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

BC and ICExpressionDescription
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)
1. Ai is the flow cross-section, Гij is the interfacial area between the fluid and solid structures, e.g., drill pipe, casing and formation, H is the well depth, Tf (r,z) is the formation temperature, Tinj is the injection temperature of the fluid, h is the heat transfer coefficient