Optimization of open-loop shallow geothermal systems using the adjoint approach

Groundwater temperature directly affects the efficiency of groundwater heat pumps, also known as open-loop shallow geothermal systems. On the other side, groundwater heat pumps cause thermal anomalies in the groundwater, which can reach neighboring systems and deteriorate their operation. Therefore, it is important to optimally position these systems to avoid negative interactions and maximize overall efficiency. Flow and heat transport in porous media can be used to describe the influences of open-loop systems numerically. The processes are described with a system of nonlinear coupled PDEs. In addition, source/sink terms, representing extraction and injection wells, are modeled by non-smooth delta functions. The underlying problem is a PDE-constrained optimization problem including control (spatial coordinates of wells) and state (groundwater temperature) constraints. In this talk, we introduce an adjoint-based approach to solve this non-smooth PDE-constrained optimization problem. Dirac delta functions are approximated with smooth bump functions, which decouples source points from the mesh and enables computation of gradients. Nonlinear state constraints are incorporated using Moreau-Yosida type regularization terms. Spatial and temporal discretization, including meshing strategies, are analyzed as well. The approach is applied on real case scenarios with different numbers of heat pumps.