A Generalized Analytical Heat Transfer Model for Enhanced Geothermal Systems: Capturing Fracture Interactions and Correcting Classical Optimistic Predictions

Numerical analytical heat transfer models play a critical role in geothermal design and feasibility studies. Classical solutions, such as those proposed by Gringarten et al. 1975, rely on simplified assumptions and systematically overestimate thermal performance, which can lead to unrealistic engineering decisions. This study presents a generalized analytical model for enhanced geothermal systems that explicitly captures thermal interactions between fractures while preserving analytical tractability. The formulation is based on Greenś functions and reproduces realistic thermal behavior under conditions representative of fractured geothermal reservoirs. The resulting solution is computationally efficient and sufficiently simple to be implemented directly in standard spreadsheets, without requiring Laplace space transformations or numerical inversion algorithms. The model is validated against numerical simulations performed using CMG STARS and Volsung software, showing close agreement in temperature evolution, including the effects of interacting fractures. Compared with classical analytical approaches, the proposed model corrects optimistic bias and provides more reliable predictions of production temperature and energy recovery. These results have direct implications for geothermal feasibility studies, well design, and power forecasting, effectively bridging the gap between legacy analytical models and numerical or commercial engineering tools. Building on the analytical framework originally introduced by Gringarten et al. 1975, the proposed formulation generalizes classical heat transfer solutions to account for fracture interaction while retaining analytical simplicity and practical applicability.

💡 Research Summary

The paper addresses a long‑standing problem in geothermal reservoir engineering: the tendency of classical analytical heat‑transfer models, epitomized by Gringarten et al. (1975), to over‑predict temperature recovery in enhanced geothermal systems (EGS). Those legacy solutions assume an infinite array of perfectly isolated vertical fractures, employ Laplace‑space transforms, and require numerical inversion, all of which hide a set of physical simplifications—most notably the neglect of thermal interaction between fractures, the assumption of extremely low fluid velocities, and the implicit requirement that the dimensionless parameter β approach zero. Consequently, the classical model predicts an unrealistically late thermal breakthrough (≈ 18 years) and an optimistic production‑temperature curve.

To overcome these limitations, the authors develop a generalized analytical model that retains the computational elegance of the Gringarten formulation while explicitly incorporating fracture‑to‑fracture heat exchange. The new framework is built on Green’s‑function theory. Heat conduction in the surrounding granitic matrix is treated as the fundamental solution, and each fracture is represented as a line source (or sink) whose strength depends on fluid flow and aperture. By integrating the Green’s function over the spatial distribution of fractures, the model derives closed‑form expressions for the temperature field that include interaction terms proportional to fracture spacing and array geometry. Importantly, the model works directly in the time domain; the final solution consists of exponential and error‑function terms that can be evaluated with ordinary spreadsheet functions, eliminating the need for Laplace inversion.

The authors adopt a set of realistic assumptions: (i) a finite number of equally spaced horizontal fractures of uniform aperture, (ii) constant rock and fluid thermal properties, (iii) conduction‑dominated heat transfer in the matrix and quasi‑steady convection in the fractures, and (iv) finite reservoir boundaries that generate asymmetric heat drainage for edge fractures. To capture boundary effects, the reservoir is partitioned into longitudinal blocks, each drained by a cluster of fractures. Classical solutions apply within each block, while inter‑block coupling is introduced through effective boundary conditions derived from volume‑averaged heat fluxes. This block‑wise approach naturally reproduces the reduced thermal contribution of edge fractures and allows designers to reposition fractures to mitigate boundary losses.

Validation is performed against three benchmarks: the original Gringarten analytical solution, CMG STARS simulations reported by Zeilani (2021), and Volsung simulations reported by Stacey (2025). Across a range of fracture spacings, fluid flow rates, and reservoir dimensions, the new model matches the numerical simulators within 5 % for temperature evolution and cumulative energy recovery. In scenarios where fracture spacing is tight, the model predicts a substantially earlier thermal breakthrough (≈ 7–9 years) compared with the 18‑year breakthrough of the classical model, aligning closely with the CMG and Volsung results. The authors also demonstrate that the optimistic bias of the classical solution can be traced to the β → 0 assumption; by retaining a finite β, the generalized model accounts for realistic thermal resistance and finite heat‑transfer rates.

From a practical standpoint, the model’s spreadsheet‑ready formulation enables rapid parametric studies during early‑stage feasibility assessments. Engineers can evaluate hundreds of fracture‑layout scenarios, fluid‑injection strategies, and reservoir‑size variations without resorting to costly commercial simulators. This capability reduces decision‑making risk, supports more accurate power‑forecasting, and informs optimal fracture stimulation designs that balance heat extraction against reservoir longevity.

In conclusion, the paper delivers a physically grounded, analytically tractable heat‑transfer model for EGS that corrects the systematic optimism of legacy analytical tools, captures essential fracture‑interaction physics, and remains accessible to practitioners through simple computational implementation. The work bridges the gap between fast but oversimplified analytical methods and accurate but resource‑intensive numerical simulations, offering a valuable new tool for geothermal engineers and decision‑makers.