Thermal non-equilibrium of porous flow in a resting matrix applicable to melt migration: a parametric study

Abstract

Abstract. Fluid flow through rock occurs in many geological settings on different scales, at different temperature conditions and with different flow velocities. Depending on these conditions the fluid will be in local thermal equilibrium with the host rock or not. To explore the physical parameters controlling thermal non-equilibrium, the coupled heat equations for fluid and solid phases are formulated for a fluid migrating through a resting porous solid by porous flow. By non-dimensionalizing the equations, two non-dimensional numbers can be identified controlling thermal non-equilibrium: the Péclet number Pe describing the fluid velocity and the porosity ϕ. The equations are solved numerically for the fluid and solid temperature evolution for a simple 1D model setup with constant flow velocity. This setup defines a third non-dimensional number, the initial thermal gradient G, which is the reciprocal of the non-dimensional model height H. Three stages are observed: a transient stage followed by a stage with maximum non-equilibrium fluid-to-solid temperature difference, ΔTmax⁡, and a stage approaching the steady state. A simplified time-independent ordinary differential equation for depth-dependent (Tf−Ts) is derived and solved analytically. From these solutions simple scaling laws of the form Tf-Ts=fPe,G,z are derived. Due to scaling they do not depend explicitly on ϕ anymore. The solutions for ΔTmax⁡ and the scaling laws are in good agreement with the numerical solutions. The parameter space PeG is systematically explored. Three regimes can be identified: (1) at high Pe (>1/G) strong thermal non-equilibrium develops independently of Pe, (2) at low Pe (<1/G) non-equilibrium decreases proportional to decreasing Pe⋅G, and (3) at low Pe (<1) and G of the order of 1 the scaling law is ΔTmax⁡≈Pe. The scaling laws are also given in dimensional form. The dimensional ΔTmax⁡ depends on the initial temperature gradient, the flow velocity, the melt fraction, the interfacial boundary layer thickness, and the interfacial area density. The time scales for reaching thermal non-equilibrium scale with the advective timescale in the high-Pe regime and with the interfacial diffusion time in the other two low-Pe regimes. Applying the results to natural magmatic systems such as mid-ocean ridges can be done by estimating appropriate orders of Pe and G. Plotting such typical ranges in the Pe–G regime diagram reveals that (a) interstitial melt flow is in thermal equilibrium, (b) melt channeling such as revealed by dunite channels may reach moderate thermal non-equilibrium with fluid-to-solid temperature differences of up to several tens of kelvin, and (c) the dike regime is at full thermal non-equilibrium.