Steady state heat flow in a random medium and the linear heat flow‐heat production relationship

Abstract

The heat transfer in the earth's crust is described by three‐dimensional steady state heat conduction in a heterogeneous medium with randomly varying thermal conductivity and heat production. The heat flow‐heat production relationship generally emerges as the regression of the surface heat flow on surface heat production. The case of normally distributed thermal conductivity and heat production in an upper heterogeneous slab yields a linear relation, the slope and intercept of which depend on the horizontal and vertical correlation scales of heterogeneity, the correlation between thermal conductivity and heat production, and the ratio of heat produced within the zone of heterogeneity to heat from deeper origin. The model is a stochastic extension of the traditional homogeneous slab model, and the intercept of the linear relationship predicts the true average heat flow at a depth equal to the slope, when this depth is smaller than the true slab thickness.