Thermal non-equilibrium of porous flow in a resting matrix applicable to melt migration: a parametric study
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 Darcy flow. By non-dimensionalizing the equations three non-dimensional numbers can be identified controlling thermal non-equilibrium: the Peclet number Pe describing the fluid velocity, the heat transfer number A describing the local interfacial heat transfer from the fluid to the solid, 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. Three stages are observed: a transient stage followed by a stage with maximum non-equilibrium fluid to solid temperature difference, ∆Tmax, and a stage approaching the steady state. A simplified time-independent ordinary differential equation for depth-dependent (Tf – Ts) is derived and analytically solved. From these solutions simple scaling laws of the form (Tf – Ts) = f (Pe, A, ϕ, H), where H is the non-dimensional model height, are derived. The solutions for ∆Tmax and the scaling laws are in good agreement with the numerical solutions. The parameter space Pe, A, ϕ, H is systematically explored. In the Pe – A – parameter space three regimes can be identified: 1) at high Pe (> 1) strong thermal non-equilibrium develops independently of Pe and A; 2) at low Pe (< 1) and low A (< 1) non-equilibrium decreases proportional to decreasing Pe; 3) at low Pe (<1) and large A (>1) non-equilbrium scales with Pe/A and thus becomes unimportant. The porosity ϕ has only a minor effect on thermal non-equilibrium. The time scales for reaching thermal non-equilibrium scale with the advective time-scale 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 A. Plotting such typical ranges in the Pe – A regime diagram reveals that a) interstitial melt flow is in thermal equilibrium, b) melt channelling as e.g. revealed by dunite channels may reach moderate thermal non-equilibrium, and c) the dyke regime is at full thermal non-equilibrium.
