and thus, qwt = 7.5 shall be used. Note that the velocity correction equation (A5b) is only applied to correct the physical velocity inside faults zones. We do so for the important reason that, the computation does not need to implement complicate grid techniques or numerical schemes, e.g., the local-refinement grid treatment with special simulation algorithms. Hence, the computational algorithm can be maintained as simple as possible, and the simulation efficiency is much improved with sufficiently accurate results.
A6 Time Scaling for Fluid Flow and Thermal Transfer
 For modeling the time scale of fluid flow and thermal transfer, we apply an “effective 1D Darcy's formation” to represent the overall heterogeneous medium flow. In the numerical unit system, the equation is considered by
where (ρLB ⋅ qLB)Overall is the overall characteristic flow moment inside the heterogeneous medium, which is solved by partial-bounce-back LB computation
 ΔPLB. Eff is the effective boundary pressure-difference, which is determined by where PH, PM, PL and NH, NM, NL are the numerical boundary pressures and boundary grid numbers at higher, middle, and lower elevation upper boundaries (Figure 5).
kLB. Eff/LLB. Eff is a ratio of the effective permeability to the effective characteristic flow length; these values are needed to be determined.
 When (ρLB ⋅ qLB)Overall, ΔPLB. Eff, and νLB are specified or computed, kLB. Eff/LLB. Eff can then be obtained by equation (A6a). Subsequently, the kLB. Eff and LLB. Eff can further be computed by the following procedures:
Using a weighting approach to evaluate the effective numerical solid fraction (ns. Eff),
where NTotal is the total grid number used for the computational domain, and is the sum of grid numbers of crust layers, and ns is the numerical solid fraction in those crust layers.
kLB. Eff is obtained by the equation for second-order partial-bounce-back LB scheme:
LLB. Eff can be computed using the kLB. Eff and (kLB. Eff/LLB. Eff) values.
 In the physical unit system, considering the logarithm relationship for the physical and numerical permeabilities, the 1-D effective Darcy's formation shall be formulated as
 In equation (A6d), νphy and ρphy are all given values and kR is computed by equation (A4b). ΔPphy. Eff can be evaluated using the approach for determining the numerical boundary pressures modeled in equation A6a, where the physical boundary pressures (PH, PM, PL) and physical upper boundary lengths (LH, LM, LL) are used to instead of the numerical pressures and grid numbers, respectively. Furthermore, the relationship between kphy. Eff and kLB. Eff shall satisfy equation (A4c), and Lphy. Eff is obtained by LLB. Eff using equation (A1). Applying this procedure, qphy can be determined using equation (A6d). Hence, the physical timescale for fluid flow and thermal transport can further be obtained by
 Note that the qLB. Overall value is directly obtained by partial-bounce-back LB simulation.
A7 Heat Flux Scaling for Crust-Bottom Boundary
 The physical crust-bottom heat flux satisfies the Fourier's Law, i.e.,
where qBtm. phy is the physical crust-bottom heat flux (with unit W/m2), krk is the thermal conductivity (W/°C m) at bottom crust, and (dT/dh)phy is the physical temperature gradient (°C/m) along vertical (y) direction When the heat flux (qBtm. phy) and thermal conductivity (krk) are given, the temperature gradient (dT/dh)phy can be determined. The numerical temperature gradient per each LB element (ΔxLB) can then be obtained by the length and temperature scaling relationships, such as
A8 Buoyancy Scaling
 For evaluation of the physical characteristic buoyant velocity (qBouy. phy: m/s) within a porous medium, the logarithm permeability relationship shall be taken into account and has a format of
where is the thermal expansion coefficient; is the reference fluid density (kg/m3), chosen by the temperature (25°C) specified at the ground surface (upper) boundaries; gphy is the physical gravity (m/sec2) along the vertical (y) direction; and vphy is physical fluid kinematic viscosity (m2/s.
 Note that this format for modeling the characteristic buoyant velocity (qBuoy) is actually identical with the format in equation (1) because of . In LB algorithm, the buoyant body force term is treated as [Kao and Yang, 2007]
where ΔtLB is the unity numerical time step, i.e., ΔtLB = 1, and ΔTLB is the numerical temperature difference between ground surface (upper) boundary and local boundary, and shall satisfy the temperature relationship of equation (A3).
 As the numerical-physical velocity relationship shall satisfy equation (A5a), the appropriate value range of (β g)LB can be found within the range of 0.00034 ~ 0.00026. For current simulations, we fix the (β g)LB = 0.0003 as a constant.
A9 Oxygen Isotope Scaling
 In isotope transport/exchange LB computations, we apply normalized (numerical) oxygen isotopes, i.e., Rf. LB and Rs. LB, to represent physical δ18O isotope components (with unit ) in the fluid phase and solid phase, respectively, which satisfy the following relationships:
 Once the numerical (Rf. LB and Rs. LB) and physical (δ18Of. phy and δ18Os. phy) oxygen isotopes are solved by LB simulation and by equations (A9a) and (A9b), the physical phase oxygen isotope ratios (R) can then be obtained by
where is the isotopic standard for oxygen with a value of 0.0020052 based on mean sea water composition (SMOW).
A10 Time Scaling for Oxygen Isotope Transport and Exchange
 Because the LB computations of fluid flow/thermal transfer and for isotope exchange are performed individually in the present investigation, a scaling analysis for the relative timescale between those transport phenomena is essential. We model the timescale ratio of the flow/thermal behaviors to isotope exchange by means of the Lewis number, which is defined by Le ≡ DThermal/DIsotope where D are the thermal and/or solute (isotope) diffusivities. Since the length scales of unit grid size (Δx) in flow/thermal field and in isotope field are identical (grid size is fixed for a non-deformable crustal topography), the Lewis number can be regarded as the timescale ratio of those transport phenomena, i.e.,
 Then a relative time factor (CTime) is defined by
where the physical and numerical Lewis numbers, i.e., Lephy and LeLB, shall be determined by their respective thermal and solute diffusivities applied in geofluid computations. Note that for the reason of computational stability using LB algorithms, those numerical properties can be specified with values much different from the real physical values.
 Finally, the timescale relationship between isotope exchange and the flow/thermal transport can be determined by
where ΔtFlow/Thermal. phy is obtained by the model formulated in section A6.
 In current investigation, we apply LeLB = 6.5439248 for LB simulations and then get CTime ≈ 1.93 × 103 for physical crust porosities range: φ = 0.03 ~ 0.08.
 We gratefully acknowledge support for this research by the National Science Foundation (NSF) under grants EAR-0838541 and EAR-0941666. M.O.S. also thanks the George and Orpha Gibson endowment for its generous support of the Hydrogeology and Geofluids research group. Finally, we thank G-cubed editor Joel Baker and two anonymous reviewers for their detailed reviews and constructive advice.