First- and second-order unconditionally stable and decoupled schemes for the closed-loop geothermal system based on the coupled multiphysics model

Abstract

In this paper, we construct first- and second-order implicit–explicit schemes for the closed-loop geothermal system, which includes the heat transfer between the porous media flow with Darcy equation in the geothermal reservoir and the free flow with Navier–Stokes equation in the pipe. The constructed fully discrete schemes are based on the exponential auxiliary variable method in time, which we have proposed in Li et al. [“New SAV-pressure correction methods for the Navier-Stokes equations: Stability and error analysis,” Math. Comput. 91, 141–167 (2022)] and the finite element method in space. These schemes are linear and uniquely solvable, decoupling not only the two flow regions but also the temperature field, and only require solving a sequence of linear differential equations with constant coefficients at each time step. In addition, we rigorously prove that the constructed first- and second-order schemes are unconditionally stable without any time step and stability parameter restrictions. Finally, some numerical simulations, including convergence tests, the benchmark problem for thermal convection in a square cavity, and the heat transfer in simplified closed-loop geothermal systems, are demonstrated to present the reliability and efficiency of the constructed schemes.