Multiscale Fracture Roughness Effects on Coupled Nonlinear Seepage and Heat Transfer in an EGS Fracture

Abstract

The seepage characteristics and heat transfer efficiency in rough fractures are indispensable for assessing the lifetime and production performance of geothermal reservoirs. In this study, a two-dimensional rough rock fracture model with different secondary roughness is developed using the wavelet analysis method to simulate the coupled flow and heat transfer process under multiscale roughness based on two theories: local thermal equilibrium (LTE) and local thermal nonequilibrium (LTNE). The simulation results show that the primary roughness controls the flow behavior in the main flow zone in the fracture, which determines the overall temperature distribution and large-scale heat transfer trend. Meanwhile, the nonlinear flow behaviors induced by the secondary roughness significantly influence heat transfer performance: the secondary roughness usually leads to the formation of more small-scale eddies near the fracture walls, increasing flow instability, and these changes profoundly affect the local water temperature distribution and heat transfer coefficient in the fracture–matrix system. The eddy aperture and eddy area fraction are proposed for analyzing the effect of nonlinear flow behavior on heat transfer. The eddy area fraction significantly and positively correlates with the overall heat transfer coefficient. Meanwhile, the overall heat transfer coefficient increases by about 3% to 10% for eddy area fractions of 0.3% to 3%. As the eddy aperture increases, fluid mixing is enhanced, leading to a rise in the magnitude of the local heat transfer coefficient. Finally, the roughness characterization was decomposed into primary roughness root mean square and secondary roughness standard deviation, and for the first time, an empirical correlation was established between multiscale roughness, flow velocity, and the overall heat transfer coefficient.

1. Introduction

With the continuous growth of global energy demand, hot dry rock (HDR) geothermal resources demonstrate increasing development potential due to their widespread distribution and high reservoir temperatures [1,2,3]. Enhanced Geothermal Systems (EGS) currently represent the primary method for extracting heat from HDR. In these systems, an artificially created fracture network enables forced convective heat exchange between the low-temperature working fluid and the high-temperature rock mass. Figure 1 illustrates the conceptual diagram of the EGS. The efficiency of heat extraction directly depends on the seepage-heat transfer coupling characteristics in the fracture structures. A comprehensive understanding of heat transport mechanisms in fractured reservoirs not only provides theoretical support for numerical simulations of geothermal systems but also guides the optimization of thermal extraction strategies in engineering practice, thereby improving both the recovery efficiency and the sustainability of geothermal resource development [4,5,6]. It is worth noting that natural fractures modified through reservoir stimulation exhibit more complex geometries, surface roughness, and spatial distributions, which make the heat transfer process show significant nonlinear characteristics [7,8]. Although extensive numerical and experimental studies have been conducted on complex fracture networks [9,10,11,12], fundamental issues related to flow and heat transfer at the single fracture scale, particularly the thermal–hydraulic–mechanical (THM) coupling mechanisms of rough fracture surfaces under realistic high-temperature and high-pressure geological circumstances, still require more accurate physical models and numerical characterization methods.

Conventional fracture seepage theory was based on the parallel plate assumption, which considered variations in fracture aperture inappreciable compared to fracture length and width. Under this assumption, fluid flow followed the classical cubic law [13,14]. However, the thermal transport behavior of rough natural and hydraulic fractures differed significantly from that of smooth fractures. The tortuosity of flow paths induced by surface roughness led to the formation of local eddies and resulted in pronounced flow heterogeneity [15,16,17,18]. These changes in flow characteristics profoundly affected water temperature distribution and heat transfer behavior within the fracture–matrix system [19,20]. Therefore, it is crucial to understand the influence of surface roughness to better explain thermal–hydraulic coupling processes in fractured rocks.

Roughness can reflect the undulations of a flat surface. Characterizing surface roughness presents significant challenges, requiring the application of diverse parameters across multidimensional assessments to capture its geometric complexity. Mandelbrot [21] innovatively introduced the concept of fractal dimension (D = 2.32 ± 0.15) to describe the self-similar characteristics of rough surfaces, and this parameter was later successfully applied by Xie [22] in cross-scale morphological modeling of rock fractures. Barton [23] established the Joint Roughness Coefficient (JRC) system, providing a practical classification of joint roughness through ten standard profiles (JRC = 0–2 to 18–20). In addition, statistical approaches to roughness characterization have continued to evolve, including peak height distribution function [24], the root mean square of the first deviation of the profile (Z₂) [25], the root mean square of the second deviation of the profile (Z₃), relative volatility, and elongation. Many researchers have studied the hydrodynamic and thermal transport characteristics within fracture–matrix systems under varying surface roughness conditions using the aforementioned parameters. Luo [26] demonstrated that thermal–hydraulic coupling processes were influenced by the spatial distribution of fracture surface asperities. Li [27] fabricated smooth and rough fractures using sandstone and granite, respectively, and conducted comparative analyses of lithological controls and roughness variations on thermal–hydraulic coupling efficiency. Their results showed that lithology had a relatively minor influence, while fracture roughness could improve overall heat transfer to some extent. However Fox [28] demonstrated that, compared with planar surfaces, the heat exchange between the fluid flowing in rough-walled fractures and the surrounding matrix is reduced. Huang [29] and Ma [30] employed JRC profiles and 3D printing techniques to create artificial rock samples, revealing anisotropic behavior induced by surface roughness. Ma [31] further analyzed these results and proposed a linear relationship between the heat transfer coefficient and the JRC value. Bai [32] used a numerical model to compute the distribution of local heat transfer coefficients in rough fractures and found that these coefficients were closely related to variations in fracture geometry. In highly rough regions, flow velocity played a greater role, and the local heat transfer coefficient was significantly higher in concave areas of the fracture surface. He [33] characterized surface roughness using fractal dimension and profile waviness, and investigated its influences on convective thermal transport in granite simple fractures through combined experimental and numerical approaches. The study showed that the local heat transfer coefficient was strongly influenced by interfacial roughness morphology, followed by aperture and flow velocity. Zhang [34] observed that the local heat transfer coefficient was closely dependent on the extent of surface undulation and fluctuation. Chen [35] analyzed the non-uniform thermal transport phenomena within geometrically complex fractures and proposed an empirical model for heat transfer coefficients that accounts for three kinds of rock fractures. Building on numerical simulations of six synthetically generated fracture geometries, Zhang [36] derives a new analytical expression correlation for the overall heat transfer coefficient.

Current studies have predominantly focused on the statistical correlation between h and JRC, aiming to reveal the macro-scale modulation of surface roughness on thermal transport performance. However, limited attention has been given to how roughness affects heat transfer through its impact on microscale flow behavior. The International Society for Rock Mechanics classified the roughness of rock surfaces into two conceptual scales: large-scale waviness (also known as primary roughness) and small-scale heterogeneity (also known as secondary roughness). Primary and secondary roughness had different mechanisms for influencing thermal–hydraulic coupling processes within a fracture [20]. The influence of multiscale roughness on the hydrodynamic and thermal transport has been investigated. Rong [37] performed a range of high-precision hydraulic experiments on artificial fracture specimens and quantified the correlation of roughness parameters on nonlinear flow behavior using the standard deviation of secondary roughness. Wang [38] employed the three-dimensional lattice Boltzmann method (LBM) to numerically investigate nonlinear flow behavior in fractures, both with and without the inclusion of secondary roughness features. The outcomes indicated that primary roughness primarily controlled the pressure profile and flow pathways on a large scale, while secondary roughness governed the nonlinear flow behavior on a local scale. With increasing pressure gradients, secondary roughness significantly amplified the complexity of the local velocity field by inducing and enlarging eddy and recirculation zones near rough surfaces. Tan [39] employed wavelet analysis to decompose original fracture geometries into reconstructed fractures without secondary roughness, in order to investigate the influence of multiscale roughness on hydrodynamic and thermal transport processes. The outcomes showed that primary and secondary roughness jointly influenced the macro-scale modulation of the overall heat transfer coefficient through two competing mechanisms. Wang [40] proposed an effective method for quantitatively identifying primary and secondary roughness according to flow behavior. The study found that the presence of secondary roughness generated more irregularly shaped eddies, which connected stagnant flow zones with active bulk flow regions, thereby enhancing local thermal exchange between the hot wall and the flowing fluid within the original fracture.

To address the above issues, this study aims to further investigate the underlying mechanisms by which multiscale roughness influences heat transfer characteristics. In this work, two-dimensional fracture–matrix models with varying levels of secondary roughness are constructed through multiscale decomposition of fracture surface roughness. A set of simulations is conducted under different flow conditions to explore fluid flow and heat transfer behavior in rough fractures. Furthermore, the relationship between nonlinear flow phenomena and local convective heat transfer coefficients is analyzed. The findings and methodologies developed in this research are anticipated to offer novel perspectives and quantitative approaches for evaluating how fracture surface roughness influences thermo-hydraulic interactions.

2. Methodology

2.1. Fracture–Matrix Model

A two-dimensional rough fracture–matrix coupled model is created using the classical JRC framework presented by Barton. The model consists of a granite matrix (with a permeability < 1 × 10⁻¹⁸ m²) intersected by a central through-going fracture. The geometric dimensions are set as follows: axial length L = 100 mm and height H = 50 mm. Three representative roughness levels are selected from the standard JRC curves, corresponding to JRC ranges of 8–10, 12–14, and 18–20. These curves are used to generate two-dimensional fracture geometries by direct lateral translation of the rough fracture surfaces, without considering surface contact or closure. The resulting models are respectively labeled M1, M2, and M3. Since the primary objective of this research is to examine the influence of secondary roughness on the hydrodynamic and thermal transport behavior within a single fracture, multiscale decomposition of the selected JRC curves is performed to isolate and analyze the impact of secondary roughness features (Figure 2).

The ten standard JRC curves proposed by Barton are originally presented in image format, and the prerequisite for roughness decomposition is digitization to obtain the coordinate values of each curve. In this study, the grayscale image processing method proposed by Yong [41] is employed to extract the coordinates of the JRC curves. Compared with other techniques, this method introduces less manual error and ensures higher accuracy. The main steps are (1) image grayscale processing; (2) binarization processing; (3) curve centerline extraction; and (4) coordinate calculation. The detailed process is illustrated in Figure 3. Here, $x_k$ and ${ y}_k$ represent the horizontal and vertical coordinates of the k-th pixel in the horizontal orientation; $\mu$ is the physical size of each pixel; $n_0$ denotes the number of pixels along the horizontal axis of the standard JRC curve; L is the physical length of the curve in the horizontal orientation; and l is the average value of the maximum and minimum row numbers of the pixels with the smaller gray scale in the kth column of the gray scale matrix.

Several approaches are available to distinguish and filter roughness components of fracture surfaces, such as image analysis, Fourier transform, mathematical morphology, and wavelet analysis [42,43]. Wavelet analysis theory, originally developed in the field of signal processing, is used to decompose non-smooth signals (acoustic waves, roughness profiles) into subcomponents in different frequency bands through operations of scaling and translation. This method has obvious advantages in analyzing the multiscale effects of roughness on the thermal–hydraulic coupling processes in rock fractures.

The Discrete Wavelet Transform (DWT) performs multiscale decomposition of surface roughness according to the multiresolution analysis framework proposed by Mallat, also known as the pyramid algorithm [44]. The core procedure involves first passing the original surface profile data through a pair of high-pass and low-pass filters to decouple it into two complementary components: the low-frequency approximation coefficients (A₁), which characterize large-scale undulating features, and the high-frequency detail coefficients (D₁), which represent small-scale random irregularities. Subsequently, A₁ is recursively decomposed as the new input signal, generating a sequence of multiscale components (Aᵢ, Dᵢ, i = 2,3, … n), where the decomposition depth n is determined by both the sampling resolution of the original data and the specific application requirements. The choice of wavelet basis function has a direct impact on the decomposition performance. Former studies have suggested that the Daubechies wavelet family, particularly the db8 wavelet available in the MATLAB 2021a Wavelet Toolbox, is well-suited for surface roughness decomposition.

In this study, the db8 wavelet from the Daubechies family is adopted as the mother wavelet to decompose and reconstruct the surface roughness, consistent with previous research. In Figure 4, using the JRC18–20 for instance, it is decomposed into eight levels, with the approximate component expressed as A1–A8 and the detail component expressed as D1–D8.

As shown in Figure 4, with increasing decomposition level, Aᵢ gradually loses the low-frequency contour features of the original fracture, while Dᵢ transitions from random noise to structured features. To accurately separate the primary and secondary roughness components, this study adopts the dual-criteria approach [20]: (1) The variance of the decomposed profile must be the same as the original fracture, thereby preserving the dominant fluctuation characteristics. A significant change in variance across decomposition levels indicates a cutoff threshold. As shown in Figure 5a, the variance of the approximation coefficients exhibits an abrupt shift at level 4. (2) The secondary roughness should exhibit characteristics of a random white noise following a Gaussian distribution with a mean close to zero. The combined detail coefficients from levels 1 to 4 satisfy this white noise criterion in Figure 5b. In summary, level 4 is determined to be the optimal cutoff level for isolating the primary and secondary roughness components. This indicates that the reconstructed fracture surfaces obtained from the first four decomposition levels retain the same large-scale geometric features dominated by the non-stationary waviness of the original profile, while exhibiting minor differences in the small-scale random roughness heights. On this basis, two-dimensional models of the original fracture and the reconstructed fractures obtained from the first four levels of wavelet decomposition are established for numerical simulation. Taking M3 as an example, the reconstructed model obtained from the i-th level of wavelet decomposition is denoted as M3-i. This establishes a solid basis for analyzing the impact of different secondary roughness.

2.2. Numerical Simulation of Fluid Flow and Heat Transfer

The study investigates thermal–hydraulic coupling behavior throughout the two-dimensional fracture–matrix system. The rock blocks are supposed to be dense and impermeable, allowing fluid flow to occur only along the fracture. The working fluid is treated as an incompressible Newtonian fluid. For simplification, the effect of temperature on the physical properties of the fluid is neglected, a common assumption adopted in many previous studies [45,46,47].

The Navier–Stokes equations (NSEs) govern the steady-state flow dynamics of an incompressible Newtonian fluid within discrete fractures, mathematically formalizing the conservation of mass and momentum. These equations are formulated as follows [48]:

$$\nabla P=\mu {\nabla }^{2}u-{\rho }_f(u\cdot \nabla )u$$

(1)

$$\nabla \cdot u=0$$

(2)

where u is the velocity vector, ${\rho }_f$ is the fluid density, P is the pressure, and $\mu$ is the dynamic viscosity of the injected fluid. The operators $\nabla$, ${\nabla }^2$, and $\nabla$ respectively are the gradient, Laplacian, and divergence operators.

Two widely adopted continuum mechanics models for simulating thermal processes in porous media are LTE and LTNE. The LTE model presumes that the temperature of the fluid in direct contact with the boundary is identical to that of the rock at the heat transfer interface. It is mainly applicable to the case where the temperature gap between the boundary fluid and the rock is minimal. However, in most practical cases, there exists a certain temperature gap between the fluid and the rock on the heat exchange boundary, leading to a mismatch between their respective temperatures. In this case, it is necessary to consider the heat transfer between the fluid and the solid at the interface, and make a certain mathematical description of the heat transfer boundary of the numerical model. In the research, two models are adopted for numerical simulation: (1) An LTE model is established to research the overall influence of multiscale roughness on single-fracture thermal–hydraulic coupling behavior. (2) An LTNE model is implemented to investigate the localized thermal behavior in rough fractures and to compare the effect of various secondary roughness on the local heat transfer coefficient.

Under local thermal equilibrium (LTE), the single energy equation for the fluid–solid mixture reads

$$\rho c_P(u\cdot \nabla T+{\displaystyle \frac{\partial T}{\partial t}})=\nabla \cdot (k\nabla T)$$

(3)

where c_p represents the effective properties.

Under local thermal non-equilibrium (LTNE), fluid and solid phases satisfy separate energy equations

$$(1-\epsilon ){\rho }_s{c}_{p,s}{\displaystyle \frac{\partial T_s}{\partial t}}=\nabla \cdot (k_s\nabla T_s)+ah(T_f-T_s)$$

(4)

$$\epsilon \rho c_p(u\cdot \nabla T_f+{\displaystyle \frac{\partial T_f}{\partial t}})=\nabla \cdot (k\nabla T_f)-ah(T_f-T_s)$$

(5)

where ε is the local porosity within the fracture control volume, a is the specific fluid–solid interfacial area density, and h is the interfacial heat-transfer coefficient. LTE is recovered when T_f ≈ T_s.

The hydraulic aperture ($e_h$), a critical parameter for characterizing fluid transport through individual fractures, is determined via the cubic law relationship:

$$e_h={({\displaystyle \frac{12\mu q}{\nabla pw}})}^{1/3}$$

(6)

It is known that with increasing flow rate, fluid flow in rough fractures may deviate from linearity, where the increase in flow rate is less than proportional to the increase in pressure gradient. This phenomenon is described as nonlinear flow [49]. The Forchheimer equation is an extensively utilized model for analyzing nonlinear flow in rough fractures, and its expression is given below [50,51,52]:

$$-\nabla P=AQ+B{Q}^2$$

(7)

$$A={\displaystyle \frac{\mu }{kw{e}_h}}={\displaystyle \frac{12\mu }{w{e}_h^3}}$$

(8)

$$B={\displaystyle \frac{\beta \rho }{w^2{e}_h^2}}$$

(9)

A is a linear coefficient representing the energy loss caused by viscous dissipation; B is a nonlinear coefficient accounting for mechanical losses associated with inertial effects. Both A and B are impacted by the roughness characteristics of fractures and the flow properties of the fluid.

The Reynolds number (Re) characterizes the flow regime, and it is defined as the ratio of inertial forces to viscous forces with the expression [53]:

$$Re={\displaystyle \frac{\rho u{e}_h}{\mu }}={\displaystyle \frac{\rho Q}{\mu w}}$$

(10)

COMSOL Multiphysics 6.2 is employed to calculate the NSE coupled with the energy conservation equations, enabling the simulation of flow behavior and heat transfer phenomena in an individual fracture. In all simulations, flow boundary conditions were prescribed as follows: a constant flow rate was imposed at the inlet, a zero-pressure condition (P = 0) was assigned at the outlet, and no-slip boundary conditions were applied along the remaining fracture walls. To minimize boundary layer effects near the inlet, laminar flow profiles were specified at the inlet boundary. For thermal conditions, the external surfaces of the surrounding rock matrix were maintained at a prescribed temperature ($T=T_{ext}$), while the inlet of the fracture was set to a constant temperature ($T=T_{in}$). The entire fracture–matrix system was initialized at a uniform temperature equal to $T_{ext}$ (Figure 6). All other boundaries of the system were assumed to be adiabatic, preventing any heat flux across them [54]. A comprehensive summary of the physical parameters and boundary specifications used in the study is provided in

Table 1. Model parameters and boundary conditions used in the numerical simulation.

Parameter	Note (Unit)	Value
Thermal conductivity of rock Matrix	ks (W∙m−1∙K−1)	2.784
Rock matrix density	ρs (kg∙m−3)	2620
Specific heat capacity of rock matrix	cp,s (J∙kg−1∙K−1)	757
Thermal conductivity of water	k (W∙m−1∙K−1)	0.662
Water density	ρ (kg∙m−3)	1000
Specific heat capacity of water	cp (J∙kg−1∙K−1)	4200
Inlet flowrate	Uin (m∙s−1)	0.01, 0.03, 0.05, 0.07, 0.1
Mechanical aperture	e0 (mm)	0.25, 0.5, 0.75, 1.0
Injection temperature	Tin (K)	293.15
Temperature of the outer surface of the rock matrix	Tout (K)	363.15

. In this study, the temperature dependence of fluid properties was neglected. Our calculations indicate that this simplification introduces an error of less than 1%, suggesting acceptable reliability of the results. However, the influence of temperature-dependent properties will be considered in future work to further improve the model accuracy.

Three systematically refined meshes (coarse/medium/fine) with refinement ratio r ≈ 1.3 were used. The key output T_out were extrapolated by Richardson’s method, and the Grid Convergence Index (GCI) at 95% confidence was computed:

$$p={\displaystyle \frac{\ln (\frac{{\phi }_3-{\phi }_2}{{\phi }_2-{\phi }_1})}{\ln r}}$$

(11)

$${\phi }_{ext}={\phi }_1+{\displaystyle \frac{{\phi }_1-{\phi }_2}{r^p-1}}$$

(12)

$$GC{I}_{12}=F_s\cdot {\displaystyle \frac{|{\phi }_1-{\phi }_2|}{\left|{\phi }_1\right|}}\cdot {\displaystyle \frac{1}{r^p-1}}$$

(13)

with safety factor F_s = 1.25. Residuals were reduced below 1 × 10⁻⁶ (momentum) and 1 × 10⁻⁸ (energy); steady state was declared when domain-integrated heat balance closed within 0.1%.

According to the above formulas, the grid convergence index (GCI₁₂) of the key parameter T_out between the fine and medium meshes is less than 1%. Therefore, the fine mesh is sufficient to ensure grid-independent solutions, and all subsequent simulations are based on this mesh configuration. The total number of grid elements in the computational model are (30∼50) × 10⁴. Coarser meshes (0.05 mm to 1.75 mm) are applied to the rock matrix, and finer meshes (0.001 mm to 0.335 mm) are used for the fracture due to its lower spatial scale and more complicated physical processes. Partial mesh refinement is implemented near the fracture surfaces to better capture the complex and unusual geometrical features. In addition, boundary layer meshes were generated along the fracture walls to resolve the multiscale surface roughness and its influence on flow behavior and heat transfer phenomena (Figure 7).

3. Results and Analysis

This section first analyzes the flow behavior in single fractures with different roughness levels, then further investigates the factors influencing their heat transfer performance.

3.1. Flow Behavior

As previously mentioned, fluid flow serves as the foundation for hydrodynamic heat transfer in fractured rock masses. Without prejudice to the generality, the numerical results of the M0, M3, M3-1, M3-2, M3-3, and M3-4 models are illustrated and compared as examples in this paper. Given the exclusion of rock matrix permeability considerations, the analysis in this section is confined to characterizing fluid flow dynamics within the fracture network.

First, Figure 8 displays the overall velocity field distribution of the six fractures at Re = 10 and Re = 100. The magnified profile in the red box highlights the x-component velocity field and streamlines. The flow field in M0 maintains a hypothetically simplified configuration exhibiting consistent velocity distribution aligned with the flow orientation. Under no-slip boundary constraints, velocity magnitudes are maximized at the central plane of the fracture, while null velocity conditions prevail at the peripheral boundaries. The maximum velocity increases proportionally with the Reynolds number. In addition, the streamlines are uniformly distributed and parallel to the main flow orientation. Under these circumstances, the x-component velocity field is identical to the total velocity field. In contrast to M0, the velocity fields and streamline layouts in M3, M3-1, M3-2, M3-3, and M3-4 are considerably more complex and exhibit significant spatial variability. The placements of the low-velocity zones and the general flow orientation are roughly similar among them, but differences exist in the velocity range and high-velocity zone distribution, especially in the vicinity of the secondary roughness, where refluxes and eddies are generated. As the decomposition level increases, the streamlines become more uniform, appearing as smooth lines nearly parallel to the boundary contours. With the increase in Re, eddy development near local protrusions becomes more pronounced.

According to the report by Lee [55], local abrupt changes in surface geometry (secondary roughness) have a major function in the formation of eddy regions, even at very low Reynolds numbers (Re < 0.1), where the influence of inertial forces is negligible. These eddy regions are the main cause of nonlinear flow behavior. As mentioned in the previous section, the Forchheimer equation is the most extensively utilized model for describing nonlinear flow in fractures and porous media. Thus, the relationship between fluid flow rate (q) and pressure gradient (∇P) under varying mechanical apertures ($e_0$) was calculated (Figure 9). It is observed that the Forchheimer equation can adequately fit their nonlinear correlation (R² > 0.98), highlighting the significant impact of nonlinear effects.

To better quantify the effect of fracture geometry on seepage mechanisms, the viscous resistance coefficient (A) and inertial resistance coefficient (B) in the Forchheimer equation are extracted based on polynomial fitting of the pressure gradient–flow rate relationship. Figure 10 illustrates the correlation between A and B with $e_0$ and the levels of roughness decomposition. Both A and B exhibit a monotonically decreasing trend with increasing $e_0$, which is closely related to the enhanced cavity connectivity and reduced tortuosity of the flow path under large-aperture conditions. At a fixed $e_0$, A decreases with increasing roughness decomposition level, indicating that the removal of secondary roughness weakens the disturbance of the viscous boundary layer. When a relatively small mechanical aperture $e_0=0.25 \mathrm{m}\mathrm{m}$ is used, fluid flow behavior is primarily controlled by roughness. In this case, coefficient B decreases with increasing roughness decomposition level, reflecting that microscale eddies induced by secondary roughness significantly increase flow resistance. However, when $e_0$ increases to 1.0 mm, the relationship between B and roughness decomposition level becomes more complex. Secondary roughness often induces more small-scale eddies near the wall, increasing flow instability. Meanwhile, under larger apertures, the overall acceleration or deceleration of the flow may lead to irreversible kinetic energy losses, thereby enhancing inertial effects.

In studies of seepage through rock fractures, the determination of hydraulic aperture is typically based on the cubic law assumption, which is calculated by inverting the correlation between the total flow rate and the pressure gradient obtained from experimental measurements or numerical simulations. This parameter essentially represents a global average of the effective width of the flow channel. However, in reality, flow behavior within rough fractures is more complex due to eddies arising from secondary roughness. The formation of eddies affects changes in the geometry and magnitude of velocities in the advection region, which in turn affects the heat transfer behavior within the fracture. Therefore, a single averaged hydraulic aperture may not sufficiently represent the system. To conduct meaningful scientific analysis, it is at least necessary to systematically investigate the local velocity and heat transfer behaviors in separately defined advective and vortical regions. Using the modeling results, the eddy aperture ($e_w$) is defined as the average thickness of the eddy region, and the effective advection aperture ($e_a$) is defined as the average thickness of the effective advection region consisting of the fluid that is not trapped by the eddy. Zhou [17] proposed to define the eddy region quantitatively as a closed region where the boundary fluxes are zero.

A cell belongs to a recirculation zone if the line integral of the normal velocity along its closed contour satisfies

$$\left|{\displaystyle \oint u\cdot ndl}\right|<\epsilon$$

(14)

We set ε = 1 × 10⁻⁴ U_in and apply 4-neighbor connectivity with minimum area filter A_min = 10 cells to remove spurious features near walls. Sensitivity to ε and A_min shows Δφ_eddy ≤ 2% over ε ∈ [10⁻⁵ U_in, 10⁻³ U_in].

The equation defines the eddy area fraction as the ratio of eddy aperture to mechanical aperture:

$$w_s={\displaystyle \frac{A_{eddy}}{A}}$$

(15)

Figure 11 gives the eddy frontier under Re = 100. AB and CD represent the eddy boundary surfaces on the upper and lower walls, respectively.

The effective advection aperture is then determined as the distance between points E and F, while the remaining portion of the mechanical aperture is determined as the eddy aperture.

Table 2. Calculated hydraulic aperture, effective advection aperture, and eddy aperture values.

Parameter (mm)	M1	M2	M3
Hydraulic aperture	0.9023	0.8680	0.7824
Effective advection aperture	0.9905	0.9859	0.9677
Eddy aperture	0.0095	0.0141	0.0323

lists the values of hydraulic aperture, effective advective aperture, and eddy aperture for models M1, M2, and M3 at Re = 100. Since the aperture values are relatively small, the values are retained to the fourth decimal place. A comparison shows that the effective advection aperture values listed in Table 2 are significantly larger than the hydraulic aperture values conventionally computed by inverting the cubic law. This is because the increased inertial term in the NSE leads to greater pressure drops, resulting in smaller hydraulic aperture values. The effective advection aperture reflects geometric variations under the combined effects of inertia and fracture roughness, making it more suitable for studying heat transfer processes.

3.2. Temperature Distribution

During fluid migration through the fracture, thermal energy transfer occurs at the fluid–rock interface, inducing spatial reorganization of thermal gradients across the fracture–matrix system. This temperature redistribution provides a diagnostic tool for evaluating coupled hydro-thermal transport mechanisms. Figure 12 shows the temperature contour maps for models M3, M3-1, M3-2, M3-3, and M3-4 at Re = 10 and Re = 100. When the fluid enters the domain, the temperature of the rock around the fracture inlet drops significantly. As the fluid continues to flow, the fracture walls begin to cool down, and the cold front gradually propagates towards the rock matrix. The temperature of the rock surrounding the fracture exhibits a gradient in the flow direction, implying that the temperature in the middle of the fissure is always the lowest at the same vertical line position. The temperature distributions of the various models are closely comparable, with only minor differences. Figure 13 is an enlarged view of the area outlined by the red dashed box in Figure 12. Magnification of the local temperature distributions reveals broader high-temperature regions adjacent to the fracture walls in areas with secondary roughness. This can be interpreted as follows: the primary roughness determines the overall temperature distribution, while the secondary roughness further increases the complexity of the local temperature fields. Comparison of temperature contour maps under varying flow rates reveals that, with increasing injection velocity, the region of the disturbed temperature field within the rock mass expands, while the average temperature of the fractured rock mass correspondingly decreases. This occurs because the increased injection velocity allows more low-temperature fluid to enter the rock mass per unit time, thereby extracting more heat from the rock. As a result, the extent and magnitude of the disturbance in the temperature field increase, leading to a lower average temperature of the rock mass.

3.3. Overall Heat Transfer Coefficients

The heat transfer process involves a range of complex mechanisms, including fluid phase transitions, coupled interactions, and nonlinear flow properties. The quantity of heat exchanged between a solid surface and the surrounding fluid can be quantified using Newton’s law of cooling.

$$Q=hS(T_r-T_f)$$

(16)

where Q represents the quantity of heat exchanged between the rock and the fluid, S is the contact area between the two phases, and h is the overall heat transfer coefficient, which is a key empirical parameter used to evaluate the heat exchange ability at the fluid–fracture interface. It can also be applied in the framework constituted by simulations of thermal–hydraulic coupling in fractured rocks based on the LTNE theory. T_r denotes the mean temperature of the rock surface within the fracture, and T_f is the mean temperature of the fluid inside the fracture channel.

According to the principle of energy conservation, the thermal energy exchange between fractured rock and the surrounding fluid must be the same as the thermal energy carried away by the fluid flowing across the fracture.

$$Q=C_{P,f}{\rho }_{f}q(T_{out}-T_{in})$$

(17)

where $C_{p,f}$ is the specific heat capacity of the injected fluid, and $T_{in}$ and $T_{out}$ represent the inlet and outlet fluid temperatures, respectively.

Combining Equations (16) and (17), h is derived from the following equation [20].

$$h={\displaystyle \frac{C_{P,f}{\rho }_{f}q(T_{out}-T_{in})}{S(T_r-T_f)}}$$

(18)

Figure 14 shows the changes in h with roughness decomposition levels under different flow conditions for each model. As shown in Figure 14, h increases with increasing flow velocity. Meanwhile, h roughly exhibits a general decreasing trend with the increase in roughness decomposition level. As previously discussed, convective heat transfer in rock fractures is strongly influenced by the characteristics of fluid flow. With increasing decomposition levels, secondary roughness progressively diminishes. Given that secondary roughness plays a crucial role in the formation of eddies, changes in eddy intensity can markedly impact the heat transfer coefficient.

Figure 15 illustrates the eddy area fraction for models with different levels of roughness decomposition and compares these values with the corresponding h. The results indicate that, as the decomposition level increases, the eddy region progressively diminishes and eventually disappears. Meanwhile, the decrease in JRC value significantly suppresses the intensity of eddy generation—the eddy area fraction shrinks by about 83% when JRC decreases from 18–20 to 8–10. Under a constant JRC, the eddy area fraction exhibits a strong positive correlation with h. This is due to eddies disrupting the laminar thermal boundary layer, enhancing mixing between the high-temperature fluid near the wall and the lower-temperature fluid in the fracture center. Consequently, more thermal energy can be derived under the equivalent initial rock temperature. This eddy induced heat transfer enhancement mechanism provides a theoretical basis for optimizing injection and production parameters in geothermal systems.

3.4. Local Heat Transfer Coefficients

Eddies typically form in localized regions within the fracture, significantly influencing both the fluid’s local temperature distribution and the associated heat transfer characteristics. Moreover, these microscale convective heat transfer processes manifest at the macroscale through their cumulative effects on the temperature distribution. Figure 16 presents the fluid temperature distributions in the orientation of flow for models M3, M3-1, M3-2, M3-3, and M3-4 at Reynolds numbers (Re) of 1, 10, and 100. The thermal evolution along the longitudinal flow axis exhibits nonlinear augmentation, particularly at low Re. At Re = 1, the fluid temperature profiles of different models are nearly identical (Figure 16). At the inlet, the fluid temperature rises rapidly due to a high thermal gradient, then gradually approaches a steady value along the flow path. Moreover, these temperature profiles exhibit minimal fluctuations, indicating that the temperature is evenly distributed along the longitudinal flow axis. This phenomenon originates from the predominance of thermal conduction at low Re, which effectively diminishes the flow field perturbations caused by surface roughness. However, remarkable differences between the models emerged as Re increased. Surface roughness enhances heterogeneous heat transfer, leading to highly nonuniform temperature distributions with large-amplitude and high-frequency fluctuations along the flow direction. With increasing decomposition level, both the amplitude and frequency of these fluctuations decrease.

The local heat transfer coefficient characterizes the localized heat exchange behavior within the fracture channel and serves as a key parameter for characterizing the fluid–rock thermal interaction along the flow direction.

$$h^{\prime }={\displaystyle \frac{C_{fp}{\rho }_f(T_2-T_1)}{2(x_2-x_1)(T_r^{\prime }-T_f^{\prime })}}$$

(19)

where $x_1$ and $x_2$ represent the x-coordinates of two sufficiently close sites (with $x_2$ − $x_1$ = 0.1 mm in this study) [39], and $T_r^{\prime }$ and $T_f^{\prime }$ respectively denote the average temperatures of the fracture surface and the fluid between these two points.

Figure 17 illustrates a significant spatial coupling between $e_w$ and $h^{\prime }$. In regions where eddies are well developed, microscale eddies enhance fluid mixing, leading to the periodic disruption and reformation of the thermal boundary layer along the fracture walls. This results in spatial heterogeneity of the local heat transfer coefficient. As the eddy aperture increases, the fluid mixing is enhanced, and the amplitude of the heat transfer coefficient is subsequently increased. Overall, the models have similar trends in heat exchange on large scales. However, compared to the higher-order decomposition models (M3-4 and M3-3), the original rough model (M3) and the primary decomposition model (M3-1) show more pronounced fluctuations with higher frequencies and amplitudes in their heat transfer coefficient curves. This is due to the microscopic bumps of the secondary roughness inducing eddy clusters, which enhance the synergistic changes of the flow velocity and temperature gradient in the near-wall region and make the local heat transfer more unsteady. Further analysis reveals a scale-dependent effect of roughness on heat transfer: while large-scale undulations govern the overall heat transfer trend, small-scale perturbations affect vortex pathways and generate localized heat transfer hotspots at the submillimeter scale.

4. Discussion

Based on the simulation results presented above, a qualitative relevancy between h and nonlinear flow behavior has been established: the presence of secondary roughness and high Reynolds numbers induces nonlinear flow, which in turn enhances h. As a key input parameter in thermo-hydraulic coupling simulations of fractured rock masses, the accurate characterization of h directly influences the reliability of geothermal reservoir heat extraction efficiency predictions. Therefore, developing a parameterized correlation model linking h with surface roughness, flow regime, and geometric features is basic for enabling predictive modeling of heat transfer at the engineering scale. Zhao [56] proposed that a power-law relationship can describe the correlation between v and h, and Tian [57] suggested a linear relationship between the h and Z₂. Primary and secondary roughness affect heat transfer by different mechanisms, and although Z₂ can characterize the effect of primary roughness on overall heat transfer, it is difficult to quantify the effect of secondary roughness induced microscale eddies and flow separation on local heat transfer behavior. In the study of Rong [37] the standard deviation of secondary roughness (${\sigma }_2$) was introduced to characterize the influence of roughness on nonlinear fluid flow. Building on this concept, the present study adopts a multiscale characterization approach: the parameter Z₂ is used to represent primary roughness, while ${\sigma }_2$ quantifies secondary roughness, aiming to investigate the correlation between fracture roughness and overall heat transfer coefficient. This dual-parameter coupling model overcomes the limitations of conventional roughness characterization and offers a novel framework for understanding the intrinsic linkage between multiscale geometric features and nonlinear heat transfer. Additionally, two more simulation cases with JRC10–12 and JRC16–18 were introduced. For models with the same JRC value but different decomposition levels, the primary roughness parameter Z₂ remains constant, whereas the standard deviation of secondary roughness varies.

Table 3. Overall heat transfer coefficients for different roughness parameters and for different flow rates.

Z2	σ2	h (W∙m−1∙K−1)
		0.01 (m/s)	0.03 (m/s)	0.05 (m/s)	0.07 (m/s)	0.1 (m/s)
0.1674	0.0564	218.41	229.45	237.4	242.07	248.97
	0.0533	217.55	231.81	232.05	243.37	247.69
	0.0491	217.65	228.09	232.05	232.98	240.71
	0.0436	217.17	225.95	227.38	229.74	233.73
0.1902	0.0451	217.68	227.31	235.26	239.62	245.57
	0.0417	218.51	226.39	232.32	236.66	241.59
	0.0376	218.41	228.09	231.38	237.52	237.53
	0.0316	217.93	227.02	228.04	230.39	233.73
0.2163	0.0611	219.17	231.16	240.75	245.98	255.33
	0.0579	217.65	231.66	231.38	242.72	247.06
	0.0547	217.55	228.38	232.72	232.98	242.61
	0.0469	217.46	226.38	228.04	231.03	235.63
0.2716	0.0777	220.89	233.74	236.78	242.54	246.26
	0.0743	219.65	230.23	236.73	239.47	243.88
	0.0714	218.22	227.59	234.72	235.57	245.15
	0.0661	218.7	227.73	232.05	236.68	242.9
0.3390	0.1029	218.51	229.16	236.73	240.12	247.06
	0.0999	218.51	228.59	234.72	236.87	256.61
	0.0978	217.46	228.09	232.05	240.77	241.98
	0.0873	217.93	226.74	232.72	232.33	238.8

summarizes the corresponding roughness parameters and h at various flow velocities:

Based on the above findings, we propose a new heat transfer coefficient model which can be expressed as

$$h=10.86\cdot (39.93-22.93\cdot Z_2)\cdot {(Z_2\cdot {\sigma }_2)}^{0.11}\cdot {\sigma }_2^{-0.05}\cdot v^{0.05}$$

(20)

The R² value of the fitted curve is 0.89, indicating a strong correlation with the actual data. Meanwhile, the relative error falls within the range of −1% to 5%. Specifically, Z₂ reflects the intensity of surface slope variation. As the slope variation increases, the heat transfer coefficient generally exhibits a decreasing trend, which may be attributed to excessive slope disrupting the stability of the main flow and thereby weakening the heat transfer performance. The influence of secondary roughness on heat transfer intensity can be understood as a competitive mechanism: while it enhances eddy formation and convective heat transfer at the local scale, it may also cause flow resistance and energy dissipation at the larger scale, thereby reducing macroscopic heat transfer efficiency. The relatively weak negative exponent of σ₂ suggests that this adverse effect is not dominant, and overall, secondary roughness still exerts an enhancing effect on the heat transfer coefficient. These findings suggest that the impact of roughness on heat transfer is complex, governed not by a single enhancement or suppression mechanism, but rather by a multiscale synergistic regulation process. Meanwhile, the flow velocity is positively correlated with the heat transfer coefficient, which is a weak effect but verifies the general law that increased velocity can promote convective heat transfer. In the design of hydraulically induced fractures in EGS, the mechanism by which secondary roughness enhances heat transfer through the induction of microscale vortices offers a novel strategy for optimizing thermal energy extraction. Looking ahead, the integration of machine learning algorithms with THM models could enable intelligent regulation of fracture morphology, providing a precise design paradigm in support of hot dry rock geothermal energy development [58,59,60]. It should be noted that this model has some limitations, with a narrow range of variation in σ₂. If extrapolated to a larger range, the accuracy of the model requires to be further confirmed.

In addition, similar to most studies, this research provides a preliminary understanding of the roughness-induced heat transfer regulation mechanisms through numerical simulations based on a two-dimensional model. However, due to the dimensional simplifications inherent in the model, it does not fully capture the complex behaviors associated with multiphysics coupling in real geological environments, particularly the three-dimensional effects such as channeling, recirculation, and the anisotropy of fracture roughness. However, the multiscale roughness effects revealed by the two-dimensional model are fundamentally controlled by NSE and the law of energy conservation, and the underlying physical principles remain applicable in three-dimensional scales. By introducing the coupling effects of three-dimensional channel flow and lateral flow, the cross-scale correlation mechanism established in the two-dimensional model can be extended to more complex three-dimensional scenarios. Therefore, future research will focus on constructing a real three-dimensional rough fracture model based on CT scanning, considering the non-uniform aperture distribution and the anisotropic characteristics of three-dimensional multiscale roughness. This will further investigate the relationship between nonlinear flow behaviors and heat transfer coefficients. We have already carried out some preliminary work on three-dimensional rough fracture surface scanning and reconstruction. It should be noted that the influence of multiscale roughness on convective heat transfer is extremely sensitive to surface details, requiring sufficiently high spatial resolution to ensure accuracy. However, increasing resolution inevitably reduces computational efficiency, and thus the key challenge lies in balancing geometric fidelity with computational efficiency.

5. Conclusions

In summary, a numerical simulation method is used to build models featuring various levels of secondary roughness through wavelet decomposition. This framework focuses on examining how multiscale surface irregularities impact fluid dynamics and thermal transport. To capture the role of flow patterns in shaping thermal behavior, the eddy area fraction is introduced as a key parameter. The overall heat transfer coefficient is evaluated, and the spatial variation of local heat transfer rates along the flow path is analyzed. The principal conclusions based on the simulation results are outlined below:

(1)

The temperature distributions of different models are generally similar, but a broader high-temperature region is observed near the wall in areas with secondary roughness. The resulting flow behavior, shaped by multiscale roughness, induces noticeable temperature non-uniformity. Moreover, as secondary roughness is progressively filtered out, the degree of temperature heterogeneity decreases accordingly.

(2)

Secondary roughness is an important factor in the generation of eddies, and a correlation between the heat transfer coefficient and the eddy area fraction is established for analyzing the effect of nonlinear flow on heat transfer. Enhanced the heat transfer coefficient in fractured media exhibits a positive correlation with eddy area fraction. As JRC increases, the eddy area fraction increases.

(3)

The local heat transfer coefficient exhibits continuous fluctuations within rough fractures, showing a strong correlation with the eddy aperture. In regions near eddy structures, the spatial heterogeneity of local heat transfer is more pronounced. With increase in the eddy aperture, enhanced mixing between high-temperature fluid near the wall and low-temperature fluid at the fracture center results in a systematic increase in the magnitude of the local heat transfer coefficient.

(4)

A novel heat transfer coefficient model incorporating both fluid velocity and multiscale surface roughness was proposed. The model demonstrates that increasing the fluid velocity enhances the heat transfer coefficient. Moreover, the synergistic effect of primary and secondary roughness greatly influences the overall heat transfer performance.

Building upon the analysis of hydrodynamic and thermal transport within single fracture systems with multiscale roughness features, this study further incorporates heat transfer characteristics under varying degrees of secondary roughness. The nonlinear flow behavior and its impact on local convective heat transfer are examined in detail. These findings advance new insights into the theoretical understanding of EGS exploitation and provide a scientific basis for practical engineering design and optimization, thereby supporting the efficient and sustainable utilization of deep geothermal energy.

Based on our findings, fracture design should not only focus on fracture opening and connectivity but also consider intentionally introducing or enhancing secondary roughness through specific measures. For instance, employing fracturing fluids or proppants that form microscopic protrusions or irregular surfaces on fracture walls during hydraulic fracturing or acidizing processes may facilitate the generation of more small-scale vortices within fractures. As demonstrated by our simulation results, these vortices significantly enhance local fluid mixing and heat transfer efficiency, thereby contributing to sustained high heat transfer levels.

However, this study still has certain limitations. The two-dimensional rough fracture model adopted here cannot fully represent the complex structural characteristics of actual three-dimensional fracture surfaces, especially in capturing fluid recirculation and local eddy behaviors. Moreover, the 2D model is insufficient to characterize preferential flow along high-conductivity pathways (large-aperture regions) or to describe detailed bypassing and stagnation phenomena in low-aperture zones. Future studies should adopt three-dimensional fracture models that focus on the complex flow behavior induced by heterogeneous aperture distributions, so as to more accurately reveal how multiscale geometric features influence heat transfer and thereby improve the consistency between simulation results and field conditions.