Causes, Geodynamic Factors and Models of Metamorphism

Abstract

This chapter considers numerical model-based relationships between metamorphism and geodynamics, discusses tectonomagmatic causes and controls of metamorphism, and makes attempt to link the geological types of metamorphism to the specific Р-Т conditions and Р-Т-t paths. Three categories of metamorphism are distinguished based on the magnitude of the heat flux: (1) metamorphism induced by a thermal gradient close to the average continental values (metamorphism associated with crustal subsidence, in depressions during continental rifting, metamorphism caused by tectonic stacking in orogeny, and metamorphism associated with Archean crust formation), (2) by a higher thermal gradient caused by the supply of additional heat by magmatic intrusions (contact metamorphism, medium-pressure zonal metamorphism) and diapirism, (3) by a lower thermal gradient during the collision of lithospheric plates and crustal blocks (metamorphism associated with overthrusting, underthrusting and subduction). Different types of metamorphism are manifested in different geodynamic regimes over different time scales and can be correlated with a specific combination of metamorphic facies. Interpretation of geodynamic and magmatic causes of different types of metamorphism using thermomechanical numerical models accounting for variable rates and mechanisms of subsidence and exhumation can be used to solve many geodynamical problems. Analysis of the problem reveals that metamorphism is a consequence and an indicator of geodynamics.

Appendix

The formulation is based on a coupled thermomechanical model of planar deformation. For twice continuously differentiable fields of the velocity vector ui, the equilibrium equations in differential form, together with the boundary conditions, have the form (here and below, the indices i and j run through 1, 2) (Korobeynikov 2000):

$$\begin{aligned} \sigma_{ij,j} + \rho g_{i} = 0{\text{ in }}V, \hfill \\ n_{i} \sigma_{ij} = f_{j}^{*} {\text{ at }}S_{f} , \hfill \\ u_{i} = u_{i}^{*} {\text{ at }}S_{u} . \, \hfill \\ \end{aligned}$$

(3.21)

Here, σ_ij are components of the symmetric Cauchy stress tensor (i, j = 1, 2); g_i the components of the gravity vector; ρ is the current mass density of the material; V is the domain occupied by the body in its current configuration; S is the closed boundary of domain V; S_u and S_f are the segments of surface S on which the components of velocity vector u_i and Cauchy stress vector $n_{i} \sigma_{ij}$ are specified; n_i are the vector components of the external unit normal vector to surface S_f; and summation goes over repeated indices from 1 to 2; the comma denotes partial derivative with respect to the corresponding coordinate, and the asterisk denotes a prescribed variable value.

For twice continuously differentiable temperature fields T, the heat balance equation may be expressed in the form (Landau and Lifshitz 1986):

$$\begin{aligned} \rho c_{p} \dot{T} = \sigma_{ij} d_{ij} + kT_{,ii} + r{\text{ in }}V, \hfill \\ kT_{,i} n_{i} = q_{n}^{*} {\text{ at }}S_{q} , \hfill \\ T = T^{*} {\text{ at }}S_{T} . \hfill \\ \end{aligned}$$

(3.22)

Here, c_p is specific heat; $d_{ij} \equiv \frac{1}{2}\left( {u_{i,j} + u_{j,i} } \right)$ are strain rate tensor components; $q_{i}$ are heat flux vector components; k is thermal conductivity; r is the heat source in a unit volume; $S_{q}$ and $S_{T}$ are the segments of surface S on which $q_{n}^{*}$ and $T^{ * }$ are specified, respectively; the dot above quantity denotes partial derivative with respect to time. Equation (3.22) should be complemented with initial conditions in the form of temperature fields specified at each point of domain V at initial time.

Note that the differential Eqs. (3.21) and (3.22) require that the functions contained in them be highly smooth. If we ignore this requirement (for instance, in the case of discontinuous fields of stress tensor components), we should introduce the internal interfaces and specify the discontinuity conditions for functions or their derivatives on these boundaries. However, with the weak formulations of equations, the internal boundaries are not required because the equations will be satisfied irrespective of these discontinuities. The finite element method, which is applied in this study for numerical modeling of geodynamical processes, is based on the weak forms of thermomechanical equations. The equilibrium equations in weak form correspond to the principle of balance of virtual works of the internal and external forces:

$$\int\limits_{V} {\sigma_{ij} \delta d_{ij} dV = } \int\limits_{V} {\rho g_{i} \delta u_{i} dV} + \int\limits_{{S_{f} }} {f_{i}^{*} \delta u_{i} dS \, \ \ \forall \delta u_{i} \left( {\delta u_{i} = 0{\text{ at }}S_{u} } \right)},$$

(3.23)

while the weak form of heat transfer equation is

$$\int\limits_{V} {\left( {\rho c\dot{T}\delta T + kT_{{{\mathbf{,}}i}} \delta T_{,i} } \right)dV} = \int\limits_{V} {\left( {\sigma_{ij} d_{ij} + r} \right)\delta TdV} + \int\limits_{{S_{q} }} {q_{n}^{*} \delta TdS \,\ \ \forall \delta T\, \left( {\delta T = 0{\text{ at }}S_{T} } \right)},$$

(3.24)

where $\delta ( \, ), \forall ( \, )$ mean variation and universal quantifier in corresponding variable.

The computational domain, including the sections of the earth’s crust and mantle lithosphere, is presented in Fig. 3.30. In the same figure, boundary conditions are given both for solving the mechanical problem (Fig. 3.30a) and for the thermal problem (Fig. 3.30b). Figure 3.30b shows the initial temperature distribution that corresponds to the stationary geothermal lithosphere of the craton with a mantle heat flux Q = 17 mW m⁻² and the thermophysical parameters of the crust and mantle specified in Table 3.8. The following boundary conditions for the mechanical problem are employed: the top boundary is a free surface, the bottom boundary and lateral boundaries of the lithospheric mantle are free slip, lithostatic pressure is assumed on the lateral boundaries of the crust. The latter condition means that all lateral boundaries of the crust move horizontally. The crust and the mantle represent two bodies in contact, and Coulomb friction with the coefficient of friction μ = 0.9 is assumed for the surface of the deformable contact. For the thermal problem, we set up the boundary conditions of the isothermal upper surface (T = 0 °C), heat-insulated lateral boundaries, the constant thermal flux at the base of the lithosphere outside the plume region and the constant temperature (1450, 1550 and 1650 °C) in the 100-km zone of the sublithospheric upper mantle plume (Fig. 3.30).

Table 3.8 Thermomechanical model parameters of the crust and mantle

The model accounts for interaction of a partially molten material with the mantle at subsolidus temperature, which requires consideration of contrasting rheological properties of the rocks. The main strain mechanism in mantle under high temperatures and pressures is described by dislocation creep (Karato and Wu 1993). In the subsolidus and partially molten state, the medium is described by non-Newtonian nonlinear viscosity. A common approach to modeling the mantle flows consists in the use of constitutive relations for the non-Newtonian viscous incompressible fluid:

$$\sigma_{ij} = - p\delta_{ij} + s_{ij} ;\quad s_{ij} = 2\eta d_{ij},$$

(3.25)

where p is the lithostatic pressure; s_ij are the components of the Cauchy deviator stress tensor; η is the nonlinear P-T-dependent viscosity expressed as (Ranalli 1995):

$$\eta = \exp \left( {\frac{ - c\phi }{n}} \right){\rm A}^{{ - \frac{1}{n}}} \left[ {\dot{\varepsilon }_{II} } \right]^{{\frac{(1 - n)}{n}}} \exp \left( {\frac{{H + pV_{a} }}{nRT}} \right),\,\,\dot{\varepsilon }_{II} \equiv \frac{1}{2}d_{ij} d_{ij},$$

(3.26)

where $\dot{\varepsilon }_{II}$ is the second invariant of the strain rate tensor; T is temperature; p is pressure; A is the coefficient before the exponent; n is the power-law exponent (n = 1 for the Newtonian liquid); $\phi$ is the melt fraction; H and Va are activation energy and activation volume, respectively; c is a given parameter in the interval from 30 to 45; and R is the universal gas constant.

In this model we used complete constitutive equations of thermal elastoplasticity with allowance for creep deformations. With the combined rheological model, the components of strain rate tensor are cast as the sum of rates of elastic $d_{ij}^{e}$, thermal $d_{ij}^{T}$, plastic $d_{ij}^{p}$ and creep strains $d_{ij}^{c}$ (Korobeynikov 2000):

$$d_{ij}^{{}} = d_{ij}^{e} + d_{ij}^{T} + d_{ij}^{p} + d_{ij}^{c}.$$

(3.27)

It is suggested that non-elastic components of the strain rate tensor correspond to the conditions of incompressibility, i.e., for the plain strain conditions, they can be written as $d_{11}^{p} + d_{22}^{p} = 0$ and $d_{11}^{c} + d_{22}^{c} = 0.$

In order to describe the rates of plastic and thermal strains, we used standard approach described in the works (Korobeynikov 2000; Polyansky et al. 2010a). As was shown by experiments on the rock strength, the tangential stresses never exceed some value corresponding to the yield stress, $\sigma_{\text{Y}}$. In this relation, our modeling was based on the rheological model of medium involving both plastic and viscous strains. The plastic strains in mantle (in nearly liquid state) can be described by the model of material with the Huber—Mises yield surface, with sufficiently low value of yield stress (Gerya and Burg 2007; Polyansky et al. 2010b). The computations show that the yield stress varied from 100 to 400 MPa for crustal material and of 0.1 to 1 MPa for mantle plume.

For the creep components of the strain rate tensor, we use the Norton steady-state creep law (Korobeynikov 2000):

$$d_{ij}^{c} = \frac{3}{2}\frac{{\dot{\bar{\varepsilon }}^{c} }}{{\bar{\sigma }}}s_{ij},$$

(3.28)

where $\bar{\sigma } = \sqrt {{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}s_{ij} s_{ij} }$ is the effective stress and $\dot{\bar{\varepsilon }}^{c} = \sqrt {{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}d_{ij}^{c} d_{ij}^{c} }$ is the effective rate of creep strain. According to (Ranalli 1995), the equation for effective rate of creep deformations has the form

$$\dot{\bar{\varepsilon }}^{c} = 3^{{ - \frac{n + 1}{2}}} 2^{1 - n} A\exp \left( {c\phi } \right)\bar{\sigma }^{n} \exp \frac{{ - \left( {H + pV_{0} } \right)}}{RT}$$

(3.29)

The experimental data on the strains in partially molten and aqueous olivine–basalt aggregates suggest the exponential decay of viscosity with increase of melt fraction: ${\eta \mathord{\left/ {\vphantom {\eta {\eta_{0} }}} \right. \kern-0pt} {\eta_{0} }} = \exp \left( {\frac{ - c\phi }{n}} \right)$ where c ≈ 45 (Mei et al. 2002).

At such parameters, the ratio of mixture viscosity to that in the absence of melt was taken within the range from 0.2 to 10⁻³. Geochemical data on mantle peridotites from xenoliths within the ancient cratons such as Siberian, Slave and Kaapval (Walter 2003), indicate a degree of melting and extraction of the melt in the range of 30–50% with an average value of 45%. Figure 3.70 shows the temperature dependence of viscosity of wet and dry mantle rocks according to (3.26) at the strain rate of 10⁻¹³ s⁻¹ based on experimental data of Chopra and Patterson (1984), Karato and Wu (1993).

Fig. 3.70

Density and viscosity distribution as a function of temperature and phase transition during melting, accepted in models. Moho depth is provisional. 1, wet quartzite (Kronenberg and Tullis 1984); 2, diabase (Carter and Tsenn 1987); 3, wet dunite (Chopra and Patterson 1984); 4, molten wet olivine-basalt (Mei et al. 2002)

The model was underlain by the reference values of parameters, which corresponded to the continental felsic crust (Ranalli 1995; Gerya and Burg 2007) and mantle with rheology of wet dunite (Chopra and Patterson 1984), dry olivine (Karato and Wu 1993) or olivine–basalt–water–melt aggregate (Mei et al. 2002). Rheological parameters of crustal and mantle rocks are shown in Table 3.8.

The equation of state is assumed in the form of the dependence of density ρ on thermal expansion α and melt fraction $\phi$:

$$\rho = \rho_{s} \left( {1 {-} \alpha T - \frac{{\rho_{s} - \rho_{m} }}{{\rho_{s} }}\phi } \right),$$

(3.30)

where $\rho_{m}$ and $\rho_{s}$ are the densities of the melt and the solid matrix, respectively. Beyond the melt region, $\phi = 0$ and density variations are only due to thermal expansion; in the melt region, the melt fraction is assumed to be constant. Figure 3.70 shows the temperature dependence of density accounting for phase transition during melting of the granite crust and peridotite mantle.