Efficient Long-Term Simulation of the Heat Equation with Application in Geothermal Energy Storage

Abstract

Long-term evolutions of parabolic partial differential equations, such as the heat equation, are the subject of interest in many applications. There are several numerical solvers marking the state-of-the-art in diverse scientific fields that may be used with benefit for the numerical simulation of such long-term scenarios. We show how to adapt some of the currently most efficient numerical approaches for solving the fundamental problem of long-term linear heat evolution with internal and external boundary conditions as well as source terms. Such long-term simulations are required for the optimal dimensioning of geothermal energy storages and their profitability assessment, for which we provide a comprehensive analytical and numerical model. Implicit methods are usually considered the best choice for resolving long-term simulations of linear parabolic problems; however, in practice the efficiency of such schemes in terms of the combination of computational load and obtained accuracy may be a delicate issue, as it depends very much on the properties of the underlying model. For example, one of the challenges in long-term simulation may arise by the presence of time-dependent boundary conditions, as in our application. In order to provide both a computationally efficient and accurate enough simulation, we give a thorough discussion of the various numerical solvers along with many technical details and own adaptations. By our investigation, we focus on two largely competitive approaches for our application, namely the fast explicit diffusion method originating in image processing and an adaptation of the Krylov subspace model order reduction method. We validate our numerical findings via several experiments using synthetic and real-world data. We show that we can obtain fast and accurate long-term simulations of typical geothermal energy storage facilities. We conjecture that our techniques can be highly useful for tackling long-term heat evolution in many applications.

Modeling and simulation of the input–output behavior of a geothermal energy storage

“…This paper extends and complements the results in Bähr et al [14, 15], where the authors focus on the numerical simulation of the long‐term behavior of the spatial temperature distribution in a geothermal storage over weeks and months and the interaction between a geothermal storage and its surrounding domain. For simplicity, charging and discharging was described by a simple source term but not by PHXs.…”

“…To this end, we recall that the temperature at the boundary grid points can be obtained by the linear algebraic equations ( 12) through ( 14) derived from the boundary conditions. Further, the values at the interface points are obtained by the interpolation formulas in (15) derived from the perfect contact condition. Thus, we can exclude these grid points from the subsequent considerations where we collect the semidiscrete approximations of the temperature T(t, x i , 𝑦 𝑗 ) at the remaining points of the grid in the vector function…”

“…Recall that 𝛼 M , 𝛽 M , 𝛾 M are given by (11), 𝛼 F± , 𝛽 F , 𝛾 F in (10). Further, 𝜓 M and 𝜓 F are given in (15) and the off-diagonal coefficients 𝛽 M I , 𝛽 F I are given in ( 16) and ( 17). The block matrices on the subdiagonals D ± ∈ R q×q , i = 1, … , N x − 1, are diagonal matrices of the form…”

“…In the present work, we focus on the computation of the of the spatial-temporal temperature distribution and the associated input-output behavior of the geothermal storage on smaller time scales as in [14,15]. This is needed for storages embedded into residential heating systems and the study of the storage's response to charging and discharging operations on time scales from a few minutes to a few days.…”

“…This is needed for storages embedded into residential heating systems and the study of the storage's response to charging and discharging operations on time scales from a few minutes to a few days. We extend the setting in [14,15] and include PHXs for a more realistic model of the storage's charging and discharging process. However, for the sake of simplicity, we do not consider the surrounding medium but reduce the computational domain to the storage depicted in Figure 2 by a black rectangle.…”