The two-dimensional domain should be defined, e.g., as a rectangle with dimensions Lx and Ly, for modelling fractured porous rocks. The discrete fracture model can consider the discrete fracture network as well as the rock matrix. Fracture modelling includes both deterministic and stochastic approaches. After the fracture geometries are defined, the rest of the domain is filled with the rock matrix. The matrix can be characterized by permeability, porosity, etc.

For deterministic fractures, two endpoints, A(x1,y1) and B(x2,y2), and the fracture aperture should be defined for each fracture. Deterministic fractures are typically used for large-scale features such as faults or fractures with well-constrained geometries, which are important for modelling flow and heat transport.

For stochastic fractures, the discrete fracture network is determined by different geometrical properties (Bonnet et al., 2001) , such as fracture length, aperture, position, and orientation (Xu and Dowd, 2010). The stochastic fracture is mainly based on the software ADFNE. Furthermore, UpsFrac is capable of modelling the truncated power-law length distribution which describes the fractal characteristics of natural fractures. The probability density function for the fracture length l can be written as (Corral and González, 2019; Hyman et al., 2016; Massart et al., 2010): 1n(l)=α-1lmin(1-(lminlmax)α-1)(lminl)α, where lmin and lmax are the lower and upper bound of the fracture length, α, ranging from 1.3 to 3.5, is α power-law exponent influenced by the growth properties of fractures (Bonnet et al., 2001).

Furthermore, the fracture aperture,w, can be correlated to fracture length in UpsFrac by the following power law expression: 2w=γlD, where γ is the coefficient related to mechanical properties of fractured rocks and D denotes the correlation exponent, which signifies the mechanical interaction among closely positioned fractures. D may vary from 0.5 to 1 resulting from observations in the field. D=0.5 represents complex open-mode fractures with a constant fracture toughness (Klimczak et al., 2010; Olson, 2003), while D=1.0 occurs in faults and shear deformation bands under constant driving stress (Vermilye and Scholz, 1995).

Users can use the ADFNE framework to create stochastic realizations for fracture orientation, density, and location. Furthermore, users can develop their models to create stochastic fractures. All the fracture geometric parameters are defined in the modelling framework. The generated fracture geometries are exported in a standardized format that ensures seamless integration with subsequent upscaling procedures.

The upscaling procedure mainly involves finding the equivalent properties on coarse-scale grid blocks based on the information from fine-scale discrete fracture models. The first step is to divide the fracture domain into a Cartesian grid with nx×ny blocks. Each fracture is geometrically clipped at grid boundaries to identify its intersection segments within individual blocks. The fracture segments within each grid block are defined by their endpoint coordinates A (x1,y1) and B (x2,y2) and aperture values (Fig. 3), providing the geometric and hydraulic parameters necessary for flow simulation. It should be noted that the dimension of the Cartesian grid is (Lx/nx) × (Ly/ny); accordingly, all fracture endpoints are constrained within the individual grid block boundaries.

Next, the clipped fractures within each Cartesian grid form local discrete fracture networks for numerical simulation. The TPFA or MPFA schemes (Sandve et al., 2012) are employed to accurately model fluid flow in these highly heterogeneous media. Each grid block maintains identical dimensions and matrix properties, with only fracture geometries varying between blocks. The mesh refinement for both fractures and the matrix is adapted based on the local fracture density and complexity to ensure numerical accuracy.

Then, numerical simulations are conducted for each fractured grid block with pressure gradients applied along both x and y axes. The resulting flux distributions and pressure fields provide the necessary information for upscaling calculations. For grid blocks without fractures, the matrix permeability is directly assigned.

Subsequently, the MFU is applied to calculate the equivalent permeability for each grid block by using the previous simulation results. The MFU uses multiple boundary expressions to calculate flux. When the linear boundary conditions are applied along the x axis, the flow rates along the x and y axes, qx and qy, are calculated on multiple boundaries: 3qx=∫0lyvr⋅nxdy+∫0lxvu⋅nxdx+∫0lxvl⋅nxdx,qy=∫0lxvu⋅nydx+∫0lxvl⋅nydx+∫0lyvr⋅nydy, where lx and ly are dimensions of the Cartesian grid in the x- and y-directions, nx and ny are unit vectors along the x and y axes, vr, vu, and vl denote the Darcy velocities on the right, upper, and lower boundaries.

Lastly, the coarse-scale equivalent permeability can be computed inversely based on the Darcy's law using the flux in the fine-scale discrete fracture model. The details of the MFU can be found in Chen et al. (2015). The resulting permeability tensor components (kxx, kxy, kyx, kyy) characterize the anisotropic flow behavior of each grid block.

The upscaling workflow is designed as a modular framework consisting of five interconnected computational stages, each addressing specific mathematical and numerical challenges in fractured porous rocks characterization. Table 2 summarizes the computational stages, their mathematical formulations, and key parameters.

The framework incorporates key algorithmic features to ensure robust upscaling: (i) stochastic fracture generation with power-law length distributions and aperture-length correlations; (ii) geometric processing with boundary intersection detection and corner point adjustment to prevent mesh singularities; (iii) automatic mixed-dimension mesh generation integrating fracture and matrix elements; and (iv) batch processing of dual orthogonal flow problems. The upscaling employs directional flux decomposition through cosine transformations and validates permeability tensors to correct non-physical values from numerical errors. This modular architecture enables users to customize individual components – such as substituting alternative fracture generation models or numerical schemes – while maintaining framework compatibility, facilitating adaptation to diverse geological scenarios. The complete algorithmic pseudocode and implementation details are provided in Appendix A1, with computational modules and data structures documented in Appendix A2 for reproducibility.

The upscaled equivalent permeability tensors are analyzed and visualized to characterize flow behavior in fractured porous rocks. The resulting equivalent permeability is a full tensor form and is not inherently symmetric for fractured rocks (Chen et al., 2016; Zijl and Stam, 1992). When a symmetric tensor is required, a symmetric permeability tensor can be obtained by averaging the off-diagonal components. Using the upscaling results, the distribution of permeability tensor components (kxx, kxy, kyy) can be visualized to characterize spatial heterogeneity and anisotropy. In addition, the equivalent permeability tensor of each grid block can also be plotted as an ellipse to visualize the directional permeability. Furthermore, to analyze the statistical properties of the equivalent permeability, histogram analysis of the permeability components is performed, which is important for building stochastic fields of equivalent permeability (Fiori et al., 2015) and supporting uncertainty quantification in reservoir modelling.

For modelling the fracture length with power law distribution, the cut-off power law is applied to create fracture length. For fractures with length l ranging from lmin to lmax, the power law distribution can be expressed as in Eq. (1). The power law distribution for fracture length is created using the rndm_powerlaw function. The values of lmin, lmax, power-law exponent α and fracture number N should be determined. For instance, when lmin= 10 m, lmax= 1000 m, α= 2.5, and N= 500, the histogram of stochastically generated fracture length is plotted in Fig. 4, showing a typical power-law decay pattern with most fractures concentrated at smaller lengths. When fitting the stochastically generated data with the maximum likelihood estimation method, the fitted α is 2.52, close to the input α 2.5. The Kolmogorov-Smirnov statistic is 0.032, indicating that the software UpsFrac generates the data set and agrees with the fitted power-law distribution.

To validate the accuracy of UpsFrac's upscaling methodology, we compare the calculated equivalent permeability against analytical solutions for a single fracture embedded in a low-permeability matrix. The validation setup consists of a 2 m × 2 m domain containing a 2 m horizontal fracture at y= 1 m spanning the entire domain width (Fig. 5a). The matrix permeability km= 9.87 × 10⁻¹⁶ m², and the fracture permeability kf varies based on aperture according to the cubic law. Linear pressure boundary conditions are applied on all four boundaries: left boundary (x= 0) with P= 2 Pa, right boundary (x= 2) with P= 0 Pa, and top/bottom boundaries with P=2-x Pa, representing a unit gradient flow field.

The analytical solution for equivalent permeability kxx under parallel flow assumption between fracture and matrix is derived from flow equivalence principle (Snow, 1969). For the single-fracture case, this can be simplified to: 4kxx=km×(1-w/Δyly)+kf×w/Δyly where kf=w2/12 based on the cubic law, and Δy is the domain height in the y-direction (Δy=ly for this single-block validation case).

We tested the upscaling accuracy by varying fracture aperture with correlation exponent D from 0 to 1 (γ= 1.2 × 10⁻⁴). The equivalent permeability calculated by UpsFrac ranged from 0.7 × 10⁻¹³ to 5.8 × 10⁻¹³ m², showing excellent agreement with the analytical solution (maximum relative error < 1 %, Fig. 5b). The MFU enables accurate handling of fractures at arbitrary angles (Chen et al., 2015), critical for complex networks with non-aligned fractures. The modular framework allows easy incorporation of alternative aperture models, including rough fractures (see Sect. 5.1).

The numerical performance of the upscaling methodology is evaluated through three complementary analyses: grid convergence studies to establish discretization requirements, parametric investigation of fracture-matrix permeability contrasts, and orientation-dependent error assessment. These analyses collectively demonstrate the method's accuracy and robustness for practical applications.

The modelling and upscaling procedure is applied to a real-scale problem by using UpsFrac. The model domain is 1000 m × 1000 m. The domain contains both large-scale fractures (faults) and small-scale fractures, referring to the measured data in the field (Massart et al., 2010). The fracture network contains both deterministic fractures and stochastically generated fractures. The deterministic fractures include three fractures with an aperture of 0.001 m. The stochastic fracture length follows a power law distribution, with lmin=10 m, lmax= 1000 m, N= 200, and α= 2.5. Fracture orientation follows Fisher distribution with κ= 0, meaning the fracture orientation is randomly distributed. Fracture location follows uniform distribution. In this study, the aperture is correlated with fracture length with γ of 2.3 ×10-5 and D of 0.5 according to the field data (Schultz and Soliva, 2012). The above geometric data are input parameters for the fracture modelling module. The rock matrix permeability is homogeneous and isotropic with 9.87 × 10⁻¹⁶ m². One realization of the discrete fracture model is shown in Fig. 11.

The grid block size for upscaling is 100 m × 100 m, which is the input parameter for the grid subdivision module. Each grid contains 3–9 fractures clipped by the grid boundaries. For the numerical simulation of each grid, the unstructured mesh size for the rock matrix is 5 m, and for fracture, it is 3.3 m. After simulating fluid flow in all grid blocks, the required information for upscaling is calculated. Through the upscaling calculation module, the equivalent permeability for each grid block is calculated.

The resulting equivalent permeability distribution is visualized in Fig. 12. The equivalent permeability distribution shows that it is highly correlated with the fracture network geometries of fractures (Fig. 12). The anisotropy in equivalent permeability components is controlled by the fracture orientation distribution.

The histograms of different components of equivalent permeability are plotted in Fig. 13. They show that all components kxx, kxy, and kyy follow a log-normal distribution. For kxy, it can be positive and negative; we use the absolute value here for plotting. For kxx and kxy, the shapes of the histograms are similar. The results indicate that even though the spatial distribution is different, the statistical properties of kxx and kxy are similar. This is mainly due to the random distribution of fracture orientations, i.e., without a preferable orientation. For kxy, it is slightly lower than kxx and kxy; this is consistent with the positive-definite nature of the permeability tensor (Lang et al., 2014).

The permeability tensor can be visualized using the ellipses. The permeability ellipses for all grid blocks are plotted in Fig. 14. They show that the long axes of an ellipse follow the fracture orientations (Fig. 11), especially for the grid with a few large fractures having high permeability (e.g., the grid block near the lower left corner of the model). The consistency between the upscaled permeability tensor ellipses and the fracture network geometries indicate that the upscaling code UpsFrac yields reasonable and accurate results. These results can be used to characterize the permeability of the fractured porous rocks and serve as input to equivalent fracture models.

A two-dimensional fractured geothermal system model was developed to evaluate the computational performance and accuracy of the EFM upscaling approach (Fig. 15). The model simulates a 500 m × 500 m geothermal reservoir with a doublet injection-production well configuration (150 m spacing), where cold water at 40 °C is continuously injected at the injection well and produced at the production well, both at 1 L s⁻¹. The domain is initialized at 80 °C and 20 MPa, with no-flow and adiabatic boundary conditions representing sealed cap and base rocks. The rock matrix has typical geothermal reservoir properties: permeability of 9.87 × 10⁻¹⁶ m², 1 % porosity, density of 2600 kg m⁻³, specific heat of 950 J (kg K)⁻¹, and thermal conductivity of 3.0 W (m K)⁻¹. The fractures feature a 500 µm mean aperture with 100 % porosity. Fluid properties include density of 1000 kg m⁻³, viscosity of 0.001 Pa s, specific heat of 4200 J (kg K)⁻¹, and thermal conductivity of 0.65 W (m K)⁻¹, representing typical values for fractured geothermal systems.

The DFM and EFM models were constructed using different meshing strategies to enable direct performance comparison (Fig. 16). For the DFM model, GMSH was employed to generate an unstructured mesh with 13 414 nodes and 30 006 elements, comprising 3820 line elements for the fracture network and 26 186 triangular elements for the rock matrix. In contrast, the EFM model utilized a structured 20 × 20 grid with uniform cell dimensions of 25 m × 25 m, resulting in 400 grid cells. The upscaling procedure using UpsFrac yielded homogeneous equivalent permeabilities of kxx≈ 4.116 × 10⁻¹³ m² and kyy≈ 4.100 × 10⁻¹³ m², with zero off-diagonal terms (kxy=kyx= 0), indicating isotropic permeability characteristics and alignment of the principal permeability directions with the coordinate axes.

Both EFM and DFM were employed to solve the coupled flow and heat transfer processes using SHEMAT-Suite and OpenGeoSys, respectively. The validation results demonstrate excellent agreement between the upscaled EFM model and the high-fidelity DFM benchmark (Fig. 17). Temperature breakthrough curves at the production well over 30 years show mean relative error of approximately 0.2 %, achieved with only 400 coarse grid cells compared to the 13 414-node DFM discretization, confirming the upscaling approach's high accuracy and robustness.

The computational performance analysis reveals substantial efficiency gains of the EFM approach (Fig. 18). The EFM upscaling process consists of four stages: Fracture-Coarse Grid Partitioning (0.186 s, 0.13 MB), Simulation Setup (28.659 s, 3.02 MB), Flow Simulation (71.245 s, 13.03 MB), and Upscaling Calculation (49.650 s, 0.67 MB), totaling 149.74 s for the upscaling process. The subsequent EFM simulation requires 540 s and 7.48 MB. In contrast, DFM using OpenGeoSys requires meshing (21.45 s, 139.41 MB) and simulation (2081 s, 75.30 MB). The complete EFM workflow required only 689.74 s versus 2102.45 s for DFM – a 67 % reduction. Total memory usage decreased from 214.71 to 24.33 MB – an 89 % reduction. These benchmarks were conducted on an Intel Core i5-9300H system with 8 GB RAM. The results confirm that the EFM upscaling methodology achieves both computational efficiency and numerical accuracy (∼ 0.2 % relative error), making it particularly suitable for large-scale fractured reservoir simulations. However, accuracy may decrease for complex fracture networks with highly irregular geometries or dense intersection patterns, where validation against fine-scale DFM solutions or field data becomes essential.

To quantify uncertainty in upscaled properties arising from stochastic DFN realizations, we performed Monte Carlo analysis with 100 independent realizations using the same geometric parameters, matrix properties, and grid configuration as described in Sect. 4.1. The ensemble analysis reveals distinct distribution characteristics for the upscaled permeability tensor components. The diagonal components (kxx and kyy) exhibit approximately log-normal distributions as confirmed by Q–Q plot analysis, while the off-diagonal component (kxy) shows normal distribution characteristics (Fig. 19). The mean upscaled permeability values across all realizations and grid cells are kxx= 1.334 × 10⁻¹³ m², kyy= 1.325 × 10⁻¹³ m², and kxy= 6.685 × 10⁻¹⁶ m² . The coefficients of variation reveal moderate variability for diagonal components (kxx: CV = 0.889, kyy: CV = 0.910) and significantly higher uncertainty for the off-diagonal component (kxy: CV = 106.288), primarily due to its very small mean value (6.685 × 10⁻¹⁶ m²) relative to its standard deviation (7.105 × 10⁻¹⁴ m²), resulting in an amplified coefficient of variation.

Convergence analysis demonstrates robust statistical behavior of the ensemble approach. The mean values of diagonal permeability, kxx and kyy, components stabilize after approximately 50–60 realizations, achieving relative errors below 5 % compared to the full 100-realization ensemble, all converging to 1.33 × 10⁻¹³ (Fig. 20). The off-diagonal component kxy converges to near-zero values. Due to the lack of pronounced directional preference in the fracture orientation distribution, and additionally, the relatively small magnitude of kxy values leads to slightly larger convergence relative errors compared to kxx and kyy. This convergence pattern indicates that 50 realizations provide sufficient statistical reliability for practical uncertainty quantification, offering guidance for balancing computational cost with accuracy requirements. The rapid convergence of ensemble statistics validates the Monte Carlo approach for capturing the stochastic variability inherent in fractured porous rocks.

The computational performance analysis reveals that the complete workflow – from fracture network generation to permeability upscaling – requires approximately 108 min total computational time for 100 realizations (1.08 min per realization) with peak memory usage of 8.2 GB. These benchmarks were conducted on a workstation equipped with an Intel Xeon W-2102 CPU @ 2.90 GHz and 16 GB RAM, running 64-bit Windows 7. The workflow consists of five stages: DFM Generation (0.19 s/realization), Fracture-Coarse Grid Partitioning (0.93 s/realization), Simulation Setup (1.43 s/realization), Flow Simulation (40.4 s/realization), and Upscaling Calculation (17.9 s/realization). Flow Simulation dominates the computational cost, accounting for 65 % of total runtime, which is expected given the need to solve pressure equations for accurate effective permeability calculations. This ensemble-based uncertainty quantification demonstrates UpsFrac's capability to efficiently perform Monte Carlo workflows for stochastic DFN analysis, transforming complex discrete fracture systems into stochastic continuum representations while preserving uncertainty information critical for risk assessment in subsurface applications.

The constant aperture assumption, while computationally efficient and widely adopted in discrete fracture modelling, represents a significant simplification of natural fracture geometries. Real fractures exhibit complex aperture distributions arising from surface roughness, stress-induced deformation, and mineral precipitation or dissolution processes (Li et al., 2022; Xue et al., 2022). To quantify the impact of this simplification, we conducted a systematic analysis comparing upscaled permeabilities computed using constant versus variable aperture distributions. The test case employs a single horizontal fracture configuration identical to the validation case (Fig. 9, θ= 0°) with a 2 m × 2 m domain and mean aperture w0= 1.2 × 10⁻⁴ m. Figure 21a demonstrates the aperture variability along a 2 m fracture for different roughness levels characterized by coefficients of variation (CV) ranging from 0.1 to 0.5, representing mild to severe aperture heterogeneity. The corresponding statistical distributions of these aperture profiles are shown in Fig. 21b through box plots, revealing increasing spread and skewness with higher CV values.

Our analysis exposes fundamental limitations of the constant-aperture assumption. Although it enables tractable simulations, accuracy degrades markedly for naturally rough fractures. As illustrated in Fig. 21c, the upscaled permeability in the x-direction (kxx) decreases substantially with increasing roughness relative to the constant aperture case. Errors increase nonlinearly with roughness. Figure 21d quantifies this degradation, showing relative errors rising from negligible (< 5 %) when CV < 0.2 to severe (> 60 %) when CV > 0.4. This occurs because the arithmetic-mean aperture cannot capture flow channeling, and the cubic law disproportionately amplifies contributions from wider segments while constrictions strongly restrict flow.

These findings have direct implications for reservoir characterization and model selection. For preliminary assessments or systems dominated by smooth, fresh tensile fractures in crystalline rocks or well-sorted sedimentary formations (CV < 0.2), where Fig. 21d indicates errors remain below 5 %, the constant-aperture model offers an efficient and sufficiently accurate approximation.In contrast, for highly rough fractures such as sheared zones and mineralized joints, explicitly representing aperture variability is essential for reliable predictions. UpsFrac's modular architecture readily accommodates this extension: supporting rough fractures requires only augmenting fracture property storage and revising local flow calculations, while preserving the existing upscaling framework.

While the current 2D implementation provides a robust foundation for method validation and pedagogical purposes, extending to 3D is essential for industrial applications. To address industrial needs, we outline a comprehensive roadmap for extending UpsFrac to 3D. Specifically, we will: (i) generalize DFN generation from 2D line fractures to 3D planar polygonal or triangulated fracture surfaces with realistic size, orientation, aperture, and connectivity distributions; (ii) extend MFU to 3D with robust treatment of fracture intersections, complex fracture–matrix exchange, anisotropy, and boundary conditions; and (iii) mitigate the substantially higher computational cost through algorithmic accelerations, including adaptive/octree meshing, hierarchical fracture-surface meshing, algebraic multigrid preconditioning, domain decomposition with parallel CPU/GPU solvers, and graph-based pruning of disconnected subnetworks. Building on our prior 3D upscaling experience (Chen et al., 2018) and established platforms such as ADFNE and MRST, we anticipate that the integration is technically feasible, though computational challenges primarily arise from the cubic scaling of 3D problems – where doubling the grid resolution increases computational cost eightfold. This provides a clear path toward evaluating industrial applicability in complex reservoir systems.

UpsFrac implements a theoretically rigorous flow-based upscaling approach. In practice, the upscaling step requires numerically solving flow in a discrete fracture model, where numerical stability must be considered for complex geometries. Achieving stable and accurate solutions may require adjusting parameters such as the mesh resolution in the fractures and the rock matrix. To prevent failed runs and facilitate correction, grid blocks that contain no fractures or that fail to converge are logged for review; by reviewing these logs, users can refine global template inputs or adjust settings for specific blocks. In the future, data-driven surrogates trained on high-fidelity DFN simulations (e.g., Almajid and Abu-Al-Saud, 2022; Yan et al., 2024b) may augment or replace direct solves to further improve robustness and efficiency.

The UpsFrac software applies MFU for calculating equivalent permeability, which is accurate as a flow-based method; other fast analytical upscaling methods (Ebigbo et al., 2016; Oda, 1985), such as Effective Medium Theory (EMT), can also easily be implemented within the UpsFrac framework to upscale permeability in fractured porous rocks. The software can be further developed to broaden its applicability and improve user convenience. Key areas for improvement include the user interface, software stability, and compatibility with other reservoir simulation tools. This can be achieved through standardization of data formats and the development of API interfaces, which will facilitate broader adoption in industrial settings (Yan et al., 2024a).