Introduction

Porphyry Cu and epithermal Au deposits are significant sources of metals in hydrothermal deposits worldwide. They form at depths < 10 km in the upper crust, with porphyry Cu ± Au ± Mo deposits centered on and around the magmatic intrusion, and vein-type Zn-Cu-Pb-Ag ± Au epithermal deposits in shallow depth above the intrusion or on the side of the intrusion (Tosdal and Richards 2001; Sillitoe 2010; Dilles and John 2021). Large porphyry Cu deposits and high-sulfidation epithermal Au deposits are mainly found in both continental and oceanic volcanic arcs, which are largely associated with magmatic activities in subduction-related scenarios; low-sulfidation epithermal deposits tend to form in extensional back-arc or rift settings, showing different tendencies of ore deposit formation associated with different tectonic environments (Fig. 1; Cooke and Hollings 2005; Richards 2013; Dilles and John 2021).

Fig. 1
figure 1

Modified from Richards (2013)

Key features of giant porphyry Cu deposits and epithermal deposits in compressional volcanic-arc settings and extensional back-arc or rift settings. Purple boxes highlight the features or processes that may extremely promote these systems to form giant deposits. 

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Geological reviews in the last decade have commonly recognized that the formation of ore bodies in porphyry and epithermal deposits is controlled not only by the geochemical evolution of magmatic-hydrothermal processes, but also by deformation structures (fractures, breccias, and faults) spatially and genetically associated with mineralizing intrusions (Tosdal and Richards 2001; Guillou-Frottier and Burov 2003; Cooke and Hollings 2005; Poulet et al. 2012; Richards 2013; Wilkinson 2013; Japas et al. 2021). The fractures caused by hot magmatic intrusion can transport ore-bearing fluids, enhance the water-rock reactions and hydrothermal alterations, and provide space for hydrothermal orebody precipitation (Tosdal and Richards 2001; Sillitoe 2010; Richards 2013; Lamy-Chappuis et al. 2020; Dilles and John 2021; Japas et al. 2021). Thus, studying the formation mechanism of these fractures in relation to mineralizing intrusions has important implications for the hydrodynamics and mineralization of hydrothermal deposits.

Extensive studies of porphyry and epithermal deposits have geochemically analyzed the sources of ore-forming fluids and elements and used chronology to infer the dynamic relationship between tectonic evolution history, magmatic activities, and ore genetic processes (Hou and Cook 2009; Richards 2011; Mao et al. 2014; Hou et al. 2015; Sun et al. 2016). This research paradigm of mineral deposit studies is particularly helpful for understanding the metallogenic processes at the crust-mantle scale; however, the analysis of the shallow metallogenic dynamic processes at the upper-crust scale is insufficient, especially the ore-bearing fracture formation and fluid-focusing mechanism around the mineralizing magmatic intrusion under different tectonic backgrounds, which could have very important influences on the emplacement of orebodies (Tosdal and Richards 2001; Richards 2013; Dilles and John 2021; Japas et al. 2021).

Besides in situ geology observations and indoor physical simulations, numerical modeling has become a quantitative and parametric research method for studying fracture formation and fluid flow patterns around intrusions at shallow depths (< 10 km) (Roselle et al. 2003; Guillou-Frottier and Burov 2003; Hurwitz et al. 2007; Rozhko et al. 2007; Zhang et al. 2007, 2012; Theissen-Krah et al. 2016). In the second half of the last century, combining the elasticity solutions of internal stress and the crack criterion is the main methods to calculate the distribution patterns of different fracture types (Concentric or radial, tensile or shear) under specific stress boundaries or around the intrusion (Hafner 1951; Koide and Bhattacharji 1975). This analytical method is easy to analyze and understand, but lack of the constraint of plastic yield criterion. Around the early twentieth century, finite element method was used to study fracture formation mechanism in elasto-plastic materials around magmatic intrusions, considering different chamber shapes, top intrusion pressures and mechanical parameters (Sartoris et al. 1990), or focusing on the thermo-mechanical processes of fracturing above plutonic apexes in porphyry ore deposits (Guillou-Frottier and Burov 2003). However, these numerical models did not consider the effect of tectonic background. In recent years, the effects of different tectonic backgrounds (compression and extension) on the development of distinct fracture patterns have been analyzed by several studies, which estimated the control of overpressure on fracture formation (Rozhko et al. 2007) and the fracture pattern in shear-zone-type gold deposits around a hot magma intrusion (Zhang et al. 2012). In the research field of hydrothermal fluid flow, the coupling of thermal and hydrological processes has been well studied by previous studies (Roselle et al. 2003; Hurwitz et al. 2007; Theissen-Krah et al. 2016), and the fluid driving of mechanical processes were received attention by recent studies (Lin et al. 2023). These numerical models did not consider some of the critical factors of the thermal-hydrological-mechanical coupling processes, that is, the influencing factors of fracture formation and fluid flow were not comprehensive in these models.

To analyze the influences of different tectonic backgrounds on fracture formation and fluid flow patterns in porphyry and epithermal deposits, we used finite element models coupled with thermal-hydrological-mechanical (THM) coupling processes to study the mechanism of fracture formation and the hydrology of fluid flowing around a hot magmatic intrusion under different structural settings (compression and extension). The numerical models used in this study have the following advantages compared to those used in the previous studies on fracture forming around intrusion: (1) We used thermal-poro-elasto-plastic materials to study the fracture formation and hydrology of fluid flowing simultaneously, and the equations were solved in a fully coupled way; (2) In order to reproduce the fluid focusing effect in fractures, we added a permeability enhancement factor linking the permeability and the fractural deformation of rock materials (Sun et al. 2020); (3) Finally, we calculated the water-rock (W/R) ratio to estimate the degree of fluid-rock interaction (Schardt and Large 2009; Takahashi et al. 2022), and analyzed how different tectonic backgrounds effect the formation of porphyry and epithermal deposits. This is the first study to analyze fracture formation and fluid flow mechanisms within and around a hot magmatic intursion in upper-crust scale using fully coupled THM model, therefore, the model results can provide a more comprehensive and enlightening analysis than the previous numerical studies.

Genetic geological model of fractures in porphyry Cu systems

Most reviews on the geology of porphyry deposits or porphyry Cu systems refer to cylindrical magmatic intrusions emplaced at shallow depths in the upper crust (< 5 km). These intrusions can be stocks with small-diameter plutonic rocks or relatively small cupolas connected to large deep batholiths (Guillou-Frottier and Burov 2003; Sillitoe 2010; Zheng et al. 2016; Dilles and John 2021). Magmatic intrusions are sources of energy, ore-genetic matter, and fluids (Richards 2013; Dilles and John 2021). Orebodies, wall-rock alterations, and ore-bearing fractures are zoned around them; thus, most reviews regard intrusions as the mineralization centers for the formation of porphyry Cu systems (Tosdal and Richards 2001; Sillitoe 2010; Dilles and John 2021; Japas et al. 2021).

The degree of fracturing plays an important role in the mineralization process; that is, the tectonic setting and structural framework may strongly influence not only the size and form of a hydrothermal mineralization system (Sillitoe 1997) but also its location (Richards et al 2001; Tosdal and Richards 2001). The fracturing structures around magmatic intrusions are not only permeable conduits of ore-forming fluids but also ore-bearing spaces for the precipitation of minerals (Sillitoe 2008, 2010; Richards 2013).

Almost all porphyry metallogenic systems can be simplified to a sketch of a shallow granite cupola intrusion in the center of these systems, with two types of fracture systems associated with the intrusion: one type is a dense fracture network surrounding and within the intrusion; the other type is vein-type fractures and breccias at the top of the intrusion (Fig. 2; Kirkham and Sinclair 1995; Guillou-Frottier and Burov 2003; Sillitoe 2010). The minerals within these two types of fracture systems differ and depend on their distance from the intrusion (Tosdal and Richards 2001). The Cu, Mo, Fe, and Au mineralization that forms at high temperatures usually occurs in the fracture network near and within the intrusions, and the veins and breccias above and around the intrusions contain Pb, Zn, Ag, and Au mineralization at low temperatures (Fig. 2; Kirkham and Sinclair 1995; Sillitoe 2010; Dilles and John 2021).

Fig. 2
figure 2

Genetic anatomy of a porphyry Cu system. This figure shows the spatial relationships of a centrally located porphyry Cu ± Au ± Mo deposit, peripheral carbonate-related deposits in a carbonate unit, and epithermal vein-type deposits surrounding the porphyry intrusion, and presents the alteration zonation and the distribution of fractural networks and veins in and alongside the porphyry intrusion. Modified from Tosdal and Richards (2001) and Sillitoe (2010)

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From a structural standpoint, there are three acknowledged processes that control fracture formation, which further enhance the permeability of rocks and make the fluid pass through and the minerals precipitate through these fractures: the thermally induced fracturing resulting from the thermal expansion of rock skeletons and pore fluid and their differences; the hydrofracturing effect of high pore pressure induced by the expansion and phase separation of pore fluid and low-permeability rock layers; and the mechanical influences of tectonic activities and the intrusion of porphyry stokes or cupolas (Tosdal and Richards 2001; Dilles and John 2021). All three processes control fracture formation in porphyry metallogenic systems, although different factors predominate at different times and spaces.

Methods of numerical modeling

Solid mechanics

The metallogenic process is the same as other geodynamic processes; it is a quasistatic process on the geological timescale. Therefore, we solve the static equilibrium equation and ignore the inertia term:

$${\sigma }_{ij,j}+{f}_{i}=0,$$
(1)

where σij is the component of the total stress tensor (i,j = x, y, z) and fi is the body force in the i direction. In this section, we define tensile stress as positive and compressive stress as negative, following the stress symbol definition of elasticity. In this model, the horizontal body force is equal to zero, and the vertical body force fz = ρmg, where ρm is the density of the whole porous material (the solid grains and the pore space) and g is the acceleration of gravity (= 9.81 m/s2).

Most of the rocks in our models were assumed to be porous materials, except for the impermeable basement rocks. Porous materials consist of solid rock grains and porous fluids. The density of the entire porous material can be expressed as

$${\rho }_{m}={\rho }_{s}\left(1-n\right)+{\rho }_{f}n,$$
(2)

where \({\rho }_{m}\), \({\rho }_{s}\), and \({\rho }_{f}\) are the densities of the entire porous material, solid grains, and pore fluid, respectively; n is the porosity of the porous rock material.

In this study, we assume that the porous materials of rocks are saturated and isotropic, and thus they follow the principle of effective stress; that is, the stresses in the rocks are sustained by both solid grain skeletons and pore fluid:

$${{\sigma }^{\prime}}_{ij}={\sigma }_{ij}+\alpha p{\delta }_{ij},$$
(3)

where \({{\sigma }^{\prime}}_{ij}\) is the component of the effective stress tensor, which represents the stress sustained in the solid grains; p is the pore fluid pressure; δij is the Kronecker delta function (i = j, \({\delta }_{ij}=1\); \(i\ne j\), \({\delta }_{ij}=0\)), and α is the Biot constant:

$$\alpha =1-\frac{K}{{K}_{s}},$$
(4)

where K is the bulk modulus of the entire porous material and KS is the bulk modulus of the solid skeleton. For incompressible grains, the value of KS tends to infinity, and the Biot constant α is equal to 1 (Thornton and Crook 2014; Luo et al. 2015; Simulia 2016).

The thermal expansion of solid grains and pore fluids is an important mechanism of fracture formation around hot magmatic intrusions (Knapp and Knight 1977; Tosdal and Richards 2001; Guillou-Frottier and Burov 2003). The constitutive equations for the coupled thermal expansion effect of the poroelastic materials are expressed as in (Jaeger et al. 2007; Zoback 2007; Bai and Li 2009):

$${{\sigma }^{\prime}}_{ij}=2G{\varepsilon }_{ij}+\lambda \theta {\delta }_{ij}-\alpha p{\delta }_{ij}-3K{\beta }_{s}\left(T-{T}_{0}\right){\delta }_{ij},$$
(5)

where \({\varepsilon }_{ij}=({u}_{i,j}+{u}_{j,i})/2\) is the component of the strain tensor determined by the displacement u; \(\theta ={\varepsilon }_{x}+{\varepsilon }_{y}+{\varepsilon }_{z}\) is the volumetric strain, βS is the linear thermal expansion coefficient of solid grains, T and T0 are the current and reference temperatures, respectively, and λ and G are the Lame’s constants of rock materials. In the theory of elasticity, λ, G, and K can be expressed in terms of the Young’s modulus E and Poisson’s ratio μ:

$$\begin{array}{c}K=\frac{E}{3(1-2\mu )}\\ \lambda =\frac{E\mu }{(1+\mu )(1-2\mu )}\\ G=\frac{E}{2(1+\mu )}\end{array},$$
(6)

The rock must enter a plastic yield state before brittle fracture occurs. We used the Drucker–Prager plastic yield criterion coupling pore pressure effect to describe the plastic mechanical behavior of porous rock materials (Alejano and Bobet 2012):

$$F=\sqrt{{J}_{2}}+{{\alpha }_{DP}{I}_{1}}^{\prime}-{k}_{DP},$$
(7)

where \({{I}_{1}}^{\prime}\) is the first invariant of the effective stress tensor and J2 is the second invariant of the deviatoric stress tensor, which can be further expressed as

$$\begin{array}{c}{{I}_{1}}^{\prime}={{\sigma }_{x}}^{\prime}+{{\sigma }_{y}}^{\prime}+{{\sigma }_{z}}^{\prime}=3{{\sigma }^{\prime}}_{m}={I}_{1}+3p=({\sigma }_{xx}+{\sigma }_{yy}+{\sigma }_{zz})+3p\\ {J}_{2}=\frac{1}{6}\left[{({\sigma }_{x}-{\sigma }_{y})}^{2}+{({\sigma }_{y}-{\sigma }_{z})}^{2}+{({\sigma }_{z}-{\sigma }_{x})}^{2}+6({\tau }_{xy}^{2}+{\tau }_{yz}^{2}+{\tau }_{zx}^{2})\right]\end{array},$$
(8)

where \({{\sigma }_{x}}^{\prime}\), \({{\sigma }_{y}}^{\prime}\), and \({{\sigma }_{z}}^{\prime}\) are the normal effective stresses in the x-, y-, and z-directions, respectively, of the rectangular coordinate system; \({{\sigma }^{\prime}}_{m}\) is the mean effective stress; and \({\tau }_{ij}\) is the shear stress component (i,j = x, y, z). In Eq. (7), αDP and kDP are constants that depend on the internal friction angle φ and the cohesion of solid grains c defined in the Mohr–Coulomb model, respectively. As the Drucker–Prager plastic yield criterion used in this study is the peripheral circle of the Mohr–Coulomb model in the deviatoric stress plane, αDP and kDP can be further expressed as in (Colmenares and Zoback 2002):

$$\begin{array}{c}{\alpha }_{DP}=\frac{2{\text{sin}}\varphi }{\sqrt{3}(3-{\text{sin}}\varphi )}\\ {k}_{DP}=\frac{6c{\text{cos}}\varphi }{\sqrt{3}(3-{\text{sin}}\varphi )}\end{array},$$
(9)

Finally, we used the equivalent plastic strain to represent the cumulative value of the plastic strain in the entire deformation history and approximated the fracture in the rock (Simulia 2016; Ding and Zhang 2017):

$$\begin{array}{c}{\overline{\varepsilon }}^{\text{pl}}=\underset{0}{\overset{t}{\int }}{\text{d}}{\dot{\overline{\varepsilon }}}^{\text{pl}}{\text{d}}t\\ {\text{d}}{\dot{\overline{\varepsilon }}}^{\text{pl}}=\sqrt{\frac{2}{3}{\text{d}}{\dot{\varepsilon }}_{ij}^{\text{pl}}{\text{d}}{\dot{\varepsilon }}_{ij}^{\text{pl}}}\end{array},$$
(10)

where \({\overline{\varepsilon }}^{\text{pl}}\) is the equivalent plastic strain, \(d{\dot{\overline{\varepsilon }}}^{\text{pl}}\) is the increment in the equivalent plastic strain rate, and \({\text{d}}{\dot{\varepsilon }}_{ij}^{\text{pl}}\) is a component of the plastic strain rate increment.

Fluid flowing control and mass conservation

The flow behavior of a pore fluid in a porous medium follows Darcy’s law (Lewis and Schrefler 1998; Craig 2004; Simulia 2016):

$$\bf v=-\frac{k}{{\eta }_{f}}\nabla \left(p-{\rho }_{f}gh\right),$$
(11)

where \(\mathbf{v}\) denotes the pore fluid flow velocity in a porous medium (also known as the Darcy velocity), k denotes the permeability of the porous medium, ηf denotes the viscosity of the pore fluid (for pure water, \({\eta }_{f}=1.0\times {10}^{-3}{\text{Pas}}\)), h denotes the depth from the top of the model to the calculated point, and \(\nabla\) denotes the gradient operator.

To account for the hydraulic effects of permeable fractures and fault zones (Caine et al. 1996; Sibson 1996), we introduced an enhancement factor that led to an increase in permeability with the development of fracture deformation (Sun et al. 2020):

$$k={k}_{0}\cdot {\lambda }_{\text{en}}({\overline{\varepsilon }}^{\text{pl}}),$$
(12)

where k0 is the initial permeability of the rock material, and λen is the enhancement factor dependent on the equivalent plastic strain, \({\overline{\varepsilon }}^{\text{pl}}\). The detailed relationship between λen and \({\overline{\varepsilon }}^{\text{pl}}\) is shown in Fig. 3c.

Fig. 3
figure 3

Model setup. (a) The shape, material distribution and the boundary conditions of the model. Three different cases (undergo no tectonic activity, tectonic compression, and tectonic extension) are established to study the tectonic effects on fracture formation and fluid flowing in and surround the magmatic intrusion, as shown by the subgraphs on the right side. (b) The meshes of model. The vertical dashed line A-A’ and the horizontal dashed line B-B’ are the profiles used to analyze the stress evolutions in Figs. 5 and 6, respectively. (c) The relationship between permeability enhancement factor and equivalent plastic strain in our model

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The mass-conservation equation for a porous medium coupled with the thermal expansion effect can be expressed as in (Selvadurai and Nguyen 1995; Bai and Li 2009; Selvadurai and Najari 2017):

$$\nabla \cdot v=-\alpha \frac{\partial \theta }{\partial t}+{\alpha }_{T}\frac{\partial T}{\partial t}-{\alpha }_{p}\frac{\partial p}{\partial t},$$
(13)
$$\begin{array}{c}{\alpha }_{p}=n(\frac{1}{{K}_{f}}-\frac{1}{{K}_{s}})+\alpha \frac{1}{{K}_{s}}\\ {\alpha }_{T}=3[n{\beta }_{f}+(\alpha -n){\beta }_{s}]\end{array},$$
(14)

where αp and αT are the coupling coefficients of pore fluid pressure and temperature, respectively, and βf is the thermal expansion coefficient of the pore fluid. In this study, it is assumed that both the solid particles and the pore fluid are incompressible; therefore, the values of KS and Kf are infinite, and the value of αp is zero.

By combining Eqs. (11) and (13), we obtain the following mass-conservation equation:

$${\alpha }_{p}\frac{\partial p}{\partial t}\text{+}\nabla \cdot \left[-\frac{k}{{\eta }_{f}}\nabla p\right]+\alpha \frac{\partial \theta }{\partial t}-3[n{\beta }_{f}+(\alpha -n){\beta }_{s}]\frac{\partial T}{\partial t}=0.$$
(15)

Heat transfer

Assuming that the temperatures of the solid grains and pore fluid in the porous rock material are equal, the heat transfer equations for the solid grains and pore fluid can be obtained respectively (Nield and Bejan 2006):

$$(1-n){\rho }_{s}{c}_{s}\frac{\partial T}{\partial t}=(1-n)\nabla \cdot ({k}_{s}^{T}\nabla T)+(1-n){Q}_{s}^{T},$$
(16)
$$n{\rho }_{f}{c}_{f}\frac{\partial T}{\partial t}+{\rho }_{f}{c}_{f}v\cdot \nabla T=n\nabla \cdot ({k}_{f}^{T}\nabla T)+n{Q}_{f}^{T},$$
(17)

where \({c}_{s}\) and cf are the specific heat of the solid grains and pore fluid, respectively; \({k}_{s}^{T}\) and \({k}_{f}^{T}\) are the thermal conductivities of the solid grains and pore fluid, respectively; \({Q}_{s}^{T}\) and \({Q}_{f}^{T}\) are the thermal source terms of the solid grains and pore fluid, respectively.

Combining Eqs. (16) and (17), the heat transfer equation of a porous medium can be expressed as

$$ {\rho }_{m}{c}_{m}\frac{\partial T}{\partial t}+{\rho }_{f}{c}_{f}v\cdot \nabla T=\nabla \cdot ({k}_{m}^{T}\nabla T)+{Q}_{m}^{T},$$
(18)
$$\begin{array}{c}{\rho }_{m}{c}_{m}=(1-n){\rho }_{s}{c}_{s}+n{\rho }_{f}{c}_{f}\\ {k}_{m}^{T}=(1-n){k}_{s}+n{k}_{f}\\ {Q}_{m}^{T}=(1-n){Q}_{s}^{T}+n{Q}_{f}^{T}\end{array},$$
(19)

where cm, \({k}_{m}^{T}\), and \({Q}_{m}^{T}\) are the specific heat, thermal conductivity, and thermal source terms of the porous material, respectively. Each term in Eq. (18) has different physical meanings: the first term represents the temperature change per time in the system, second and third terms are the convective and conductive terms, respectively.

Evaluation of fluid-rock interaction

To estimate the effect of fracture formation and permeability enhancement on the fluid-rock interaction of ore-forming processes and hydrothermal alteration, we introduced the W/R ratio to evaluate the intensity of the fluid-rock interaction (Schardt and Large 2009; Takahashi et al. 2022):

$$W/{R}_{t}=W/{R}_{(t-1)}+\frac{(\left|{v}_{x}\right|{A}_{x}+\left|{v}_{y}\right|{A}_{y}+\left|{v}_{z}\right|{A}_{z})ndt}{{d}_{x}{d}_{y}{d}_{z}},$$
(20)

where the subscript t is the step number of the calculation; vi is the component of the fluid velocity in the i direction; dx, dy, and dz are the projected lengths of the grid element boundaries on the x-, y-, and z-axes, respectively; A is the grid element area in the cross-section perpendicular to the rectangular coordinate system; and dt is the time increment between adjacent computation steps.

Model setup

The commercial finite element software Abaqus was used to construct numerical models in this study. Abaqus is a finite element numerical simulation software that has been widely used in engineering and scientific research fields, such as soil mechanics (Helwany 2007), oil and gas drilling engineering (Chang et al. 2020), biomechanical modeling (Wu et al. 1998), and earth sciences (Poulet et al. 2012). Abaqus has a complete solid mechanics calculation module, which can simulate various complex mechanical behaviors (elastoplastic, viscoelastic, fracture, etc.), and can take physical factors such as pore fluid pressure, fluid seepage, temperature change, thermal expansion effect, and fluid thermal convection effect into complex solid mechanics models, showing good multi-field coupling calculation ability. In recent years, Abaqus has been gradually applied to geological studies of geomechanics (Luo et al. 2012) and metallogenic dynamics (Poulet et al. 2012; Zhang et al. 2012; Chang and Luo 2021), showing its ability to solve complex mechanical problems of geological systems.

Model geometry

We constructed a conceptual and simplified two-dimensional (2D) plane strain finite element model to study the fracture formation and hydrodynamic evolution around the hot magmatic intrusion (Fig. 3a). This model has a small thickness (100 m) in the out-of-plane direction, but we defined the front and back surfaces of this model as fixed in normal displacement and having no water flow or heat flux. Therefore, this model is an equivalent 2D plane strain finite element model coupling thermal-hydrological-mechanical processes. The horizontal length and vertical depth of the model were 6 and 5 km, respectively. The key analysis area of this model ranges from the model surface to a depth of 4 km, with a hot magmatic intrusion centrally embedded in the wall rock. The top width of the magmatic intrusion was 500 m, and the distance between the model surface and the top of the intrusion was 1.5 km. The other rock materials in this area are wall rocks (Fig. 3a). At the bottom of the model are basement rocks with a thickness of 1 km, which are impermeable and have relatively high mechanical strength (Fig. 3a). The basement rock in our models represents the precursor plutons (Sillitoe 2010), which are considered as the mid- to upper-crustal crystallization sites of mafic to felsic magmas that ascended from deeper reservoirs before porphyry Cu systems were developed (Richards 2003). The addition of the basement rock makes the temperature boundary conditions easy to set, and also makes the simulation calculation easier to converge. As basement rocks are not the main minerogenetic area in porphyry deposits, we only show the model results for wall rocks and magmatic intrusion (i.e., depths from 0–4 km of the model) in the following sections.

The meshes of the models were shown in Fig. 3b. The meshes in and above the magmatic intrusion are denser than those in wall rocks because of the reasons below. (1) The mesh of finite element model has continuity. The wall rocks near the magmatic intrusion are the key analyzing area on fracture formation and fluid flow, especially the wall rocks near the top of the magmatic intrusion. If the mesh density of wall rocks near the top of the magmatic intrusion should be denser, the same goes for the magmatic intrusion, because they share the same material boundary of the top of the magmatic intrusion. (2) The meshes within the intrusion has the same mesh density as the boundary between the wall rock and the magmatic intrusion, because we used nearly structured meshes to improve the convergence of calculations inside the intrusion. (3) The fracture formation inside the magmatic intrusion is also an important result of our model simulation besides the surrounding rock, and different tectonic stresses have certain influences on the fracture formation and fluid flow within the intrusion.

Gravity balancing step

The model calculations were divided into two steps. The first step is the gravity balancing step. After performing this step, we obtained the initial stress field for the second step of the THM coupling analysis. In this step, the initial ratio between the horizontal and vertical effective stresses is set to 0.5, that is

$${K}_{0}=\frac{{{\sigma }^{\prime}}_{h}}{{{\sigma }^{\prime}}_{v}}=\frac{{{\sigma }{\prime}}_{oop}}{{{\sigma }{\prime}}_{v}}=0.5,$$
(21)

where K0 is the initial ratio of the horizontal and vertical effective stresses, \({{\sigma }^{\prime}}_{h}\), \({{\sigma }{\prime}}_{v}\), and \({{\sigma }^{\prime}}_{oop}\) are the horizontal, vertical, and out-of-plane effective stresses, respectively.

THM coupling step, initial and boundary conditions

The second step involves THM coupling analysis. The temperature of the magmatic intrusion is set at 600 ℃ (Fig. 3a), between the maximum temperature of copper precipitation and that of large batholith in upper crust, i.e., in the range of 550–700 ℃ (Sillitoe 2010; Richards 2011). The initial temperature of wall rock is 20 ℃ at the top. With increasing depth, the initial temperature of wall rock increases with a geothermal gradient of 25 ℃/km, and finally reaches 170 ℃ at the bottom of the model (Fig. 3a). The initial pore pressure of the top surface of the wall rock is one atmosphere (0.1 MPa), and the initial pore pressure of the wall rock at other depths is hydrostatic plus atmospheric (Fig. 3a).

Previous studies have shown that the thermal evolution of porphyry metallogenic systems is highly complex and that the intrusion of molten magma and the formation of porphyry deposits are accompanied by multiple or continuous magma injections (Sillitoe 2010; Owen-Smith and Ashwal 2015; Buret et al. 2016). In order to simplify this complex process, this model does not consider the large deformation process of magmatic intrusion and mainly simulates the physical field evolution processes of the surrounding rock heated by the intrusion after the completion of high-temperature magma emplacement; thus, the intrusive magmatic rock always maintains a high temperature of 600 ℃ during the simulation duration, to approximate the continuous action of high-temperature magma.

For boundary conditions, the temperature on the model surface is fixed at 20 ℃, and the temperatures at the lateral and bottom boundaries are calculated by surface temperature and geothermal gradient (Fig. 3). The pore fluid pressure boundary on the model surface is fixed at 0.1 MPa, and the pore pressure at the lateral and bottom boundaries are hydrostatic plus atmospheric pressure.

The displacement boundaries of the model were the focus of comparison. In order to study the model results under different tectonic stress backgrounds, we established three cases of this model with different boundary conditions at the left and right sides: Case 1 (no tectonic effect), the displacements of both lateral sides are fixed in normal direction and free in tangential direction; Case 2 (tectonic compression), we apply the compressional velocity of 0.3 mm/y to the left and right edges of the model, that is, the model will undergo 1‰ shortening of width during 10,000 y; Case 3 (tectonic extension), we apply the extensional velocity of 0.3 mm/y to the left and right edges of the model, that is, the model will undergo 1‰ elongation of width during 10,000 y (Fig. 5a, Table 1). A previous study of present-day plate motion and plate boundary deformation, which was based on 3,000 geodetic velocities, suggested that the strain rates of the active plate margins are in the range of 1.0 × 10–15/s to 1.0 × 10–13/s (Kreemer et al. 2003). The compressional and extensional velocities of our model cases are 0.3 mm/y, that is, our model undergoes 1‰ deformation in width during 10,000 y, equal to a strain rate of 3.17 × 10–15/s and within the reasonable range of strain rates of the active plate margins (Kreemer et al. 2003). The displacement boundary of the model surface was free in both the normal and tangential directions, and the displacement was fixed in the normal direction and free in the tangential direction at the bottom of the model.

Table 1 Material properties in this model
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The total time for each model case was 10,000 y. The time steps of the simulation increments were adaptively adjusted by the program based on the convergence of the results, and the time steps ranged from a few seconds to several decades. When the convergence of the model results was satisfactory, the time step increased. When the convergence of the model results is poor, the time step decreases (Simulia 2016).

The material properties

The material properties of the model are listed in Table 1. The wall rock may contain sedimentary or volcanoclastic rocks; therefore, we assumed that the wall-rock material in our model had a relatively low mechanical strength and the highest porosity. As the magmatic rocks intrude and consolidate into the shallow part of the Earth’s crust, the magmatic intrusion material in our model exhibited medium mechanical strength and relatively low porosity, compared to the wall rock material. We assumed that the permeability of magmatic intrusion is relatively higher than that of wall rock; because the ore-forming hydrothermal fluids of porphyry deposits and epithermal deposits are mainly related with magmatic source, relatively low permeability within magmatic intrusion will render it difficult for the ore-forming fluid to flow outward from the intrusion. The basement rock material is impermeable and has relatively high mechanical strength. We used the pore fluid properties at 20 ℃ (indoor temperature) to simulate the fluid flowing behavior and ignored the fluid properties changing with temperature, because the focus of this study was to determine how the different tectonic settings influence the fracture formation and fluid flow around upper-crust magmatic intrusion, rather than the changing of fluid properties with temperature influences the fracture formation and fluid flow around upper-crust magmatic intrusion. All the material properties were referred to the Handbook of Rock Mechanics and previous numerical simulation studies (Guillou-Frottier and Burov 2003; Ingham and Pop 2005; Nield and Bejan 2006; Jaeger et al. 2007; Rozhko et al. 2007; Zoback 2007; Schon 2011; Zhang et al. 2012).

Model results

Fracture formation

Case 1 was a model that did not contain tectonic stresses (normal displacement was fixed at the left and right edges of the model). In the early stage of the simulation (1,000 y), fractures formed at the contact zones between the intrusion and the wall rocks, and several high-angle vein-type fractures formed above the top of the intrusion (Fig. 4a). After another 1,000 y, small-scale conjugate fractures occurred in the magmatic intrusion and tensile fractures occurred near the model surface above the intrusion (Fig. 4b). Over time, the fractures in the magmatic intrusion grew further (Fig. 4c), and finally, a fracture system of a certain scale and depth was formed in the magmatic intrusion (Fig. 4d). The tensile fractures that occurred near the model surface developed further, and two low-angle fractures occurred laterally from the vein-type fractures (Fig. 4c, d).

Fig. 4
figure 4

Evolution of equivalent plastic strain (fracture formation) in all cases of the model

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Case 2 was the model that contained horizontal tectonic compression at the left and right edges. Small-scale conjugate fractures in the intrusion were formed during the early 1,000 y of simulation, as well as fractures at the contact zones between the intrusion and wall rocks and high-angle vein-type fractures above the top of the intrusion (Fig. 4e). In the middle stage of the simulation (2,000–5,000 y), the conjugate fractures in the intrusion grew deeper, and several low-angle fractures occurred laterally from the early formed high-angle fractures (Fig. 4f, g). Finally, several conjugate fracture zones that formed from the left and right edges of the model were in contact with the early low-angle fractures, showing the maximum tectonic compression effect of the model (Fig. 4h).

Case 3 was the model that contained horizontal tectonic extension at the left and right edges. The fractures that formed in the first 1,000 y of the simulation were similar to those in Case 1 (Fig. 4i). However, after other 1,000 y, the two high-angle fractures above the intrusion grew further upward and intersected near the model surface (Fig. 4j). At a simulation time of 5,000 y, more high-angle fractures grew upward above the intrusion (Fig. 4k), and as the degree of fractural deformation increased, a fracture system with an inverted conical shape finally occurred above the magmatic intrusion (Fig. 4l). The conjugate fractures in the intrusion also developed during the middle and final stages of the simulation; however, the scale of the fracture system in the intrusion in Case 3 was smaller than those in Cases 1 and 2 (Fig. 4k, l).

Comparing the model results of all three cases, we found that tectonic compression has an effect on the formation of fractures lateral to the hot intrusion and is beneficial for the fracture system within the intrusion to grow deeper (compare Fig. 4e–h with 4 a–d). On the contrary, tectonic extension has advantages in forming a high-angle fracture system right above the intrusion and has an inhibitory effect on the formation of fractures within the intrusion (compare Fig. 4i–l with 4 a–d).

Stress analysis

As mentioned in Sect. 3.1, we used the Drucker-Prager plastic yield criterion to couple the pore pressure effect to simulate the fracturing behavior of rocks (Eq. (7)). When analyzing the stresses, we preferred to define the compressive stress as positive and the tensile stress as negative, as the compressive stress in the model is relatively large and widely distributed. Thus, in this section, the Drucker-Prager plastic yield criterion (Eq. (7)) is expressed as:

$$F=\sqrt{{J}_{2}}-{{\alpha }_{DP}{I}_{1}}^{\prime}-{k}_{DP}=\sqrt{{J}_{2}}-{\alpha }_{DP}({I}_{1}-3p)-{k}_{DP},$$
(22)

where the sign leftward of the \({{\alpha }_{DP}{I}_{1}}^{\prime}\) becomes negative compared with Eq. (7). In this model, we use ideal elastic-plastic constitutive relation, that is, when F < 0, the material is elastic and does not yield; when F = 0, the material yields plastically, and the plastic law limits the differential stress and keeps F = 0 or F < 0.

There are three variables of stress that can affect the fracture forming processes in Drucker–Prager plastic yield criterion, that is, \(\sqrt{{J}_{2}}\), I1, and p. We used three common equivalent stresses to represent these three quantities:

$$\begin{array}{c}{\sigma }_{Mises}=\sqrt{3{J}_{2}}=\sqrt{\frac{1}{2}\left[{({\sigma }_{x}-{\sigma }_{y})}^{2}+{({\sigma }_{y}-{\sigma }_{z})}^{2}+{({\sigma }_{z}-{\sigma }_{x})}^{2}+6({\tau }_{xy}^{2}+{\tau }_{yz}^{2}+{\tau }_{zx}^{2})\right]}\\ {\sigma }_{m}=\frac{1}{3}{I}_{1}=\frac{1}{3}({\sigma }_{x}+{\sigma }_{y}+{\sigma }_{z})\\ {p}_{over}=p-\rho gh\end{array}$$
(23)

where σMisses is the von Mises stress (characterizing the shear effect), σm is the mean pressure (characterizing the compressive or extensive effect), and pover is the pore overpressure (current pore pressure minus the hydrostatic pressure). We plotted these three variables along vertical and horizontal profiles (A-A’ and B-B’ in Fig. 3b, respectively) to estimate the different fracture-forming mechanisms and stress evolutions of various tectonic stress backgrounds around and within the magmatic intrusion (Figs. 5 and 6).

Stress analysis along vertical profile

We chose a vertical profile (A-A’ in Fig. 3b) to analyze the fracture-forming mechanisms and the distribution of stresses with depth across the intrusion and overlying rocks (Fig. 5).

Fig. 5
figure 5

Plots of Mises stress, mean pressure, and over pore pressure with depth along the vertical profile (A-A’ in Fig. 3b) of the model. The yellow bars at the y axis indicate that the analyzing points are in the magmatic intrusion

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In early stage (1,000 y), the σMises in Case 3 is larger than those in Cases 1 and 2 from the surface to 1.25 km, showing that the extensional tectonic setting enhances the shear effects above the intrusion; on the contrary, the σMises in Case 2 was larger than those in Cases 1 and 3 from 1.5–2 km and from 2.5–4 km, showing that the compressive action enhances the shear effects within the intrusion (Fig. 5a). However, the σMises in Case 2 from 2–2.5 km is smaller than those in Cases 1 and 3; furthermore, in this range of depth, the curve of σMises fluctuated (Fig. 5a) because the mechanical conditions in this location reach plastic yield (see Fig. 4e, only Case 2 has equivalent plastic strain within the intrusion in 1,000 y), and plastic laws impose limitations on differential stresses (Eq. (22); Luo et al. 2012). The fluctuations in the other curves in Figs. 5 and 6 can be explained similarly. The peak value of the σMises at the top of the intrusion was same in all three cases during the early stages (Fig. 5a).

In the intermediate stage (5,000 y), the value of σMises in Cases 1 and 3 fluctuated above the intrusion (0–1.25 km) because fractures form above the intrusion, and the plastic law causes σMises to decrease (Figs. 5b and 4 c, k). However, the σMises in Case 2 was also the smallest at depths of 0–1.25 km but became maximal at the top of the intrusion (Fig. 5b). σMises in Case 2 was the largest within the intrusion (Fig. 5b). In the late stage (10,000 y), the σMises in Case 3 decreased and the σMises in Case 2 increased at all depths, making the differences between all three cases increase in the peak value and within the intrusion (Fig. 5c).

The mean pressure σm of all three cases was similar at all depths in the early stage, except for the small fluctuation of σm in Case 2 because of the fracture forming within the intrusion (Figs. 5d and 4 e). With time, the σm in Case 2 became larger and the σm in Case 3 became smaller at all depths in the intermediate and late stages (5,000 and 10,000 y), showing that the compressive action enhances the compressive stresses, and the extensive action inhibits them, especially the tensile stresses occurring above the intrusion (Fig. 5e, f).

The pore-over-pressure pover of Case 2 was the largest, and the pover of Case 3 was the smallest at all depths in the early and intermediate stages (1,000 and 5,000 y), implying that the pore fluid endured a part of the compressive action by compressional tectonic setting, and the extensional action decreased the pore pressure (Fig. 5g, h). pover was small in all three cases over 10,000 y, implying that the systems became near-hydrostatic in the late stage of the simulation (Fig. 5i).

Stress analysis along horizontal profile

We chose a horizontal profile (B-B’ in Fig. 3b) to analyze the fracture-forming mechanisms and the distribution of stresses above and around the magmatic intrusion (Fig. 6).

Fig. 6
figure 6

Plots of Mises stress, mean pressure, and over pore pressure with distance from the horizontal center along the horizontal profile (B-B’ in Fig. 3b) of the model. The orange bars at the x axis indicate that the analyzing points are right above the magmatic intrusion

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Near the top of the intrusion (0–0.75 km from the center), the σMises of all cases were similar in the early stage (1,000 y) (Fig. 6a). Over time, the σMises of Case 3 became the largest in the intermediate stage (5,000 y) (Fig. 6b). In the late stage, although the σMises of Case 3 decreased near the center (0–0.25 km from the center), the peak of σMises in Case 3 was the highest near the intrusion (0–0.75 km from the center) (Fig. 6c). At the distance far from the model center above the intrusion (0.75–3 km from center), the σMises of Case 2 was larger than that for other two cases in all the stages (1,000, 5,000, and 10,000 y) and increased over time, showing the enhancement of shear effect by the compressional tectonic settings around the intrusion (Fig. 6a–c).

The σm of all three cases increased over time along the horizontal profile, but the magnitude of the increase differed between the cases (Fig. 6d–f). The σm of Case 2 increased the most, followed by the σm of Case 1, and the smallest increase was that of Case 3 (Fig. 6d–f). Of note, the σm of Case 2 near the top of the intrusion (0–0.5 km from the center) increased more than that distant from the intrusion (0.5–3 km from the center) (Fig. 6e–f), which is why fractures did not occur in Case 2 near the top of the magmatic intrusion, but were well developed on the lateral sides of the intrusion (Fig. 4e–h).

The pover of all cases along the horizontal profile was similar to that along the vertical profile (compare Figs. 6g–i to 5 g–i). The pore-over-pressure pover of Case 2 was the largest, and the pover of Case 3 was the smallest at all distances in the early and intermediate stages (1,000 and 5,000 y) (Fig. 6g, h). In the late stage (10,000 y), the pover became small, and the pore pressures became hydrostatic in all cases (Fig. 6i). The pover near the intrusion was larger than that distant from the intrusion, indicating the thermal expansion effect of the pore fluid (Fig. 6g–i).

Mechanical mechanism of fracture formation

By combining the fracture patterns (Fig. 3) and stress analyses (Figs. 4 and 5), we conclude that:

In Case 2, tectonic compression enhanced the formation of fractures lateral to the hot intrusion and was beneficial for the fracture system within the intrusion to develop deeper (Fig. 4e–h), mainly because of the high σMises lateral to the intrusion (Fig. 6b, c) and the high σMises in the intrusion (Fig. 5b, c), respectively. These results indicate that tectonic compression enhanced the shear effects lateral to and within the intrusion, resulting in fracture formation at these positions. The fractures immediately above the intrusion were absent in Case 2 (Fig. 4e–h) because of the high σm immediately above the intrusion (Fig. 5e, f), implying that tectonic compression inhabits the fracture forming immediately above the hot intrusion.

In Case 3, tectonic extension has advantages in forming the high-angle fracture system right above the intrusion (Fig. 4i–l) because of the high σMises (Fig. 6b, c) and low σm right above the intrusion (Fig. 6e, f), which are tensile because of the negative σm right above the intrusion (Fig. 5e, f). However, tectonic extension inhibited the formation of fractures within the intrusion (Fig. 4i–l), mainly because of the low σMises within the intrusion (Fig. 5b, c).

Fluid flow patterns

Subsequently, we analyzed the fluid flow patterns for all three cases. Due to the permeability enhancement effect of the fracture formation, the fluid in the fractures was more active than that in the unfractured wall rocks, and the fluid velocity in the fracturing zones was larger (compare Figs. 7 and 4). The fluid velocities of all three cases decreased with time, which was controlled by Darcy’s law (Eq. (11)) and was consistent with the pore pressure evolution (Figs. 5g–i and 6 g–i).

Fig. 7
figure 7

Vectors of pore fluid flow velocity in all the cases of model

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In Case 1 (no tectonic settings), in the early stage (1,000 y), the fluid flowed rapidly in the fractures above the intrusion and the contact zones between the intrusion and wall rocks, forming a wide high fluid velocity zone above the intrusion (Fig. 7a). In the intermediate stage (5,000 y), the high-velocity fluid mainly flowed in the two large fractures (or faults) above the intrusion; however, the fluid velocities within the magmatic intrusion increased, accompanied by the formation of internal fractures of the intrusion (Fig. 7b). In the late stage (10,000 y), only the fluid velocities in the two high-angle faults and the contact zones between the intrusion and wall rocks were relatively high, but all the magnitudes of the fluid velocities decreased over time (Fig. 7c).

Comparing the results of the fluid velocity in Case 1, the fluid in Case 2 (tectonic compression) preferred to flow laterally around the intrusion (Fig. 7d–f). The fluid flow pattern above the intrusion of Case 2 in the early stage was similar to that of Case 1; however, the fluid velocities within the intrusion in Case 2 were larger than those in Case 1 (Fig. 7d). In the intermediate stage (5,000 y) of Case 2, the fluid flowed laterally along the nearly plane fractures above the intrusion, and the fluid velocities within the intrusion largely increased with the fracture evolution within the intrusion (Fig. 7e). In the late stage (10,000 y), the fluid velocities within the intrusion decreased, but the velocities of the lateral-flowing fluid were still relatively high, and the fluid flowed away from the intrusion (Fig. 7f).

The high fluid velocities in Case 3 were not as widely distributed as those in Cases 1 and 2 in the early stage of the simulation (1,000 y) (Fig. 7g). However, the pore fluid flowed upward and was more concentrated above the magmatic intrusion in the intermediate and late stages (5,000 and 10,000 y, respectively), resulting in the formation of a central high-velocity zone above the top of the intrusion (Fig. 7h, i). In this zone, the fluid velocities were concentrated and focused (particularly in the intermediate stage), which is beneficial for the water-rock interaction and ore precipitation (Fig. 7h, i).

W/R ratio

Figure 8 shows the results of the W/R ratio for three different cases. To a certain extent, W/R represents the cumulative amount of fluid flux over time (Eq. (20)); therefore, it is directly related to the entire evolution process of the fluid velocity. Moreover, our model considered the permeability enhancement effect of rock fracturing (Eq. (12)), resulting in the formation of fractures that strengthen the velocity of fluid in the fracturing zones (Fig. 7) and indirectly affect W/R (Fig. 8).

Fig. 8
figure 8

Evolution of water-rock ratio (W/R) in all the cases of model

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In Case 1 (no tectonic setting), the high values of W/R were mainly distributed in the fractures above the intrusion and in the contact zones between the intrusion and wall rocks in the early stage (Fig. 7a, b). Over time, the fracture system above the intrusion developed further, creating a high W/R area (or net) above the intrusion (Fig. 7c, d). Moreover, relatively small W/R zones occurred within the magmatic intrusion in the intermediate stage and developed further in the late stage (Fig. 7c, d). Although the W/R within the intrusion was smaller than that in the fractures above the intrusion, it indicates the formation of porphyry ore bodies in the intrusions, as the formation temperatures of Cu and Mo ores are relatively high (300–500 ℃) (Sillitoe 2010).

The high W/R zones in Case 2 (tectonic compression), compared to the results in Case 1, were mainly distributed on both lateral sides of the intrusion and extended far from it (Fig. 7g, h). Moreover, the W/R zone within the intrusion of Case 2 was deeper than that of Case 1 (comparing Fig. 7g, h with Fig. 7a–d), indicating that the compressional tectonic settings were beneficial for the fluid-rock interaction and the formation of porphyry ore bodies within the magmatic intrusion.

In Case 3 (tectonic extension), the highest value of W/R was distributed in the centered fracture nets above the intrusion, and the high W/R zones above the intrusion in Case 3 were more concentrated than those in Cases 1 and 2 (Fig. 7i–l). However, the W/R ratio was smaller, and the W/R zone was narrower within the intrusion in Case 3 than that in Cases 1 and 2 (comparing Fig. 7i–l with Fig. 7a–h), showing that the extensional tectonic settings inhibit the fluid-rock interaction effects within the magmatic intrusion.

Discussion

Implications for porphyry Cu systems and epithermal Au deposits in tectonic compression

According to geological reviews of porphyry Cu systems, they are generated mainly in convergent tectonic environments related to subduction- and collision-associated scenarios, such as volcanic, continental, island, and post-collisional arcs (Hou et al., 2009; Japas et al. 2021; Groves et al., 2022). The structural morphology and deposit distribution of these tectonic environments are subject to a spectrum of regional-scale stress regimes ranging from moderately extensional through oblique slip to contractional (Tosdal and Richards 2001).

Nevertheless, large high-grade porphyry Cu deposits have a significant empirical relationship with contractional or compressional settings marked by thickening of the crust and surface uplift in post-collisional arcs and rapid exhumation in island arc settings (Tosdal and Richards 2001; Sillitoe 2010; Groves et al., 2022). Examples include the latest Cretaceous to Paleocene metallogenic province of southwestern North America, the Eocene to Pliocene metallogenic belts of the central Andes, Iran, New Guinea, and the Philippines, and the Cenozoic collisional orogen of Tibetan China (Hill et al. 2002; Perelló et al. 2003; Cooke and Hollings 2005; Hou and Cook 2009; Sillitoe 2010; Richards 2013). From the perspective of magmatism, crustal thickening may heighten the level of magma fertility because the dehydration melting of deep-crust arc cumulates in the enriched Melting, Assimilation, Storage, and Homogenization (MASH) reservoir (Richards 2013). The results of our numerical model suggest that regional tectonic compression may enhance the form scale of fractures within the shallow porphyry intrusion, cause the fractures to form deeper and denser, and increase the fluid velocities and degree of water-rock interaction within the shallow porphyry intrusion for porphyry deposit formation (Figs. 4e–h, 7 d–f, 8 e–h). These results may explain the favorable influence of the tectonic compression environment on porphyry deposits from the perspective of solid mechanics and dynamics at shallow depths in the upper crust.

Notably, large, high-sulfidation epithermal Au deposits also prefer to form in tectonic compressional settings at the top of thickened crustal sections, although not necessarily together with porphyry deposits (Sillitoe 2008; Richards 2013; Dilles and John 2021). Most high-sulfidation epithermal Au deposits are associated with acidic magmatic-hydrothermal fluids (Dilles and John 2021), and moderate tectonic compression may promote the development of large magma chambers in the middle to upper crust (Takada 1994), which enhances the fractionation of ore-forming elements from magma and the generation and release of magmatic fluids (Sillitoe 2010). The results of our model show that in the case of tectonic compression (Case 2), fractures tend to occur laterally around the intrusion and at the top of the intrusion, and there is a lack of high-angle fractures or faults that penetrate the shallow rock layers and communicate with the model surface (Fig. 4e–h). This situation resulted in an increase in fluid velocities and the degree of water-rock interaction laterally around the intrusion (Figs. 7d–f, 8 e–h). According to Chiaradia and Caricchi (2022), supergiant porphyry copper deposits require large amounts of magma volume and magma injection rates, which are equal to those typical for large volcanic eruptions from rift, hot spot, and subduction-related settings; thus, they suggested that supergiant porphyry copper deposits can be considered as failed large eruptions. Our model results suggest that moderate tectonic compression is conducive to sealing the magmatic system and may suppress surface eruption, which is beneficial for the accumulation of magmatic-hydrothermal fluids and the formation of large porphyry Cu deposits and high-sulfidation epithermal Au deposits dominated by mineralizing hydrothermal fluids of magmatic origin (Dilles and John 2021).

Implications for porphyry Cu systems and epithermal Au deposits in tectonic extension

It has long been recognized that regions that undergo large-scale extensional strain are absent from significant porphyry Cu systems (Tosdal and Richards 2001; Sillitoe 2010; Richards 2013; Dilles and John 2021). These strongly extensional settings, typified by bimodal basalt-rhyolite magmatism, preclude the formation of magmatic arcs, or decrease their duration, which is considered unfavorable for the formation of porphyry Cu deposits (Tosdal and Richards 2001; Sillitoe 2010). The results of our model in Case 3 suggest that regional tectonic extension at the deposit scale can restrain fracture formation within the magmatic intrusion (Fig. 4i–l) and further decrease the fluid velocities and the degree of water-rock interaction within the intrusion (Figs. 7g–i, 8 i–l), which prejudices the ore-forming processes of porphyry deposits.

Moreover, our model results of extensional settings show that a funnel-shaped tensile fracture system occurred through the region at the top of the magmatic intrusion and connected the deep intrusion to the surface (Fig. 4i–l). This penetrating fracture system, on one hand, provides highly permeable paths for magma ascent, resulting in the direct rising and eruption of asthenosphere-derived magmas, and thus the extensional settings, such as rifting arcs, are related to the lack of porphyry Cu deposits (Tosdal and Richards 2001). On the contrary, this penetrating fracture system can enhance the fluid flow and the water-rock interaction at a shallow depth right above the intrusion (Figs. 7g–i, 8 i–l), in favor of the formation of low-sulfidation epithermal Au deposits associated with large, vein-type fractures or faults and relatively low-temperature metallogenic processes at shallow depths of extensional magmatic-hydrothermal systems (Richards 2013; Dilles and John 2021).

Intensity of tectonic activity

Despite our model results showing that tectonic compression promotes the development of deep, high-temperature porphyry Cu systems and that tectonic extension promotes the development of shallow, low-temperature hydrothermal deposits, not all intensities of tectonic activity are favorable for the formation of magmatic-hydrothermal deposits. According to Takada (1994), large differential stresses are not conducive to the formation of large magmatic chambers in the upper crust. Tosdal and Richards (2001) also suggested that in intense compressional tectonics, magmas penetrate fault zones at high overpressure and erupt violently without significant residence in upper crustal magma chambers, whereas in intense extensional tectonics such as tensional arcs, asthenosphere-derived magmas may rise directly to the surface, lacking the distillation and crustal interaction processes for the development of porphyry Cu-prospective magmas. Thus, we suggest that moderate-intensity tectonic activity was conducive to the formation of ore-bearing fractures and the strengthening of water-rock interactions in magmatic-hydrothermal deposits, similar to the viewpoint of Tosdal and Richards (2001).

The possible effects of fluid density and viscosity variations with temperature

Our study focuses on the influence of different tectonic settings on fracture formation and fluid flow around upper-crustal magmatic intrusion. Under the effect of tectonic extrusion and extension, the stress state in the system changed over time, this change of stress state resulted in permanent deformation (plastic strain, similar to fracture), and the reconstructions of rock permeability and fluid flow pattern by this permanent deformation are the main research objects of our study. However, the models above did not consider the effects of fluid density and viscosity variations with temperature and pressure, which may lead to significant differences in fluid behavior.

According to the IAPWS Formulation 1995 (Harvey 1998), the water density decreases from 1009.74 kg/m3 to 533.69 kg/m3 with temperature ranged from 0 ℃ to 360 ℃ at constant pressure of 20 MPa, and increases from 998.17 kg/m3 to 1007.13 kg/m3 with pressure ranged from 0.01 MPa to 20 MPa at constant temperature of 20 ℃. Similarly, the water viscosity decreases significantly with temperature increasing and increases slightly with pressure increasing (Sengers and Watson 1986). Given that the temperature effects on water density and viscosity are more dramatic than the pressure effects, we only discuss the situation that adding fluid density and viscosity variations with temperature in our models. With the hot magmatic intrusion heating pore fluid, water density and viscosity decrease significantly. The fluid near the magmatic intrusion is affected by larger buoyancy and has higher mobility, and may flow faster and reach shallower parts further away from the magma chamber than that with constant water density and viscosity. The fluid may accumulate at shallow parts of wall rocks, make the pore pressure increase, which resulting in further hydrofracturing of shallow wall rocks and enhancing of ore-bearing spaces.

Model limitations

Our model coupled the thermal-hydrological-mechanical processes associated with the mechanism of fracture formation and hydrothermal fluid flow in porphyry Cu systems; however, there are some processes that we have not considered that may influence the dynamic processes of fracturing and fluid flow in the systems. For example, the fluid density and viscosity changes with temperature and pressure (Sengers and Watson 1986; Harvey 1998), and water will undergo phase transitions under different temperature and pressure conditions (Weis et al. 2012; Heinrich 2005); the increases of rock porosity and permeability, and the decreases of tensile strength and compressive strength with the temperature increasing, resulting from the escaping of adhered water, bound water and structural water (Weis et al. 2012; Zhang et al. 2016). However, since the major purpose of this paper is to study the influence of tectonic stress on the fracture formation and fluid flow in the magmatic intrusions and their surrounding rock, the temperature effects on the physical properties of surrounding rock is not taken into account in our models. These effects should be considered in future studies, and the influence of the size, shape and temperature evolution of the magma chamber on the fracture and fluid flow around the intrusion can also be studied.

The fluid source of magmatic water is not added to our model for the following reasons: (1) The magmatic water concentration is strongly dependent on the dynamic conditions of the magmatic system, and can be changed on large amount during a short time. According to the thermodynamic numerical experiments by Petrelli et al. (2018), an initial batch of magma containing ~ 3–4.5 wt.% of water, stored at mid- to deep-crustal levels (~ 20–35 km), can evolve to a water-rich (H2O equal to ~ 6–9 wt.%) residual melt in timescales ranging from a few to several thousand years. (2) If we considered the magmatic water dissolved from magmatic crystallization, the role of multiphase fluid needs to be considered because of large temperature variation (Heinrich 2005). However, the calculation of multiple fluids in solid mechanics is complex and hard to converge. Given the complexity of magmatic water production process during magmatic crystallization, we set the temperature of magmatic intrusion at 600 ℃ below the solidus temperature of magma (700 ℃) and assumed pore fluid saturation at 100% in all porous rock materials. This simplification allows our models to consider and focus on the effects of tectonic stress on the fracture and fluid flow around the magmatic intrusion.

Conclusions

To explore the influences of different tectonic settings (compression and extension) on ore-bearing fracture formation and the hydrodynamics of hydrothermal fluid in porphyry Cu systems, we established a 2D plane strain finite element model coupling thermal-hydrological-mechanical processes and divided it into three cases to study the fracturing mechanism and fracture distribution, fluid flow patterns, and degree of water-rock interaction within a hot magmatic intrusion and the surrounding wall rocks under different tectonic settings. After analyzing the model results and comparing them with geological reviews, we summarized the following conclusions:

  1. (1)

    On one hand, the compressional tectonic settings can increase the degree of fracturing and the depth of fracture distribution within the porphyry intrusion, enhancing the hydrothermal fluid velocities and the degree of water-rock interaction within the intrusion. These results indicate that tectonic compression promotes the formation of the porphyry Cu deposits.

  2. (2)

    On the other hand, the model case of tectonic compression lacks the fractures connecting the surface and deep intrusion, but enhances the fractures and the water-rock interaction laterally around the intrusion, suggesting that tectonic extrusion is conducive to the sealing and accumulation of a magmatic hydrothermal system in the shallow upper crust, which may be beneficial to the formation of porphyry Cu deposits and high-sulfidation epithermal Au deposits.

  3. (3)

    The extensional tectonic settings inhibit fracture formation, fluid velocities, and the water-rock interactions within the magmatic intrusion, thus confirming the long-known phenomenon that regions undergoing large-scale tectonic extension are absent from significant porphyry Cu systems.

  4. (4)

    In contrast, a funnel-shaped fracture system occurs immediately above the intrusion in the model case of tectonic extension, penetrates the overlying rocks, connects the surface and deep magmatic intrusion, and enhances the fluid activity at shallow depths above the intrusion. These results suggest that tectonic extension may promote the formation of shallow, low-temperature hydrothermal deposits, such as low-sulfidation epithermal deposits.