In this paper, a new time-dependent 2D/3D stochastic closed-loop geothermal system with multiplicative noise is developed and studied. This model considers heat transfer between the free flow in the pipe region and the porous media flow in the porous media region. Darcy’s law and stochastic Navier-Stokes equations are used to control the flows in the pipe and porous media regions, respectively. The heat equation is coupled with the flow equation to describe the heat transfer in these both regions. In order to avoid sub-optimal convergence, a new mixed finite element method is proposed by using the Helmholtz decomposition that drives the multiplicative noise. Then, the stability of the proposed method is proved, and we obtain the optimal convergence order \(o(\Delta t^{\frac{1}{2}}+h)\) of global error estimation. Finally, numerical results indicate the efficiency of the proposed model and the accuracy of the numerical method.

本文开发并研究了一种新的具有乘性噪声的时间相关 2D/3D 随机闭环地热系统。该模型考虑管道区域中的自由流动与多孔介质区域中的多孔介质流动之间的传热。达西定律和随机纳维-斯托克斯方程分别用于控制管道和多孔介质区域中的流动。热方程与流动方程相结合来描述这两个区域的传热。为了避免次优收敛，通过使用驱动乘性噪声的亥姆霍兹分解，提出了一种新的混合有限元方法。然后证明了该方法的稳定性，得到了全局误差估计的最优收敛阶数\(o(\Delta t^{\frac{1}{2}}+h)\) 。最后，数值结果表明了所提出模型的效率和数值方法的准确性。