Table 1 Boundary and initial conditions of the thermal–hydraulic models

IC

\(T_{i} (r,z,t = 0) = T_{f} (r,z)\), \(i = 1,2,3,4\)

\(\forall r\), \(0 \le z \le H\)

The initial temperature is equal to the formation temperature

BC1

\(v_{{z_{i} }} = \frac{{\dot{m}_{i} }}{{\rho A_{i} }}\), \(z = 0\), \(i = 1,3\)

The velocity of the drill pipe fluid and the annulus fluid is calculated according to the mass flow rate at the wellhead

BC2

\(q = \left. { - \lambda \left( {\frac{\partial T}{\partial r}} \right)} \right|_{{\varGamma_{ij} }} = h(T_{i} - T_{j} )\), on \(\varGamma_{12}\), \(\varGamma_{23}\)\(\varGamma_{34}\)

Heat flux across the solid–fluid interface is determined by the heat transfer coefficient times the temperature difference between fluid and solid wall

BC3

\(- \lambda \left( {\frac{{\partial T_{1} }}{\partial r}} \right) = 0\) at \(z = H,z = 0\)

No thermal gradient at the surface and bottom of the reservoir

BC4

\(T_{4} (r = \infty ,z,t) = T_{f} (r,z)\) at \(r = \infty\)

Formation temperature at the far-field remains undisturbed

BC5

\(T_{1} (r,z = 0,t) = T_{\text{inj}}\) at \(0 < r < r_{1} ,z = 0\)

The temperature at the well-head equals the injection temperature

BC6

\(T_{1} (z = H,t) = T_{3} (z = H,t)\)

The fluid temperature of the drill pipe fluid and the annulus fluid at the bottom hole is equal. This is only validated for the counterflow scenario (mud circulation)