Discrete Fracture Network Research Articles

A hybrid multiscale model for fluid flow in fractured rocks using homogenization method with discrete fracture networks

Fluid flow in subsurface tight reservoirs containing pores, microcracks and macrocracks is notably influenced by the characteristics of macro/micro-cracks. A novel hybrid multiscale model is proposed to address the response of macrocracks and pores/microcracks in different spatial scales. Specifically, an equivalent macroscopic model (EMM) deduced from locally periodic representative element volume (REV) is developed using the asymptotic homogenization method to represent the poroelastic behavior of porous medium with microcracks. Simultaneously, the macrocracks are modeled explicitly using the discrete fracture model (DFM), where the hydraulic properties of cracks influenced by fluid pressure gradient is represented by the nonlinear opening/closure behavior. The obtained hybrid model takes into account the heterogeneous nature of fractured rock masses containing pores, micro/macro-cracks, which is fundamental to describe fluid flow behavior in fracture-matrix system. Specialized finite elements, regular meshing technique and adaptive time stepping algorithm are adopted to improve the computational efficiency. The hybrid multiscale model is firstly validated step by step to demonstrate the accuracy and then used to simulate fluid flow in fractured rock reservoir, shedding light on the underlying mechanisms of the enhanced flow capacity resulting from microcrack distribution, connectivity, and macrocrack stimulation.

Optimizing Production Well Geometry in the Utah FORGE Geothermal Project Using Machine Learning and Fluid Flow Modeling

This study addresses the critical challenge of optimizing well placement in Enhanced Geothermal Systems (EGS), specifically within the framework of the Utah FORGE geothermal project, as part of the 2023 Society of Petroleum Engineers (SPE) Geothermal Datathon. Effective well placement is essential for enhancing geothermal production efficiency and maximizing resource utilization. We employed a discrete fracture network (DFN) modeling approach, utilizing the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm from the Scikit-Learn library to analyze microseismic event location data. Through rigorous simulations conducted in the open-source GeoDT fluid flow simulator, we identified an optimal production well configuration characterized by a spacing of 400 m, an injection rate of 0.03 m³/s, and alignment parameters that significantly improve thermal recovery. The results indicate a projected net present value (NPV) of $75 million over a 20-year operational horizon, underscoring the economic potential of optimized well placement strategies. This study offers valuable insights for the operation of the FORGE geothermal site. More importantly, it exclusively utilizes open-source tools, enhancing accessibility and adaptability for the broader geothermal community.

Numerical Simulation of CBM Seepage Characteristics Based on Fracture Network Images

The natural fracture network within the coal body serves as the main pathway for gas migration, with its geometric characteristics significantly impacting coalbed methane flow. In order to enhance the numerical model for simulating coalbed methane flow based on fracture network images, we define porosity and permeability functions for these images and improve upon existing methods. By employing a pixel probability decomposition algorithm, we establish a geometric model of a rough discrete fracture network, which is imported into COMSOL Multiphysics to build a numerical model of gas flow. We analyze the impact of different fracture structures on coal seam permeability and find that gas primarily flows through interconnected fractures at much higher velocities compared to matrix pores. Furthermore, we observe that fracture network permeability increases with increasing porosity (0.0635–0.164), fractal dimension (1.571–1.755), maximum fracture branch length (0.0111–0.0249 m), and connectivity (0.808–2.789). Conversely, it decreases with an increasing fracture dip angle (1.61–88.39°) and tortuosity fractal dimension (1.0018–1.0195). Our simulation method based on fracture network imaging provides a simple yet feasible approach to simulate gas extraction while accurately capturing various stages in the extraction process, including the temporal and spatial evolution of gas velocity and pressure as well as differences between fractures and the coal matrix.

Integration of Geological Model and Numerical Simulation Technique to Characterize the Remaining Oil of Fractured Biogenic Limestone Reservoirs

AbstractFine geological modeling leads to accurate reservoirs numerical simulations. Fractured biogenic limestone has abundant storage spaces and flow paths to accumulate oil and gas. The complexity and diversity of fractured biogenic limestone also lead to challenges in accurately characterizing its pore volume and remaining oil. This investigation aimed to enhance the understanding of fractured biotite reservoir properties via geological modeling. Numerical simulations were used to characterize the remaining oil during the late stage of field development. Considering the differences in porosity and permeability between fractures and matrix, a facies-controlled stochastic modeling technique was used to establish a dual-porosity and dual-permeability (DPDP) model for numerical simulation. Core information, logging data, and multiple seismic attributes were combined to guide low-level sequence fault interpretation for tectonic refinement. Based on classified seismic inversion, sedimentary phases were reconstructed. A discrete fracture network (DFN) model was obtained based on fracture occurrences and density models. The optimized discrete adjoint (ODA) algorithm was utilized to calibrate model parameters. The findings revealed that dense tectonic fractures develop in thick biogenic limestone areas. Combined with advanced reservoir simulation technology, these findings suggest that areas of thicker biogenic limestone were consistent with areas of higher fracture matrix conductivity multipliers. The remaining oil distribution patterns were investigated, and to deploy new wells was guided. Therefore, it is essential to better understand the tectonic characteristics of fractured biogenic limestone reservoirs and their remaining oil distribution patterns by integrating multiple sources of information and mastering advanced reservoir simulation technology for oilfield development.

Research Progress of Three-dimensional Geological Modeling and Multi-field Coupling of Fractured Oil and Gas Reservoirs

Fractured oil and gas reservoirs play an important role in oil and gas exploration and development due to their complex fracture network structure. As the main fluid channel in the reservoir, fractures have a significant impact on the migration, accumulation and exploitation efficiency of oil and gas. This paper summarizes the key geological characteristics of fractured reservoirs, including fracture types, distribution rules and main controlling factors of fracture development, and discusses the influence of fractures on reservoir physical properties. Furthermore, the deterministic and stochastic modeling methods of three-dimensional geological modeling of fractured reservoirs and the latest progress of discrete fracture network modeling technology are introduced in detail. In addition, this paper also discusses in detail the application of multi-field coupling model in the numerical simulation of fluid flow in fractured reservoirs, including continuous medium model and discrete medium model, and their advantages and disadvantages in simulating fluid flow in fractured reservoirs. Finally, this paper summarizes the research progress of multi-physics coupling model of fractured reservoirs, and points out the direction and challenges of future research. Through these studies, it can provide theoretical support and technical guidance for oil and gas exploration and development, in order to achieve more effective development and management of fractured reservoirs.

Size-Dependent Mechanical Properties and Excavation Responses of Basalt with Hidden Cracks at Baihetan Hydropower Station through DFN–FDEM Modeling

Basalt is an important geotechnical material for engineering construction in Southwest China. However, it has complicated structural features due to its special origin, particularly the widespread occurrence of hidden cracks. Such discontinuities significantly affect the mechanical properties and engineering stability of basalt, and related research is lacking and unsystematic. In this work, taking the underground caverns in the Baihetan Hydropower Station as the engineering background, the size-dependent mechanical behaviors and excavation responses of basalt with hidden cracks were systematically explored based on a synthetic rock mass (SRM) model combining the finite-discrete element method (FDEM) and discrete fracture network (DFN) method. The results showed that: (1) The DFN–FDEM model generated based on the statistical characteristics of the geometric parameters of hidden cracks can consider the real structural characteristics of basalt, whereby the mechanical behaviors found in laboratory tests and at the engineering site could be exactly reproduced. (2) The representative elementary volume (REV) size of basalt blocks containing hidden cracks was 0.5 m, and the mechanical properties obtained at this size were considered equivalent continuum properties. With an increase in the sample dimensions, the mechanical properties reflected in the stress–strain curves changed from elastic–brittle to elastic–plastic or ductile, the strength failure criterion changed from linear to nonlinear, and the failure modes changed from fragmentation failure to local structure-controlled failure and then to splitting failure. (3) The surrounding rock mass near the excavation face of underground caverns typically showed a spalling failure mode, mainly affected by the complex structural characteristics and high in situ stresses, i.e., a tensile fracture mechanism characterized by stress–structure coupling. The research findings not only shed new light on the failure mechanisms and size-dependent mechanical behaviors of hard brittle rocks represented by basalt but also further enrich the basic theory and technical methods for multi-scale analyses in geotechnical engineering, which could provide a reference for the design optimization, construction scheme formulation, and disaster prevention of deep engineering projects.

Research on DFN Fracture Modeling and Fracturing Network Expansion Simulation Method for Shale Reservoirs

Abstract The degree of natural fracture formation and their link to hydraulic fracturing fractures are critical factors in production in the current large-scale fracturing development of shale gas reservoirs. Any magnitude of fractured reservoir heterogeneity may be described using the Discrete Fracture Network (DFN) model. This paper builds on earlier research by providing an additional summary of the multiscale DFN fracture modeling method, analyzing the fracture occurrence using statistical analysis techniques, analyzing the fracture scale using fractal theory, and determining the fracture development degree by combining the fracture attributes in imaging logging and three-dimensional seismic data. Different-scale fracture network models were developed. Simulating the growth of hydraulic fractures accurately using Mohr Coulomb and Barton Bandis criteria, this study also takes into account the interaction between natural and hydraulic fractures, as well as the physical properties of the rock (e.g., elastic modulus, Poisson’s ratio), closure pressure, crustal stress field, and other factors. The fracture network information, sitmulated reservoir volume, stress shadow and other attributes obtained from fracturing simulation can further analyze and evaluate the effect of fracturing simulation, and optimize the well location deployment plan and construction plan in combination with geological engineering conditions.

Automatic interpretation of geometric information of discontinuities and its influence on the stability of highly-jointed rock slopes

The discrete fracture system of a rock mass plays a crucial role in controlling the stability of rock slopes. To fully account for the geometric shape and distribution characteristics of jointed rock masses, terrestrial laser scanning (TLS) was employed to acquire high-resolution point-cloud data, and a developed automatic discontinuity-identification technology was coupled to automatically interpret and characterize geometric information such as orientation, trace length, spacing, and set number of the discontinuities. The discrete element method (DEM) was applied to study the influence of the geometric morphology and distribution characteristics of discontinuities on slope stability by generating a discrete fracture network (DFN) with the same statistical characteristics as the actual discontinuities. Based on slope data from the Yebatan Hydropower Station, a simulation was conducted to verify the applicability of the automatic discontinuity identification technology and the discrete fracture network-discrete element method (DFN-DEM). Various geological parameters, including trace length, persistence, and density, were examined to investigate the morphological evolution and response characteristics of rock slope excavation under different joint combination conditions through simulation. The simulation results indicate that joint parameters affect slope stability, with density having the most significant impact. The impact of joint parameters on stability is relatively small within a reasonable range but becomes significant beyond a certain threshold, further validating that the accuracy of field geological surveys is critical for simulation. This study provides a scientific basis for the construction of complex rock slope models, engineering assessments, and disaster prevention and mitigation, which is of great value in both theory and engineering applications.

An extended two-parameter mixed-dimensional model of fractured porous media incorporating entrance flow and boundary-layer transition effects

We develop an enhanced reduced model for single-phase flow in fractured porous media capable of incorporating more realistic interface conditions at the fracture terminations. In addition to the traditional dimensional model reduction, where the elements of the discrete fracture network are treated as lower dimensional manifolds embedded in the porous matrix, we explore the microscale behavior of the boundary layer flow at the entrances of a fracture bounded by two parallel plates to construct a new set of interface conditions of Robin-type, giving rise to localized pressure jumps at the fracture edges. Within this enriched description, sharper reduced flow and tracer transport mixed-dimensional models are constructed in the asymptotic limit ruled by two small parameters related to the ratio between fracture aperture and entrance developing length and a macroscopic length scale. The discrete flow/transport mixed-dimensional model is discretized by a new discontinuous Galerkin(dG)-based formulation. An adequate version of the Galerkin-Newton method is developed for the numerical treatment of the non-linear Robin interface condition. Considering several fracture arrangements, numerical results illustrate the sharper description of the model proposed herein in predicting flow and tracer transport patterns in fractured media.

Modelling the Flow in the Utah FORGE Wells Disrete Fracture Network

The focus of this paper is the efficient numerical solution of the fluid flow in the Utah Frontier Observatory for Research in Geothermal Energy (FORGE) reservoir. In this study, the public data available for Discrete Fracture Networks (DFN) around well 58-32 is used to represent the DFN. In this research, a novel computationally efficient method called Hele-Shaw (HS) approximation is used for modeling fluid flow in FORGE well. An analysis of the influence of fracture intensity in a network is carried out using the HS method. The HS method was validated by solving the full Navier–Stokes equations (NSE) for a network of eight fractures. A good agreement was observed between the evaluated results (average deviation of 0.76%).

Coupling stress and transmissivity to define equivalent directional hydraulic conductivity of fractured rocks

A DFN (Discrete Fracture Network) modelling approach is developed to couple stresses with fracture transmissivities and to evaluate large scale rock mass hydraulic conductivity. The transmissivity-stress coupling relies on a negative exponential correlation between normal stress acting on a fracture and fracture transmissivity, bounded by residual and maximal apertures. The remote stresses and the local stress fluctuations induced by the fractures themselves are combined in a semi-analytical approach to compute the normal stress acting on each fracture of a DFN. Directional equivalent hydraulic conductivities are numerically computed in all directions from a spherical permeameter setup. The resulting properties are first a cloud of points, where each point defines a direction and an equivalent hydraulic conductivity. The distribution of equivalent hydraulic conductivities is analyzed to define mean values, preferential directions and anisotropy ratio. The entire workflow is developed in the numerical platform DFN.lab. The capacity of the method to investigate the impact of the in-situ stresses on the rock mass hydraulic conductivity is illustrated for fracturing and in-situ stress conditions similar to the conditions at the Forsmark site in Sweden. We find that the stress fluctuations induced by the fractures have a significant impact on the resulting hydraulic conductivity field. They limit the anisotropy ratio to values close to a factor of 3 while the transmissivity distribution is correlated to orientations and spans several orders of magnitude. Sensitivity analyses, performed by changing the parameters of the transmissivity-stress law, show quantitatively how the directional hydraulic conductivities are rather controlled by the orientations of the in-situ stresses or by the underlying connectivity structure of the DFN.

Study on Discrete Fracture Network Model and Rock Mass Quality Evaluation of Tunnel Surrounding Rock

Study on Discrete Fracture Network Model and Rock Mass Quality Evaluation of Tunnel Surrounding Rock

A study of rock mass properties based on discrete fracture network modeling and compression damage process

A study of rock mass properties based on discrete fracture network modeling and compression damage process

Investigation of nonlinear flow in discrete fracture networks during shear based on XFEM method

To characterize fractured rock masses, self-developed programs are utilized to generate fracture networks. A nonlinear flow model considering the shear dilatancy effect is established, and a numerical solution method for modelling nonlinear flow in fractured rock masses during shear is proposed on the basis of extended finite element analysis. The contour plots reveal distinct patterns in the water pressure and flow distributions within fractures. The reduction in the lateral pressure coefficient and increase in the shear stiffness of the joints facilitate a more homogeneous distribution of the water pressure gradient. Under the same vertical stress, increasing the lateral pressure coefficient or decreasing the shear stiffness leads to a more pronounced shear dilatancy effect on fractures. Consequently, an increase in fracture aperture and permeability occurs, and the flow of the fractured rock mass is enhanced. With the same vertical stress, an increase in horizontal stress and a decrease in shear stiffness lead to a gradual reduction in the linear and nonlinear coefficients of Forchheimer’s law. Specifically, the influence of the lateral pressure coefficient on the linear coefficient is greater than that on the nonlinear coefficient.

HVPS-DFN-DL: Intelligent capture and characterization of geological fracture outcrops based on a hybrid vision-photogrammetric system and discrete fracture network

The main objective of this article is to provide a framework for intelligent capture-acquisition analysis of geometric information from geological outcrops. By combining deep learning methods with photogrammetric data from unmanned aerial vehicles (UAVs), FPV drones, and terrestrial cameras acquired by a hybrid vision-photogrammetric system (HVPS), intelligent fracture detection and geometric information segmentation of multiscale field geological outcrops were achieved. The extraction results were subsequently used to generate a three-dimensional discrete fracture network (DFN) of real rock masses for studying the influence of the spatial connectivity of discontinuity structural planes on the mechanical and hydrodynamic characteristics of rock masses. By testing data collected in situ from a variety of field rock masses in several regions of China, this framework was shown to be a very efficient method for geostatistical work, exhibiting very low measurement errors. Furthermore, this framework is extremely safe for geologists and applicable to a wide range of site geological environments. It is also suitable for field geological surveys, geometry acquisition of outcropping lithologies, obtaining tunnel face and surrounding fissure statistics, and geological stability assessment of unstable rock masses. This framework can also provide a method for unmanned topographic-geological exploration. Furthermore, the fracture network realism and the data acquisition efficiency have been greatly improved, and the difficulty of developing field measurements and validating the DFN model has been overcome.

Comparative investigation of heat extraction performance in 3D self-affine rough single fractures using CO2,N2O and H2O as heat transfer fluid

The type and thermophysical properties of heat transfer fluids significantly impact the heat extraction rate of Enhanced Geothermal Systems (EGS). Studies based on field-scale stochastic discrete fracture network geometric models have certain limitations and uncertainties. To compare the heat extraction performance of heat transfer fluids H2O, CO2, and N2O in a core-scale (φ50 × 100 mm) rough single fracture in hot dry rock, we first constructed the fracture geometry with rock surface roughness characteristics using a fractal self-affine function, with a joint roughness coefficient (JRC) of 12.38. Then, at temperatures ranging from 37 to 300°C and a pressure of 30 MPa, we established a thermo-hydraulic (TH) coupling model based on the thermophysical parameter equations of the three heat transfer fluids. Finally, we conducted a comparative study of the changes in the outlet mean temperature and heat extraction rate after flowing through the fracture, as well as the temperature distribution on the fracture surface, under conditions of constant inlet flow velocity with varying inlet temperature and constant inlet temperature with varying inlet flow velocity. The results indicate that (1) with the increase in inlet flow velocity and temperature, the outlet mean temperature and heat extraction rate of CO2 and N2O at the fracture outlet are essentially the same and lower than those of H2O. (2) When the inlet flow velocity increases from 0.005 m/s to 0.025 m/s, the thermal convection effect from the rock matrix to the fluid in the fracture predominates, and flow velocity is the main factor affecting the heat transfer process between the fluid and the rock matrix. When the inlet temperature increases from 37°C to 57°C, the thermal conduction effect from the rock matrix to the fluid in the fracture predominates, and temperature is the main factor affecting the heat transfer process between the fluid and the rock matrix.(3)When maintaining constant inlet temperature, increasing inlet flow velocity from 0.005 m/s to 0.025 m/s results in CO2 experiencing a maximum heat extraction rate increase of 118.85 W. In contrast, N2O experiences 117.05 W, and H2O experiences 266.4 W.(4) When the inlet flow rate is constant, the maximum increase in heat extraction rate for CO2 is 15.34 W as the inlet temperature increases from 37 °C to 57 °C; for N2O, it is 13.9 W; and for H2O, it is 45.45 W.

Comparative Assessment of U-Net-Based Deep Learning Models for Segmenting Microfractures and Pore Spaces in Digital Rocks

Summary Segmentation of high-resolution X-ray microcomputed tomography (µCT) images is crucial in digital rock physics (DRP), affecting the characterization and analysis of microscale phenomena in the porous media. The complexity of geological structures and nonideal scanning conditions pose significant challenges to conventional image segmentation approaches. Motivated by the recent increasing popularity of deep learning (DL) techniques in image processing, this work undertakes a comparative study of DL models, specifically U-Net and its variants, for segmenting multiple targets with distinguished features in digital rocks, including discrete fracture networks (DFNs), pore spaces, and solid rock. Particularly, DFNs have a smaller volumetric fraction over others, bringing in a substantial challenge of imbalanced segmentation. The primary focus is to evaluate the architecture and feature enhancement strategies of various DL models, including U-Net, attention U-Net, residual U-Net, U-Net++, and residual U-Net++. The models were designed as 2.5D, utilizing a central 2D image and its two adjacent upper and lower 2D images as input to provide a pseudo-3D context. In addition, because the ground truth of segmentation was unknown for real-world digital rocks, we created a benchmark data set following the inverse operations of segmentation. The data synthesis started from the label images (i.e., solid rock, pore spaces, and DFNs), followed by simulating partial volume blurring, adding random background noise, and introducing ring artifacts to mimic real raw X-ray µCT images. The data set, which included various rock types (i.e., sandstone and artificial data), scanning resolution, and magnitudes of noise and artifacts, was divided into training and testing data sets with a 90% and 10% ratio, respectively. Moreover, in addition to the conventional pixel-wise evaluation metrics, the physics-based metric of the lattice-Boltzmann method (LBM) simulated permeability provided more comprehensive assessments. The results demonstrated that the residual connections, nested architectures, and redesigned skip connections contribute to the model performance and give the residual U-Net++ the highest accuracy. The improvements were mainly on the boundaries and small targets, especially the DFNs, which dominate the interconnectivity and therefore affect the permeability greatly. This study also rigorously evaluated the efficiency and generalization of each model, demonstrating that the sophisticated architectures achieved excellent practicability and maintained robust performance on completely unseen data, ensuring their suitability for diverse and challenging DRP applications.

Stimulation characteristics of natural reservoir system under high-voltage electropulse-assisted fluid injection

This study elucidates the findings of a computational investigation into the stimulation characteristics of natural reservoir systems enhanced by high-voltage electropulse-assisted fluid injection. The presented methodology delineates the comprehensive rock-fracturing process induced by electropulse and subsequent fluid injection, encompassing the discharge circuit, plasma channel formation, shockwave propagation, and hydro-mechanical response. A hydromechanical model incorporating an anisotropic plastic damage constitutive law, discrete fracture networks, and heterogeneous distribution is developed to represent the natural reservoir system. The results demonstrate that high-voltage electropulse effectively generates intricate fracture networks, significantly enhances the hydraulic properties of reservoir systems, and mitigates the adverse impact of ground stress on fracturing. The stimulation-enhancing effect of electropulse is observed to intensify with increasing discharge voltage, with enhancements of 118.0%, 139.5%, and 169.0% corresponding to discharge voltages of 20 kV, 40 kV, and 60 kV, respectively. Additionally, a high-voltage electropulse with an initial voltage of U0 = 80 kV and capacitance C = 5 μF has been shown to augment the efficiency of injection activation to approximately 201.1% compared to scenarios without electropulse. Under the influence of high-voltage electropulse, the fluid pressure distribution diverges from the conventional single direction of maximum stress, extending over larger areas. These innovative methods and findings hold potential implications for optimizing reservoir stimulation in geo-energy engineering.

Sensitivity Analysis in the Presence of Intrinsic Stochasticity for Discrete Fracture Network Simulations

AbstractLarge‐scale discrete fracture network (DFN) simulators are standard fare for studies involving the sub‐surface transport of particles since direct observation of real world underground fracture networks is generally infeasible. While these simulators have successfully been used in several engineering applications, estimates of output quantities of interest (QoI) — such as breakthrough time of particles reaching the edge of the system — suffer from two distinct types of uncertainty. A run of a DFN simulator requires several parameters to be set that dictate the placement and size of fractures, the density of fractures, and the overall permeability of the system; uncertainty on the proper parameters will lead to uncertainty in the QoI, called epistemic uncertainty. Furthermore, since these input settings to DFN simulators control the stochastic processes which place fractures and govern flow, understanding how this randomness affects the QoI requires several runs of the simulator at distinct random seeds. The uncertainty in the QoI attributed to different realizations (i.e., different seeds) of the same random process (i.e., identical input parameters) leads to a second type of uncertainty, called aleatoric uncertainty. In this paper, we perform a Sensitivity Analysis, which directly attributes the uncertainty observed in the QoI to the epistemic uncertainty from each input parameter and to the aleatoric uncertainty. Beyond the specific takeaways on which input variables influence uncertainty in the QoI the most, a major contribution of this paper is the introduction of a statistically rigorous workflow for characterizing the uncertainty in DFN flow simulations that exhibit heteroskedasticity.

REV and Three-Dimensional Permeability Tensor of Fractured Rock Masses with Heterogeneous Aperture Distributions

This study performed a representative elementary volume (REV) and 3D equivalent continuum study of rock fractures based on fluid simulations of 3D discrete fracture networks (DFNs). A series of 3D DFNs with heterogeneous aperture distributions (the DFN-H model) and uniform apertures (the DFN-I model) were established, in which the fractures were oriented according to the geological field mapping of a high-level radioactive waste candidate site in China. The 3D DFNs of the different model sizes were extracted and rotated in a number of directions to check whether there was a tensor quality of the permeability at a certain scale. The results show that aperture heterogeneity increases the REV size and results in a necessarily larger model size to reach an equivalent continuum behavior, and this effect is more obvious when the fracture density is smaller. The shape of the 2D permeability contour is irregular, with some breaks when the model size is small. As the model size increases, its shape gradually tends to become smooth and approaches an ellipse. The shape of the permeability contours of the DFN-H model is slender compared to the DFN-I model, indicating a larger difference between the minimum and maximum values of the permeability. For the DFN-H model, there is no appropriate approximation for the equivalent permeability tensor over the studied model size range, whereas a good fit of the permeability ellipsoid is obtained for the DFN-I model, and the 3D directional permeability is calculated at this model scale. The corresponding magnitude and direction of the principal permeability are obtained, which can be viewed as the equivalent permeability tensor for the approximated continuum medium.