A Hydromechanical Model for Lower Crustal Fluid Flow

Abstract

Metamorphic devolatilization generates fluids at, or near, lithostatic pressure. These fluids are ultimately expelled by compaction. It is doubtful that fluid generation and compaction operate on the same time scale at low metamorphic grade, even in rocks that are deforming by ductile mechanisms in response to tectonic stress. However, thermally-activated viscous compaction may dominate fluid flow patterns at moderate to high metamorphic grades. Compaction-driven fluid flow organizes into self-propagating domains of fluid-filled porosity that correspond to steady-state wave solutions of the governing equations. The effective rheology for compaction processes in heterogeneous rocks is dictated by the weakest lithology. Geological compaction literature invariably assumes linear viscous mechanisms; but lower crustal rocks may well be characterized by non-linear (power-law) viscous mechanisms. The steady-state solutions and scales derived here are general with respect to the dependence of the viscous rheology on effective pressure. These solutions are exploited to predict the geometry and properties of the waves as a function of rock rheology and the rate of metamorphic fluid production. In the viscous limit, wavelength is controlled by a hydrodynamic length scale δ, which varies inversely with temperature, and/or the rheological length scale for thermal activation of viscous deformation l _A , which is on the order of a kilometer. At high temperature, such that δ < l _A , waves are spherical. With falling temperature, as δ → l _A , waves flatten to sill-like structures. If the fluid overpressures associated with viscous wave propagation reach the conditions for plastic failure, then compaction induces channelized fluid flow. The channeling is caused by vertically elongated porosity waves that nucleate with characteristic spacing δ. Because δ increases with falling temperature, this mechanism is amplified towards the surface. Porosity wave passage is associated with pressure anomalies that generate an oscillatory lateral component to the fluid flux that is comparable to the vertical component. As the vertical component may be orders of magnitude greater than time-averaged metamorphic fluxes, porosity waves are a potentially important agent for metasomatism. The time and spatial scales of these mechanisms depend on the initial state that is perturbed by the metamorphic process. Average fluxes place an upper limit on the spatial scale and a lower limit on the time scale, but the scales are otherwise unbounded. Thus, inversion of natural fluid flow patterns offers the greatest hope for constraining the compaction scales. Porosity waves are a self-localizing mechanism for deformation and fluid flow. In nature these mechanisms are superimposed on patterns induced by far-field stress and pre-existing heterogeneities.

Appendix: Steady-State Porosity Waves in a Viscous Matrix

This appendix presents a steady-state wave solution for flow of an incompressible fluid through a viscous matrix composed of incompressible solid grains. Geological compaction literature invariably assumes Newtonian behavior for the viscous mechanism; however, lower crustal environments may well be characterized by power-law viscous rheology (e.g., Kohlstedt et al. 1995). Accordingly, the solution derived here is general with respect to the dependence of the viscous rheology on effective pressure. Aside from this modification, the mathematical formulation of the governing compaction equations is identical to that of Connolly and Podladchikov (2000, 2007).

Conservation of solid and fluid mass requires

$$ \frac{{\partial (1 - {\phi} )}}{{\partial t}} + \nabla \cdot \left( {\left( {1 - {\phi} } \right){{{\mathbf{v}}}_{\rm{s}}}} \right) = 0 $$

(14.35)

and

$$ \frac{{\partial {\phi} }}{{\partial t}} + \nabla \cdot \left( {{\phi} {{{\mathbf{v}}}_{\rm{f}}}} \right) = 0, $$

(14.36)

where subscripts f and s distinguish the velocities, v, of the fluid and matrix. From Darcy’s law, force balance between the matrix and fluid requires

$$ {\mathbf{q}} = {\phi} \left( {{{{\mathbf{v}}}_{\rm{f}}} - {{{\mathbf{v}}}_{\rm{s}}}} \right) = - \frac{k}{\mu }\left( {\nabla {{p}_{\rm{f}}} - {{{\rho} }_{\rm{f}}}{\hbox{g}}{{{\mathbf{u}}}_{\rm{z}}}} \right). $$

(14.37)

In one-dimensional compaction of a vertical column, mean stress is identical to the load

$$ \bar{\sigma } = \int\limits_0^z {\left( {\left( {1 - {\phi} } \right){{{\rho} }_{\rm{s}}} + {\phi} {{{\rho} }_{\rm{f}}}} \right) } {\hbox{g}}{{{\mathbf{u}}}_{\rm{z}}}{\hbox{d}}z. $$

(14.38)

Thus, in terms of effective pressure, ${{p}_{\rm{e}}} = \bar{\sigma } - {{p}_{\rm{f}}},$ Eq. 14.37 is

$$ {\phi} \left( {{{{\mathbf{v}}}_{\rm{f}}} - {{{\mathbf{v}}}_{\rm{s}}}} \right) = \frac{k}{\mu }\left( {\nabla {{p}_{\rm{e}}} - \left( {1 - {\phi} } \right)\Delta {\rho} {\hbox{g}}{{{\mathbf{u}}}_{\rm{z}}}} \right). $$

(14.39)

The divergence of the total volumetric flux of matter is the sum of Eqs. 14.35 and 14.36

$$ \nabla \cdot \left( {{{{\mathbf{v}}}_{\rm{s}}} + {\phi} \left( {{{{\mathbf{v}}}_{\rm{f}}} - {{{\mathbf{v}}}_{\rm{s}}}} \right)} \right) = 0, $$

(14.40)

and substituting Eq. 14.39 into Eq. 14.40

$$ \nabla \cdot \left( {{{{\mathbf{v}}}_{\rm{s}}} + \frac{k}{\mu }\left( {\nabla {{p}_{\rm{e}}} - \left( {1 - {\phi} } \right)\Delta {\rho} {\hbox{g}}{{{\mathbf{u}}}_{\rm{z}}}} \right)} \right) = 0. $$

(14.41)

Matrix rheology is introduced with Eq. 14.16 by observing that the divergence of the solid velocity is the dilational strain rate of the matrix

$$ \nabla \cdot {{{\mathbf{v}}}_{\rm{s}}} = \frac{{\phi} }{{1 - {\phi} }}{{\dot{\varepsilon }}_{\phi}} = - {c}_{\sigma}{{f}_{\phi}}A{{\left| {p_{\rm{e}}} \right|}^{{{{n}_{\sigma }} - 1}}}p_{\rm{e}}^{{}} $$

(14.42)

where ${{f}_{\phi}} = {\phi} \left( {1 - {\phi} } \right)/{{\left( {1 - {{{\phi} }^{{1/{{n}_{\sigma }}}}}} \right)}^{{{{n}_{\sigma }}}}}$(Wilkinson and Ashby 1975). As the functional form of Eq. 14.42 may vary depending on the magnitude of the porosity and the viscous mechanism (Ashby 1988), the subsequent analysis assumes f _φ is an unspecified function of porosity.

To avoid the unnecessary complication associated with the use of vector notation for a one-dimensional problem, in the remainder of this analysis vector quantities are represented by signed scalars and the gradient and divergence operators are replaced by ∂/∂z. Supposing the existence of a steady state solution in which fluid explusion is accomplished by waves that propagate with unchanging form through a matrix with background porosity φ₀ filled by fluid at zero effective pressure, then, in a reference frame that travels with the wave, integration of Eq. 14.35 gives the matrix velocity as

$$ {{v}_{\rm{s}}} = {{v}_{\infty }}\frac{{1 - {{{\phi} }_0}}}{{1 - {\phi} }} $$

(14.43)

where v _∞ is the solid velocity in the limits φ → φ₀ and p _e → 0, i.e., at infinite distance from the wave. After substitution of Eq. 14.43, the integrated form of Eq. 14.41 can be rearranged to

$$ \frac{{\partial {{p}_{\rm{e}}}}}{{\partial z}} = \left( {{{q}_{\rm{t}}} - {{v}_{\infty }}\frac{{1 - {{{\phi} }_0}}}{{1 - {\phi} }}} \right)\frac{\mu }{k} + \left( {1 - {\phi} } \right)\Delta \rho {\hbox{g}} $$

(14.44)

where q _t = φv _f + (1 − φ)v _s is the constant, total, volumetric flux of matter through the column, which evaluates in the limit φ → φ₀ and p _e → 0 as

$$ {{q}_{\rm{t}}} = {{v}_{\infty }} - \left( {1 - {{{\phi} }_{{0}}}} \right)\frac{{{{k}_0}}}{\mu }\Delta \rho {\hbox{g}} $$

(14.45)

where k ₀ is the permeability at φ₀. Thus, Eq. 14.44 can be rewritten

$$ \frac{{\partial {{p}_{\rm{e}}}}}{{\partial z}} = \Delta {\rho} {\hbox{g}}\left( {1 - {\phi} - \left( {1 - {{{\phi} }_0}} \right)\frac{{{{k}_0}}}{k}} \right) - {{v}_{\infty }}\frac{\mu }{k}\frac{{{\phi} - {{{\phi} }_0}}}{{1 - {\phi} }}. $$

(14.46)

Likewise, substitution of Eq. 14.43 into Eq. 14.42 yields

$$ \frac{{\partial {\phi} }}{{\partial z}} = - \frac{{{{{\left( {1 - {\phi} } \right)}}^2}}}{{1 - {{{\phi} }_0}}}{{f}_{\phi}}\frac{{{{c}_{\sigma }}A{{{\left| {p_{\rm{e}}} \right|}}^{{{{n}_{\sigma }} - 1}}}p_{\rm{e}}^{{}}}}{{{{v}_{\infty }}}} $$

(14.47)

If permeability is an, as yet unspecified, function of porosity, then Eqs. 14.46 and 14.47 form a closed system of two ordinary differential equations in two unknown functions, φ and p _e. As v _∞ is the solid velocity at infinite distance from a steady-state wave, if the reference frame is changed to that of the unperturbed matrix, the phase velocity of the wave is v _φ = −v _∞.

For notational simplicity Eqs. 14.46 and 14.47 are now rewritten as

$$ \frac{{\partial {{p}_{\rm{e}}}}}{{\partial z}} = {{f}_1} $$

(14.48)

$$ \frac{{\partial {\phi} }}{{\partial z}} = {{f}_2}\frac{{{{c}_{\sigma }}A}}{{{{v}_{\phi}}}}{{\left| {p_{\rm{e}}} \right|}^{{{{n}_{\sigma }} - 1}}}p_{\rm{e}}^{{}} $$

(14.49)

where f ₁ is the dependence of Eq. 14.46 on φ and v _φ, and f ₂ isolates the dependence of Eq. 14.47 on φ. Combining Eqs. 14.48 and 14.49 to eliminate z, and rearranging, yields

$$ 0 = \frac{{{{c}_{\sigma }}A}}{{{{v}_{\phi}}}}{{\left| {p_{\rm{e}}} \right|}^{{{{n}_{\sigma }} - 1}}}p_{\rm{e}}^{{}}{\hbox{d}}{{p}_{\rm{e}}} - \frac{{{{f}_1}}}{{{{f}_2}}}{\hbox{d}}{\phi}, $$

(14.50)

which must be satisfied by the φ-p _e trajectory of any steady-state solution. Defining a function H such that

$$ H \equiv - \int {\frac{{{{f}_1}}}{{{{f}_2}}}{\hbox{d}}{\phi} }, $$

(14.51)

the integral of Eq. 14.50 yields a function

$$ U \equiv \frac{{{{c}_{\sigma }}A}}{{{{v}_{\phi}}}}\frac{{{{{\left| {p_{\rm{e}}} \right|}}^{{{{n}_{\sigma }} - 1}}}p_{\rm{e}}^{{^2}}}}{{{{n}_{\sigma }} + 1}} + H $$

(14.52)

whose φ-p _e contours explicitly define the φ-p _e trajectory for all steady-state solutions as a function v _φ. Because U increases monotonically, and symmetrically, with positive or negative p _e at constant φ, and H is independent of p _e, the stationary points of U must occur at p _e = 0 and correspond to extrema in H, i.e., the real roots of ${{{\partial H}} \left/ {{\partial \phi = - {{{{{f}_1}}} \left/ {{{{f}_{{2}}}}} \right.} = 0}} \right.}.$ Moreover, as f ₂ must be finite if the matrix is coherent, the roots of ${{{\partial H}} \left/ {{\partial \phi = 0}} \right.}$ are identical to the roots of f ₁ = 0. Therefore φ₀ is always a stationary point, with the character of a focal point if ${{{\partial {{f}_1}}} \left/ {{\partial \phi \, < \,0}} \right.}$ and that of a saddle point if ${{{\partial {{f}_1}}} \left/ {{\partial \phi > \,0}} \right.}$. When φ₀ is a focal point, the steady-state wave solutions correspond to periodic waves that oscillate between two values of porosity on either side of φ₀, characterized by equal H, at which p _e vanishes (Fig. 14.6b). The case of greater interest is a solitary wave (Fig. 14.6a), in which the porosity rises from φ₀ to a maximum, at which H(φ_max) = H(φ₀), and then returns to φ₀. This solution requires both the existence of a focal point at φ > φ₀ and that φ₀ is a saddle point. For the rheological constitutive relation employed here (Eq. 14.42), the first condition is always met when φ₀ is a saddle point. Thus, the critical velocity for the existence of the solitary wave solution, i.e., the bifurcation at which φ₀ switches from focal to saddle point, is

$$ v_{{\phi} }^{\rm{crit}} = - \frac{{{{k}_0}}}{\mu }\left( {1 - {{{\phi} }_0}} \right)\Delta \rho g\left( {\frac{{\left( {1 - {{{\phi} }_0}} \right)}}{{{{k}_0}}}{{{\left. {\frac{{\partial k}}{{\partial {\phi} }}} \right|}}_{{{\phi} = {{{\phi} }_0}}}} - 1} \right), $$

(14.53)

which is obtained by solving ${{{\partial {{f}_1}}} \left/ {{\partial {\phi} = 0}} \right.}$ for v _φ. Substituting the explicit function for permeability given by Eq. 14.17 into Eq. 14.53 yields

$$ v_{{\phi} }^{\rm{crit}} = - \frac{{{{k}_0}}}{{{{{\phi} }_0}\mu }}\left( {1 - {{{\phi} }_0}} \right)\Delta {\rho} g\left( {\left( {1 - {{{\phi} }_0}} \right){{n}_{\phi}} - {{{\phi} }_0}} \right) = {{v}_{{0}}}\left( {\left( {1 - {{{\phi} }_0}} \right){{n}_{\phi}} - {{{\phi} }_0}} \right). $$

(14.54)

Equation 14.54 implies that, in the small-porosity limit, the minimum speed at which steady solitary waves exist is n _φ times the speed of the fluid through the unperturbed matrix.

The relation between solitary wave amplitude (maximum porosity) and v _φ is obtained by solving

$$ H\left( {{{{\phi} }_{{\max }}}} \right) - H\left( {{{\phi }_0}} \right) = - \int\limits_{{{{\phi }_0}}}^{{{{\phi }_{{\max }}}}} {\frac{{{{f}_1}}}{{{{f}_2}}}{\hbox{d}}\phi } = 0. $$

(14.55)

The resulting expressions are cumbersome, but, in the small-porosity limit of Eqs. 14.17 and 14.42, the solution of Eq. 14.55 is

$$ {{v}_{\phi }} = - \frac{{{{c}_{\phi }}\phi_0^{{{{n}_{\phi }} - 1}}\Delta {\rho} {\text{g}}}}{\mu }\left( {{{n}_{\phi }} - 1} \right)\frac{{\phi_0^{{{{n}_{\phi }}}} + \phi_{{\max }}^{{{{n}_{\phi }}}}\left[ {{{n}_{\phi }}\ln \left( {\frac{{{{\phi }_{{\max }}}}}{{\phi_0}}} \right) - 1} \right]}}{{\phi_0^{{{{n}_{\phi }} - 1}}\left[ {{{n}_{\phi }}{{\phi }_{{\max }}} - \phi_0\left( {{{n}_{\phi }} - 1} \right)} \right] - \phi_{{\max }}^{{{{n}_{\phi }}}}}}. $$

(14.56)

From Eq. 14.56 it follows that n _φ > 1 is a necessary condition for the existence of solitary waves. Equation 14.56 also has the surprising implication that amplitude is not a function of n _σ, for large porosity the function f _φ, in the exact form of Eq. 14.42, gives rise to a weak dependence of amplitude on n _σ. For a solitary wave with specified phase velocity, the effective pressure is obtained as an explicit function of φ from the definite integral of Eq. 14.50, which can be rearranged to

$$ p_{\rm{e}} = \pm \root{{{n}_{\sigma}} + 1}\of{\left( {{n}_{\sigma} + 1} \right)\frac{{{{v}_{\phi}}}}{{{{c}_{\sigma}A}}\int\limits_{{{{{\phi} }_0}}}^{{\phi} } {\frac{{{{f}_1}}}{{{{f}_2}}}{\hbox{d}}{\phi} } }}, $$

(14.57)

where the signs have been dropped in view of the symmetry of the solution. And finally, substituting Eq. 14.57 into Eq. 14.47, inverting the result, and integrating yields the depth coordinate relative to the center of a wave as a function of φ

$$ z = \pm \sqrt[{{{n}_{\sigma }} + 1}]{{\frac{{{{v}_{\phi }}}}{{{{c}_{\sigma }}A{{{\left( {{{n}_{\sigma }} + 1} \right)}}^{{{{n}_{\sigma }}}}}}}}}{{\int\limits_{{{{\phi }_{{\max }}}}}^{\phi } {\frac{1}{{{{f}_2}}}\left( {\int\limits_{{{{\phi }_0}}}^{\phi } {\frac{{{{f}_1}}}{{{{f}_2}}}{\text{d}}\phi } } \right)} }^{{ - \frac{{{{n}_{\sigma }}}}{{{{n}_{\sigma }} + 1}}}}}{\text{d}}\phi . $$

(14.58)

To demonstrate that $z \to \pm \infty$ as $\phi \to {{\phi }_0}$, the inner integral and its factor in Eq. 14.58 are approximated by the first non-zero terms of Taylor series expansions about φ = φ₀ to obtain

$$ z \approx \pm {{\left( {\frac{{{{v}_{\phi }}}}{{{{c}_{\sigma }}A{{{\left. {{{f}_2}} \right|}}_{{\phi = {{\phi }_0}}}}}}{{{\left( {\frac{{{{n}_{\sigma }} + 1}}{2}{{{\left. {\frac{{\partial {{f}_1}}}{{\partial \phi }}} \right|}}_{{\phi = {{\phi }_0}}}}} \right)}}^{{ - {{n}_{\sigma }}}}}} \right)}^{{\frac{1}{{{{n}_{\sigma }} + 1}}}}}\int\limits_{\Phi }^0 {{{\Phi }^{{ - \frac{{2{{n}_{\sigma }}}}{{{{n}_{\sigma }} + 1}}}}}{\hbox{d}}\Phi } $$

(14.59)

where Φ = φ − φ₀. In the limit $\Phi \to 0$, the integral in Eq. 14.59 is finite only if n _σ < 1, from which it is concluded that solitary waves have infinite wavelength in a linear or shear thinning viscous matrix, but may have finite wavelength in the peculiar case of a shear thickening viscous matrix. Rewriting the integral in Eq. 14.59 in terms of dlnφ, and differentiating yields

$$ \frac{{\partial z}}{{\partial \ln \Phi }} \approx {{\left( {{{\Phi }^{{1 - {{n}_{\sigma }}}}}\frac{{{{v}_{\phi }}}}{{{{c}_{\sigma }}A{{{\left. {{{f}_2}} \right|}}_{{\phi = {{\phi }_0}}}}}}{{{\left( {\frac{{{{n}_{\sigma }} + 1}}{2}{{{\left. {\frac{{\partial {{f}_1}}}{{\partial \phi }}} \right|}}_{{\phi = {{\phi }_0}}}}} \right)}}^{{ - {{n}_{\sigma }}}}}} \right)}^{{\frac{1}{{{{n}_{\sigma }} + 1}}}}} $$

(14.60)

the depth interval over which porosity decays from eφ₀ to φ₀ within a porosity wave. This interval is taken here as the characteristic length scale for variations in porosity, i.e., the viscous compaction length. The derivative on the right-hand side of Eq. 14.60,

$$ {{\left. {\frac{{\partial {{f}_1}}}{{\partial \phi }}} \right|}_{{\phi = {{\phi }_0}}}} = \Delta \rho g\left( {\frac{{\left( {1 - {{\phi }_0}} \right)}}{{{{k}_0}}}{{{\left. {\frac{{\partial k}}{{\partial \phi }}} \right|}}_{{\phi = {{\phi }_0}}}} - 1} \right) + \frac{{{{v}_{\phi }}\mu }}{{{{k}_0}\left( {1 - {{\phi }_0}} \right)}}, $$

(14.61)

is zero at ${{v}_{\phi }} = v_{\phi }^{\rm{crit}}$, but decreases monotonically with v _φ; thus dropping the first term in Eq. 14.61, substituting ${{v}_{\phi }} = v_{\phi }^{\rm{crit}}$ and Φ = (e − 1) φ₀ in Eq. 14.60, and expanding f ₂ at φ₀ as$\left( {1 - {{\phi }_0}} \right){{\left. {{{f}_{\phi }}} \right|}_{{\phi = {{\phi }_0}}}}$yields

$$ \delta = {{\left[ {{{{\left( {\left( {{\hbox{e}} - 1} \right){{\phi }_0}\Delta \rho {\hbox{g}}\left( {\left[ {1 - {{\phi }_0}} \right]{{{\left. {\frac{{\partial k}}{{\partial \phi }}} \right|}}_{{\phi = {{\phi }_0}}}} - {{k}_0}} \right)} \right)}}^{{1 - {{n}_{\sigma }}}}}{{{\left( {\frac{{2{{k}_0}}}{{{{n}_{\sigma }} + 1}}} \right)}}^{{{{n}_{\sigma }}}}}\frac{1}{{{{c}_{\sigma }}A\mu {{{\left. {{{f}_{\phi }}} \right|}}_{{\phi = {{\phi }_0}}}}}}} \right]}^{{\frac{1}{{{{n}_{\sigma }} + 1}}}}}, $$

(14.62)

a length scale that provides a lower bound on wavelength. For a linear-viscous matrix with shear viscosity η = 1/(3A), the constitutive relation given by Eq. 14.42, and the small-porosity limit, Eq. 14.62 simplifies to

$$ \delta = \sqrt{{\frac{4}{3}\frac{\eta }{{{\phi}_0}}\frac{{k_0}}{\mu}}} $$

which, accounting for differences in the formulation of the bulk viscosity of the matrix, is identical to the viscous compaction length of McKenzie (1984). For a non-linear viscous matrix, making use of constitutive relations given by Eqs. 14.17 and 14.42, in the small-porosity limit the compaction length is

$$ \delta = C\phi_0^{{\frac{{{{n}_{\phi }} - 1}}{{{{n}_{\sigma }} + 1}}}}\sqrt[{{{n}_{\sigma }} + 1}]{{{{{\left( {\frac{2}{{{{n}_{\sigma }} + 1}}} \right)}}^{{{{n}_{\sigma }}}}}\frac{{{{c}_{\phi }}}}{{{{c}_{\sigma }}A\mu {{{\left( {\Delta \rho {\text{g}}} \right)}}^{{{{n}_{\sigma }} - 1}}}}}}}, $$

(14.63)

where

$$ C = \root{{n}_{{\sigma}} + 1}\of{{{{{\left[ {{{n}_{\phi}}\left( {{\hbox{e}} - 1} \right)} \right]}}^{{1 - {{n}_{{\sigma}}}}}}}}. $$

(14.64)

The factor C represents two non-general assumptions of the analysis: that the phase velocity is n _φ v ₀; and that the porosity decay is from eφ₀ to φ₀. In the spirit of dimensional analysis, this factor (~2.27 for n _σ = n _φ = 3) is neglected in the text.