Optimization of deep hydrothermal systems via the adjoint approach

The exploitation of Deep Hydrothermal Systems (DHS) offers significant potential for decarbonizing district heating networks by providing renewable baseload energy. However, the sustainable management of these resources requires balancing conflicting physical phenomena. DHS operations extract hot fluid from deep aquifers and reinject heat-depleted water. This process creates a hydraulic head loss near production wells, increasing pumping costs, while simultaneously driving the cold-water plume toward the producer. While placing injection wells nearby mitigates hydraulic pressure drop, it accelerates thermal breakthrough, reducing the system’s thermal capacity.

In this work, we present a computational framework to maximize the net energy extraction of DHS—defined as thermal energy production minus pumping energy consumption—by optimizing well flow rates and positions. We formulate the problem as a PDE-constrained optimization governed by a coupled thermo-hydraulic (TH) model. To solve this efficiently, we utilize the Finite-Element Method (FEM) combined with the adjoint approach to compute gradients, allowing for the use of the Interior Point Optimizer (IPOPT). This is paired with a multi-start strategy to approximate global optimality. Compared to gradient-free algorithms, this gradient-based method offers superior convergence rates, making the optimization of large-scale systems computationally tractable.

We validate the proposed framework by benchmarking against analytical solutions for homogeneous reservoirs, subsequently demonstrating its efficacy through numerical examples in 2D aquifers with heterogeneous hydraulic properties. The results illustrate how optimal well configurations shift based on subsurface permeability structures. Ultimately, this gradient-based framework provides a computationally efficient foundation for optimizing DHS in structurally complex 3D geothermal reservoirs.