Fracture Heat Resistance

A fracture interrupts direct heat conduction through the intact matrix. Heat may still cross the discontinuity through gas, liquid, or solid material in the fracture, as well as through microscopic asperity contacts between its faces. These mechanisms can be represented by an effective thermal resistance.

Series-resistance model

Consider a fracture of aperture \(b\) filled with a material of thermal conductivity \(k_f\). For one-dimensional steady conduction normal to the fracture, the resistance per unit area of the filling layer is

\[R''_{f}=\frac{b}{k_f}.\]

If the two fracture faces also have interfacial resistances \(R''_{c,1}\) and \(R''_{c,2}\), the total resistance per unit area is

\[R''_{\mathrm{tot}}=R''_{c,1}+\frac{b}{k_f}+R''_{c,2}.\]

The corresponding effective conductance is \(h_f=1/R''_{\mathrm{tot}}\), and the heat flux from side 1 to side 2 is

\[q''_{1\rightarrow2} =h_f(T_1-T_2) =\frac{T_1-T_2}{R''_{\mathrm{tot}}}.\]

Here \(T_1\) and \(T_2\) are the temperatures on the opposing sides of the fracture. For a fracture with negligible filling-layer resistance, \(b/k_f\) may be omitted; for a dry, open fracture, the effective conductance can instead be calibrated to account for gas conduction and radiation if those mechanisms are included in the intended model.

Interpretation and assumptions

The series model assumes heat flows approximately normal to a locally planar fracture and that the layer and interface resistances act in series. The resistance has units of \(\mathrm{m^2\,K/W}\), while conductance has units of \(\mathrm{W/(m^2\,K)}\). For a contact area \(A\), the total heat transfer rate is \(\dot Q=Aq''_{1\rightarrow2}\). A larger aperture or lower filling conductivity increases the layer resistance and reduces heat transfer, all else being equal.

This effective fracture resistance represents heat transmission across the fracture itself. It is distinct from the contact-thermal exchange law used when two discrete elements or blocks are in thermal contact. In a numerical model, the effective conductance should reflect the chosen representation so that fracture resistance and contact resistance are not inadvertently counted twice.