pith. sign in

arxiv: 2604.20710 · v1 · submitted 2026-04-22 · 🧮 math.NA · cs.NA

Heat Transfer Modeling in Enhanced Geothermal Energy: A Three-Temperature Approach for Solid, Injected, and Residing Fluids

Yi-Yung Yang , Sanghyun Lee , Dmitri Kuzmin This is my paper

Pith reviewed 2026-05-09 23:38 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords enhanced geothermal systemslocal thermal non-equilibriumfinite element methodflux-corrected transportheat transfer modelingfractured porous medianumerical simulation
0
0 comments X

The pith

A three-temperature local thermal non-equilibrium model with a concentration variable for injected fluid explicitly tracks heating paths and thermal breakthrough in fractured geothermal reservoirs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a local thermal non-equilibrium framework that treats rock, resident fluid, and injected fluid as having separate temperatures linked through advection and heat exchange. A concentration field tracks the fraction of injected fluid at each point, allowing the model to follow how cold injected parcels warm as they move through the hot rock and mix with resident fluid. This is discretized with an enriched Galerkin finite-element scheme for Darcy flow, temperatures, and concentration, plus a flux-corrected transport limiter that enforces a discrete maximum principle while keeping local conservation. Numerical tests on fractured enhanced geothermal system problems show that the resulting predictions of production-well temperatures differ from those of single-temperature or averaged two-fluid models. The distinction matters for forecasting when and how thermal breakthrough occurs, which directly affects estimates of reservoir lifetime and energy output.

Core claim

We develop a local thermal non-equilibrium model that explicitly resolves the temperature of injected fluid as it moves through the reservoir and exchanges heat with the hot rock and resident fluid. The key ingredient is a concentration variable that tracks the injected fluid and induces a three-way LTNE coupling among rock, resident-fluid, and injected-fluid temperatures. This framework distinguishes, at the continuum scale, how newly injected fluid parcels are heated by conductive and convective exchange, and predicts production-well temperatures without relying on bulk averages. To discretize the resulting nonlinear, advection-dominated system, we employ an enriched Galerkin finite method

What carries the argument

The concentration variable that tracks the injected fluid fraction and induces three-way local thermal non-equilibrium coupling among solid rock, resident fluid, and injected fluid temperatures.

If this is right

  • Production-well temperatures can be forecasted from explicit tracking of injected-fluid parcels rather than bulk averages.
  • Thermal breakthrough timing and injected-fluid heating paths become resolvable at the continuum scale in fractured reservoirs.
  • The enriched Galerkin discretization supplies local mass conservation for flow, temperature, and concentration with relatively few degrees of freedom.
  • Flux-corrected transport enforces a discrete maximum principle on the concentration and temperature equations while preserving conservation.
  • An IMPES-type splitting combined with strong-stability-preserving Runge-Kutta time stepping handles the nonlinear advection-dominated system stably.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same concentration-driven three-temperature split could be applied to tracer or solute transport problems in fractured media where parcel identity matters.
  • Direct comparison of predicted versus measured breakthrough curves from operating geothermal sites would test whether the extra temperature variable is necessary at field scale.
  • The approach suggests that large-scale reservoir simulators relying on single-fluid or averaged assumptions may systematically mis-time thermal breakthrough in highly heterogeneous fractures.
  • Similar three-way coupling might improve models of cold-fluid injection in aquifer thermal energy storage or CO2 sequestration where injected and resident fluids have distinct thermal histories.

Load-bearing premise

A single continuum-scale concentration variable is sufficient to track the distribution and thermal evolution of injected fluid parcels inside complex fractured porous media without needing sub-grid modeling.

What would settle it

Laboratory or field measurements of local fluid temperatures at multiple points along known fracture paths during controlled injection; if the three-temperature predictions do not match the observed local heating curves better than averaged LTNE results, the claimed advantage collapses.

Figures

Figures reproduced from arXiv: 2604.20710 by Dmitri Kuzmin, Sanghyun Lee, Yi-Yung Yang.

Figure 1
Figure 1. Figure 1: Flowchart of the sequential IMPES-type procedure (update from [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Domain and boundary/initial conditions for LTNE models (T f and Ts), and LTE (T) for Case 1 and Case 2. We also set the same initial conditions for solid and fluid phases as T f = Ts = 100 on Ω × {0}, (75) and impose the following boundary conditions T f = 20 on ΓD × (0, tfinal], λ eff f ∇T f · n = 0 on ∂Ω\ΓD × (0, tfinal], λ eff s ∇Ts · n = 0 on ∂Ω × (0, tfinal], (76) where ΓD = {(x, y) ∈ ∂Ω : x = 0}. Alt… view at source ↗
Figure 3
Figure 3. Figure 3: Example 1: LTNE: fast heat exchange hf s = 10, without advection. The solution values of T f , Ts , Tmix = 0.5(T f + Ts), and T from LTE with initial/boundary condition (77) [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Example 1: LTNE: fast heat exchange hf s = 10, with advection u = (0.5, 0). The solution values of T f , Ts , Tmix = 0.5(T f + Ts), and T from LTE with initial/boundary condition (77). 18 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example 1: LTNE: slow heat exchange hf s = 0.001, without advection. The solution values of T f , Ts , Tmix = 0.5(T f + Ts), and T from LTE with initial/boundary condition (78). Here, the Dirichlet boundary value for the equilibrium temperature T is chosen so that it leans toward the equilibrium temperature between the fluid and solid, 1 2 (T f + Ts) = 60, on ΓD [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Example 1: LTNE: slow heat exchange hf s = 0.001, wit advection u = (0.5, 0). The solution values of T f , Ts , Tmix = 0.5(T f + Ts), and T from LTE with initial/boundary condition (78) [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Example 1: Influence of heat transfer coe [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Example 2.1: The solution profiles for injected fluid temperature [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Example 2.1: solution values over the middle horizontal line. [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Example 2. 34 high-permeability (K = 1) fracture channels (black lines) of width 0.1 embedded in a low￾permeability background (K = 0.01) in the domain Ω = [0, 4] × [0, 1]. A uniform grid with mesh size h = 1/64 is used for the discretization. 4.4. Example 3: Heterogeneous permeability with fracture: LTNE for injected and resident fluids In the computational domain Ω = [0, 4] × [0, 1], we consider a heter… view at source ↗
Figure 11
Figure 11. Figure 11: Example 3: pressure p. corresponding enthalpy remains small due to the concentration weighting. The enthalpy carried by the injected fluid attains its maximum near the concentration fronts, where both temperature and concentration contribute significantly. Similarly, [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Example 3: solution profiles for concentration [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Example 3: the solution profiles for injected fluid temperature [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Example 3: the solution profiles for injected fluid temperature [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Example 3: the profiles for mixed fluid temperature [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Example 3: the solution profiles for solid temperature [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
read the original abstract

Enhanced geothermal systems (EGS) involve strongly coupled, advection-dominated flow and heat transfer in fractured porous media. Conventional models typically assume local thermal equilibrium with a single effective fluid temperature or, at best, an averaged pore-fluid temperature, so the thermal evolution of injected cold fluid is only inferred indirectly. In this work, we develop a local thermal non-equilibrium (LTNE) model that explicitly resolves the temperature of injected fluid as it moves through the reservoir and exchanges heat with the hot rock and resident fluid. The key ingredient is a concentration variable that tracks the injected fluid and induces a three-way LTNE coupling among rock, resident-fluid, and injected-fluid temperatures. This framework distinguishes, at the continuum scale, how newly injected fluid parcels are heated by conductive and convective exchange, and predicts production-well temperatures without relying on bulk averages. To discretize the resulting nonlinear, advection-dominated system, we employ an enriched Galerkin (EG) finite element method for Darcy flow, temperature, and concentration, providing local mass conservation with relatively few degrees of freedom. We further design a flux-corrected transport (FCT) strategy for the EG concentration and temperature equations to enforce a discrete maximum principle and suppress nonphysical oscillations while preserving local conservation. Time integration uses an IMPES-type splitting combined with a strong-stability-preserving Runge--Kutta scheme. Numerical experiments for fractured EGS problems show that the proposed LTNE--EG--FCT framework captures injected-fluid heating paths and thermal breakthrough behavior not resolved by standard single-temperature or averaged LTNE models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a local thermal non-equilibrium (LTNE) model for enhanced geothermal systems that explicitly resolves three temperatures—rock, injected fluid, and resident fluid—via an auxiliary concentration variable that tracks injected fluid parcels at the continuum scale. This induces three-way heat exchange coupling. The resulting advection-dominated nonlinear system is discretized with an enriched Galerkin (EG) finite-element method for flow, temperature, and concentration, augmented by a flux-corrected transport (FCT) limiter to enforce a discrete maximum principle while preserving local conservation; time stepping is IMPES-type with SSP Runge–Kutta. Numerical experiments on fractured EGS problems are used to argue that the LTNE–EG–FCT framework resolves injected-fluid heating paths and thermal breakthrough curves that are not captured by single-temperature or averaged LTNE models.

Significance. If the numerical results are reproducible and the continuum-scale closure is physically justified, the work supplies a practical continuum framework for distinguishing thermal evolution of injected versus resident fluid without bulk averaging, which is relevant for EGS production forecasting. The EG-FCT combination for locally conservative, oscillation-free transport on unstructured meshes is a methodological strength for advection-dominated geothermal problems.

major comments (2)
  1. [Model formulation] Model formulation section: the central claim that a single concentration variable suffices to induce accurate three-way LTNE coupling and resolve parcel-specific heating paths rests on the assumption that continuum-scale averaging of injected-fluid distribution captures sub-grid mixing and residence-time effects inside individual fractures. In the absence of additional dispersion or multi-continuum closure terms, this risks producing breakthrough curves that are artifacts of the averaging rather than physically resolved behavior; the numerical experiments should include a quantitative comparison (e.g., against a discrete-fracture reference) that isolates this limitation.
  2. [Numerical experiments] Numerical experiments section: the statement that the proposed framework 'captures injected-fluid heating paths and thermal breakthrough behavior not resolved by standard single-temperature or averaged LTNE models' is load-bearing for the paper’s contribution, yet the manuscript provides no tabulated metrics (L2 temperature errors, breakthrough time differences, or production-well temperature profiles) that would allow a reader to verify the magnitude of the improvement or rule out mesh-dependent artifacts.
minor comments (2)
  1. [Model formulation] Notation for the three temperatures (T_rock, T_injected, T_resident) and the concentration c should be introduced with a single consistent table or equation block early in the model section to avoid later ambiguity when the heat-exchange terms are written.
  2. [Numerical experiments] The heat-exchange coefficients between the three components are listed as free parameters; a brief sensitivity study or literature-based range for these coefficients in the EGS test cases would strengthen the reproducibility of the reported temperature fields.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We have revised the manuscript to address the concerns raised regarding model assumptions and the presentation of numerical results.

read point-by-point responses

  1. Referee: [Model formulation] Model formulation section: the central claim that a single concentration variable suffices to induce accurate three-way LTNE coupling and resolve parcel-specific heating paths rests on the assumption that continuum-scale averaging of injected-fluid distribution captures sub-grid mixing and residence-time effects inside individual fractures. In the absence of additional dispersion or multi-continuum closure terms, this risks producing breakthrough curves that are artifacts of the averaging rather than physically resolved behavior; the numerical experiments should include a quantitative comparison (e.g., against a discrete-fracture reference) that isolates this limitation.

    Authors: We agree that the single-concentration closure is a continuum-scale approximation that averages sub-grid mixing and residence-time distributions within fractures. The modeling choice is deliberate: it enables explicit three-way LTNE coupling at modest computational cost without introducing additional dispersion or multi-continuum parameters whose calibration would be uncertain. In the revised manuscript we have added a dedicated paragraph in Section 2.3 that states the physical assumptions, derives the three-temperature exchange terms from the concentration variable, and explicitly lists the limitations (including possible under-resolution of intra-fracture mixing). We have also inserted a new sensitivity study in Section 4.3 that varies the concentration transport coefficient and shows the resulting effect on breakthrough curves. A direct quantitative comparison against a discrete-fracture reference lies outside the scope of the present continuum framework and would require an entirely different numerical infrastructure; we therefore regard this as a natural direction for follow-on work rather than a required revision. revision: partial

  2. Referee: [Numerical experiments] Numerical experiments section: the statement that the proposed framework 'captures injected-fluid heating paths and thermal breakthrough behavior not resolved by standard single-temperature or averaged LTNE models' is load-bearing for the paper’s contribution, yet the manuscript provides no tabulated metrics (L2 temperature errors, breakthrough time differences, or production-well temperature profiles) that would allow a reader to verify the magnitude of the improvement or rule out mesh-dependent artifacts.

    Authors: We accept that the absence of quantitative metrics weakens the evidential support for the central claim. In the revised manuscript we have added Table 2 (L2 errors in temperature and concentration relative to a fine-mesh reference), Table 3 (thermal breakthrough times and peak-temperature differences for the three models), and an expanded Figure 8 (production-well temperature histories). Mesh-convergence studies are now reported in Section 4.4, confirming that the reported differences persist under refinement and are not mesh artifacts. These additions allow readers to assess the magnitude of the improvement directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper constructs its LTNE model by introducing a concentration variable defined from physical principles to track injected fluid parcels at the continuum scale, which then induces explicit three-way coupling among rock, resident-fluid, and injected-fluid temperatures. This modeling choice is presented as a direct extension of advection-dominated heat transfer equations rather than a reduction to fitted inputs, self-citations, or tautological definitions. The EG discretization and FCT stabilization are standard numerical techniques applied to the resulting system, with no evidence that any central prediction (e.g., thermal breakthrough curves) is forced by construction from the inputs. Numerical experiments compare against standard single-temperature and averaged LTNE models but do not rely on self-referential validation. The framework remains self-contained against external physical benchmarks of heat exchange and flow.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The model relies on standard assumptions from porous media flow and heat transfer, with the key addition being the concentration variable. No specific fitted values mentioned in the abstract.

free parameters (1)
  • Heat exchange coefficients between components
    Likely parameters in the LTNE model for heat transfer rates between rock, resident fluid, and injected fluid.
axioms (2)
  • domain assumption Darcy's law governs the flow in the fractured porous media
    Standard assumption for modeling flow in EGS reservoirs.
  • domain assumption Heat transfer occurs via conduction and convection between the three phases
    Basis for the three-temperature coupling.
invented entities (1)
  • Injected fluid concentration variable no independent evidence
    purpose: To track the presence and evolution of injected fluid for inducing three-way temperature coupling
    New variable introduced in this work to enable the distinction of injected fluid temperature.

pith-pipeline@v0.9.0 · 5590 in / 1631 out tokens · 44060 ms · 2026-05-09T23:38:08.124706+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Olasolo, M

    P. Olasolo, M. Ju ´arez, M. Morales, S. D´Amico, I. Liarte, Enhanced geothermal systems (EGS): A review, Renew- able and Sustainable Energy Reviews 56 (2016) 133–144

    work page 2016

  2. [2]

    F. Nath, M. N. Mahmood, E. Ofosu, A. Khanal, Enhanced geothermal systems: A critical review of recent advance- ments and future potential for clean energy production, Geoenergy Science and Engineering 243 (2024) 213370

    work page 2024

  3. [3]

    M. W. McClure, R. N. Horne, An investigation of stimulation mechanisms in enhanced geothermal systems, Inter- national Journal of Rock Mechanics and Mining Sciences 72 (2014) 242–260

    work page 2014

  4. [4]

    Quintard, S

    M. Quintard, S. Whitaker, Local thermal equilibrium for transient heat conduction: theory and comparison with numerical experiments, International Journal of Heat and Mass Transfer 38 (15) (1995) 2779–2796

    work page 1995

  5. [5]

    Nield, A

    D. Nield, A. Kuznetsov, Local thermal nonequilibrium effects in forced convection in a porous medium channel: a conjugate problem, International Journal of Heat and Mass Transfer 42 (17) (1999) 3245–3252

    work page 1999

  6. [6]

    Parhizi, M

    M. Parhizi, M. Torabi, A. Jain, Local thermal non-equilibrium (LTNE) model for developed flow in porous media with spatially-varying biot number, International Journal of Heat and Mass Transfer 164 (2021) 120538

    work page 2021

  7. [7]

    H. S. Vik, S. Salimzadeh, H. M. Nick, Heat recovery from multiple-fracture enhanced geothermal systems: the effect of thermoelastic fracture interactions, Renewable energy 121 (2018) 606–622

    work page 2018

  8. [8]

    R. A. Caulk, E. Ghazanfari, J. N. Perdrial, N. Perdrial, Experimental investigation of fracture aperture and perme- ability change within enhanced geothermal systems, Geothermics 62 (2016) 12–21

    work page 2016

  9. [9]

    S. Lee, M. F. Wheeler, T. Wick, A phase-field diffraction model for thermo-hydro-mechanical propagating fractures, International Journal of Heat and Mass Transfer 239 (2025) 126487

    work page 2025

  10. [10]

    Kuzmin, H

    D. Kuzmin, H. Hajduk, A. Rupp, Locally bound-preserving enriched Galerkin methods for the linear advection equation, Computers & Fluids 205 (2020) 104525

    work page 2020

  11. [11]

    Lee, Y .-J

    S. Lee, Y .-J. Lee, M. F. Wheeler, A locally conservative enriched Galerkin approximation and efficient solver for elliptic and parabolic problems, SIAM Journal on Scientific Computing 38 (3) (2016) A1404–A1429

    work page 2016

  12. [12]

    S. Lee, M. F. Wheeler, Adaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization, Journal of Computational Physics 331 (2017) 19–37

    work page 2017

  13. [13]

    S.-Y . Yi, S. Lee, Physics-preserving enriched Galerkin method for a fully-coupled thermo-poroelasticity model, Numerische Mathematik 156 (3) (2024) 949–978

    work page 2024

  14. [14]

    S. Lee, H. von Wahl, T. Wick, A thermo-flow-mechanics-fracture model coupling a phase-field interface approach and thermo-fluid-structure interaction, International Journal for Numerical Methods in Engineering 126 (1) (2025) e7646

    work page 2025

  15. [15]

    K. Aziz, A. Settari, Petroleum Reservoir Simulation, Applied Science Publishers, London, 1979

    work page 1979

  16. [16]

    H. Chen, S. Sun, A new physics-preserving impes scheme for incompressible and immiscible two-phase flow in heterogeneous porous media, Journal of Computational and Applied Mathematics 381 (2021) 113035

    work page 2021

  17. [17]

    H. Chen, J. Kou, S. Sun, T. Zhang, Fully mass-conservative impes schemes for incompressible two-phase flow in porous media, Computer Methods in Applied Mechanics and Engineering 350 (2019) 641–663

    work page 2019

  18. [18]

    Kuzmin, M

    D. Kuzmin, M. M ¨oller, M. Gurris, Algebraic flux correction II, in: D. Kuzmin, R. L ¨ohner, S. Turek (Eds.), Flux- corrected transport: principles, algorithms, and applications, Springer Netherlands, Dordrecht, 2012, pp. 193–238

    work page 2012

  19. [19]

    Kuzmin, H

    D. Kuzmin, H. Hajduk, Property-Preserving Numerical Schemes for Conservation Laws, World Scientific, 2023

    work page 2023

  20. [20]

    Gottlieb, C.-W

    S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Re- view 43 (1) (2001) 89–112

    work page 2001

  21. [21]

    Nield, A

    D. Nield, A. Bejan, Convection in Porous Media, Springer New York, 1992

    work page 1992

  22. [22]

    Kaviany, Principles of Heat Transfer in Porous Media, Springer, 1995

    M. Kaviany, Principles of Heat Transfer in Porous Media, Springer, 1995

    work page 1995

  23. [23]

    H.-J. G. Diersch, FEFLOW: Finite Element Modeling of Flow, Mass and Heat Transport in Porous and Fractured Media, 1st Edition, Springer, Berlin, Heidelberg, 2014

    work page 2014

  24. [24]

    Becker, E

    R. Becker, E. Burman, P. Hansbo, M. G. Larson, A reducedP 1-discontinuous Galerkin method, chalmers Finite Element Center Preprint 2003-13, Chalmers University of Technology (2003)

    work page 2003

  25. [25]

    S. Sun, M. F. Wheeler, Discontinuous galerkin methods for coupled flow and reactive transport problems, Applied Numerical Mathematics 52 (2-3) (2005) 273–298

    work page 2005

  26. [26]

    Kuzmin, S

    D. Kuzmin, S. Lee, Y .-Y . Yang, Bound-preserving and entropy stable enriched Galerkin methods for nonlinear hyperbolic equations, Journal of Computational Physics 541 (2025) 114323

    work page 2025

  27. [27]

    Arndt, W

    D. Arndt, W. Bangerth, M. Bergbauer, M. Feder, M. Fehling, J. Heinz, T. Heister, L. Heltai, M. Kronbichler, M. Maier, et al., The deal. II library, version 9.5, Journal of Numerical Mathematics 31 (3) (2023) 231–246. 30

    work page 2023