Random Fracture Networks Research Articles

Quantifying the impact of upscaled parameters on radionuclide transport in three-dimensional fracture-matrix systems

Assessing the long-term safety of geological repositories for high-level radioactive waste is critically dependent on understanding radionuclide transport in multi-scale fractured rocks. This study explores the influence of upscaled parameters on radionuclide movement within a three-dimensional fracture-matrix system using a discrete fracture-matrix (DFM) model. The developed numerical simulation workflow includes creating a random discrete fracture network, meshing of the fractures and matrix, assigning upscaled parameters, and conducting finite element simulations. We simulated the spatiotemporal evolution of radionuclide concentrations in the fractures and matrix over a century, revealing significant spatial heterogeneity driven by a heterogeneous seepage field. Employing geostatistics-based upscaling methods, we predicted the effective ranges of crucial solute transport parameters at the field scale. The matrix diffusion coefficient, matrix distribution coefficient, and longitudinal dispersivity were upscaled by factors of 2.0–3.0, 2.5–4.0, and 10–104, respectively, based on laboratory-scale measurements. Incorporating these upscaled parameters into the DFM model, we analyzed their impact on radionuclide transport. Our findings demonstrate that an upscaled matrix diffusion coefficient and matrix distribution coefficient result in a delayed transport of radionuclides in fractures by enhancing mass transfer between the fractures and rock matrix, while an upscaled longitudinal dispersivity accelerates transport by advancing the positions of concentration peaks. Sensitivity analysis revealed that the matrix distribution coefficient is the most impactful, followed by dispersivity and matrix diffusion coefficient. These insights are important for minimizing parameter uncertainties and enhancing the accuracy of predictions concerning radionuclide transport in multi-scale fractured rocks.

Data-driven hydraulic property analysis and prediction of two-dimensional random fracture networks

The large parameter space and uncertainty of the discrete fracture networks (DFNs) make the hydraulic characteristics have multiple possibilities. Statistical analysis of hydraulic characteristics based on a large amount of data is a standard practice and central to DFN methodology. The present study developed a two-dimensional (2D) DFN numerical seepage model based on Matlab and Comsol from modeling and parameter calculation to simulation to investigate the influence of parameters involved fracture network on seepage characteristics of DFNs. In total, 1728 2D DFN models were generated with increasing fracture density, fracture length, power-law exponent, fracture orientations, and fracture included angles, and 1728 × 3 = 5184 numerical simulations of fluid flow along the x-axis, y-axis, and center through the DFN models are conducted using a self-developed numerical code. The influence of fracture geometric features on connectivity and permeability are estimated, respectively. And to realize the prediction of permeability tensor by known geological data, a theoretical model based on the Snow model and a data mining model based on the back-propagation model are discussed. The analysis helps researchers and engineers can make informed decisions and develop effective strategies for addressing the challenges posed by uncertainty hydraulic property of the fractured rock masses.

The hidden structure of hydrodynamic transport in random fracture networks

We study the large-scale dynamics and prediction of hydrodynamic transport in random fracture networks. The flow and transport behaviour is characterized by first passage times and displacement statistics, which show heavy tails and anomalous dispersion with a strong dependence on the injection condition. The origin of these behaviours is investigated in terms of Lagrangian velocities sampled equidistantly along particle trajectories, unlike classical sampling strategies at a constant rate. The velocity series are analysed by their copula density, the joint distribution of the velocity unit scores, which reveals a simple, albeit hidden, correlation structure that can be described by a Gaussian copula. Based on this insight, we derive a Langevin equation for the evolution of equidistant particle speeds. In this framework, particle motion is quantified by a stochastic time-domain random walk, the joint density of particle position, and speed satisfies a Klein–Kramers equation. The upscaled theory quantifies particle motion in terms of the characteristic fracture length scale and the distribution of Eulerian flow velocities. That is, it is predictive in the sense that it does not require the a priori knowledge of transport attributes. The upscaled model captures non-Fickian transport features, and their dependence on the injection conditions in terms of the velocity point statistics and average fracture length. It shows that the first passage times and displacement moments are dominated by extremes occurring at the first step. The presented approach integrates the interaction of flow and structure into a predictive model for large-scale transport in random fracture networks.

A New Method to Three-dimensional Dual Medium Geometry Model Construction

Dual medium geometric model is widely used in soil and rock mass environment, which can only reflect the material exchange between cracks and matrix. The dual medium geometric model ignores the hydraulic relationship between the pores and cracks. The paper develops a new method to construct the three-dimensional dual medium geometry model. This method is completely written by Matlab scripting language, can describe the real spatial distribution state of fracture in bedrock. In the method, three-dimensional random fracture network geometric models with different parameters and three-dimensional porous media geometric models with Berlin Noise characteristics are constructed respectively, and then use the three-dimensional direct superposition method to construct the dual medium geometric model. The paper introduces the construction of three-dimensional random fracture network geometry model, three-dimensional porous media geometric model and the dual medium geometric model by direct superposition method. This research is of great significance to the construction of dual medium model and numerical simulation in related fields.

A new numerical well test method of multi-scale discrete fractured tight sandstone gas reservoirs and its application in the Kelasu Gas Field of the Tarim Basin

The cretaceous gas reservoir in Kelasu Gas Field of the Tarim Basin is a rare ultra-deep and ultra-high pressure fractured tight sandstone gas reservoir where multi-scale discrete fractures of matrix, fracture and fault are developed, so its development cannot be conducted just based on static and dynamic reservoir description. In order to solve this problem, this paper establishes a numerical well test model of vertical wells based on matrix, fractures and faults (large fractures and small faults) by combining the random generation of natural fracture networks with the unstructured discrete fracture modeling method to break through the traditional continuous medium well test model. In addition, the model is solved by using the finite element method with mixed element, and the typical well test type curves under different random fracture networks are obtained. And the following research results are obtained. First, based on the observed data, the fracture network distribution modes of fractured tight sandstone gas reservoirs are classified into three categories. The influence of random generation of fracture networks on typical well test type curves is discussed. The results of discrete fracture well test model are compared with those of the traditional continuous medium well test model, and the applicable conditions of the traditional continuous medium well test model is determined. Second, there are great differences between the results of discrete fracture model and those of dual porosity medium model. 1 The dual porosity medium model is a special case of the discrete fracture model, in which the fractures are evenly distributed within infinitely small spacing. Third, the characteristics of well test type curves under three fracture network distribution modes are discussed. The well test type curves that cannot be interpreted by the conventional dual/triple porosity continuous medium model are successfully interpreted by using the established well test interpretation model of random discrete fracture. The curve matching effect is ideal and the interpreted parameters are reasonable. In conclusion, the new model and the new method reveal the development mechanism of step-by-step production and coordinated gas supply between media of different scales, explain the development characteristics of large inter-well productivity difference and abnormal rapid inter-well pressure response, and provide a reference for the development of similar gas reservoirs.

The Propagation of Hydraulic Fractures in a Natural Fracture Network: A Numerical Study and Its Implications

Natural fractures play a significant role in creating a fracture network simulation treatment. In this work, global cohesive elements were incorporated into the cohesive zone method to realize the unprompted propagation of a hydraulic fracture. The step-by-step propagation patterns of hydraulic fractures in a random natural fracture network were discussed. An effective area was defined to quantitively assess the influenced area of induced fractures. The results showed that the hydraulic fracture tips were attracted by local natural fractures when the horizontal stress difference was low. Bifurcations and secondary fractures occurred at the natural fracture intersections, which contributed to the complexity of the induced fracture network on a local scale. The length of the main hydraulic fracture reached the maximum when the in situ stress ratio was 1.12. The influence of natural fractures on the overall trend of fracture propagation was limited when the in situ stress difference increased. It also suggested that a lower rock tensile strength and natural fractures cementation strength improved the main fracture length. A higher tensile strength of rock increased the initiation pressure of the induced fracture, while the cementing strength of the natural fractures showed no impact on it. The results presented in this paper could improve the basic understanding of the fracture development in a natural network and help to predict a complex fracture network in a real situation.

Performance of enhanced geothermal system with varying injection-production parameters and reservoir properties

The injection-production parameters and reservoir properties significantly affect the heat extraction performance of the enhanced geothermal system (EGS), while quantitative rankings of the effecting parameters have been yet unknown. This paper is devoted to investigating numerically the heat extraction performance of EGS by three-dimensional reservoir models under the influence of varying injection-production parameters and reservoir properties. A thermal-hydro-mechanical (THM) coupled model is used to quantify the complicated heat extraction process of EGS. Three-dimensional geothermal reservoir models containing random fracture networks are established based on the Monto Carlo algorithm to quantify the effect of fracture numbers in the process of heat extraction. The parameter sensitivity of the injection-production parameters (injection mass flow rate and injection temperature) and the reservoir properties (rock matrix porosity, fracture permeability, and rock matrix permeability) on the heat extraction performance are ranked and evaluated over the Qiabuqia enhanced geothermal system. It is found the heat extraction performance shows non-monotonic relationships with the fracture numbers in the three-dimensional reservoir model, which indicates the necessity of EGS fracture optimization. The injection mass flow rate turns out the first while the fracture permeability and injection temperature are the secondary dominating factors affecting the heat extraction performance among the evaluated parameters. The outcomes are beneficial to the high-efficiency construction, operation, and optimization of the EGS especially those with similar conditions to the Qiabuqia EGS.

Investigation of Nonlinear Flow in Discrete Fracture Networks Using an Improved Hydro-Mechanical Coupling Model

Fractures commonly exist in rock masses; the coalescence of fractures provides fluid flow pathways in a fractured rock mass and greatly increases the flow capacity of fractured rock. This work aims to study the characteristics of nonlinear flow in fractures. A series of tests were conducted and indicated that the Forchheimer law performed well when describing the nonlinear relationship between hydraulic gradient and flow. The test results also indicate that higher water pressure may induce stronger nonlinearity. Additionally, the linear and nonlinear coefficients of the Forchheimer law increase with a decrease in the particle size of the filling material in fractures. On the basis of the laboratory results, the classical Forchheimer law was modified by considering the influence of stress on the variation of fracture aperture. A hydro-mechanical coupling model for fractured rock masses was built and programmed with a subroutine through ABAQUS. Furthermore, a random discrete fracture network was generated and simulated to prove that a high flow velocity will result in a nonlinear flow, not only in a single fracture, but also in a fracture network. The numerical results from fractured rock masses show that a ratio of the flow to the hydraulc gradient will change the flow from linear to weak nonlinearity and, finally, to strong nonlinearity with an increase in the hydraulic gradient. It also shows that the linear and nonlinear coefficients increase with an increase in the confining pressure and that they decrease with an increase in the aperture. Due to the complexity of fracture channels, a nonlinear flow is likely to occur in a fractured rock mass. Finally, the developed model was applied to simulate the flow behavior of underground engineering; the results show that the smaller the hydraulic aperture is, the higher the water pressure is required to be in order to change the flow regime from linear to nonlinear.

Pressure transient response with random distributed fracture networks using a semi-analytical method for fractured horizontal wells in a rectangular closed reservoir

The focus of this paper is to establish a general model, which is capable of dynamic simulation of discrete fractures at any position and arbitrary distribution in rectangular closed reservoir. In this article, a semi-analytical model for fractured horizontal wells with random distributed fracture networks is proposed to facilitate pressure transient response using a double node method. A rectangular closed reservoir model and fracture network flow model are established respectively, and then coupled at the fracture wall. In order to deal with the flow distribution at the fracture intersection, a simple adaptive material balance method is proposed in this paper, which successfully solves the flow distribution at the intersection. Using advanced mathematical means, the model is successfully solved in Laplace space, and then the pressure response solution in real space is obtained by Stehfest numerical inversion method. The obtained solutions are verified and compared with the numerical simulation results. The calculation results show that the flow characteristics of the orthogonal fracture network can be roughly divided into six stages, namely, bi-linear flow, first linear flow, first radial flow, second linear flow, second radial flow and boundary dominated flow. However, these flow characteristics will deviate or even be missing, depending on the complexity of the fracture network. Finally, the effects of some important parameters (such as dimensionless fracture network conductivity, main fracture length, fracture network density, reservoir area and eccentric position) on pressure response and pressure field distribution are discussed in detail.

The investigation of fracture pattern effect on fluid transport and production prediction in M field

Due to the discontinuity and multi-scale of fracture porous media (FPM), fluid transport in the naturally fractured reservoir (NFR) is more complex than in the conventional reservoirs. Based on the embedded discrete fracture model (EDFM), the effect of fracture pattern (i.e. fracture orientation, intersection relationship, numbers and distribution) on fluid transport is investigated, and the paper also conducts research on pressure distribution and flow behavior. A series of fracture models are built to get a detailed analysis of fracture pattern influences. Results show that when the fracture orientation tends to parallel the main stream direction, the seepage flow through the fracture is more distinct and the pressure drop at two sides of fracture becomes larger. As the intersecting fracture cannot serve as part of main-stream channels, the intersection is almost invalid. In the multi-scale fracture model, the pressure drop in the large-scale fractures network is higher than in the small-scale fractures. Thus, the simulation results show that the fracture pattern has a significant impact on the seepage flow and pressure distribution. Also, this model is simulated for M field which has random natural fracture network to study the flow and transport characteristics in fractured porous media. The comprehensive simulation results the fracture patterns have impact effect on production.

Grout penetration process simulation and grouting parameters analysis in fractured rock mass using numerical manifold method

Grouting is a commonly used technique in rock engineering to enhance the joint strength and improve the stability of surrounding rock. Grout penetration characteristic is controlled by grouting parameters and has a significant role on practice. In the study, a numerical manifold method (NMM) for grout penetration process simulation in fractured rock mass is firstly proposed. The fluid flow behaviour of the grout is assumed to be a Bingham fluid and control equations are established using discrete fracture network model. The global discretization equation, element sub-matrixes and NMM simulation algorithm for grouting are presented. Then, numerical tests for grouting process in a single fracture and a regular fracture network are conducted firstly to verify the proposed NMM grouting model by comparing with analytical solutions and experimental results. Furthermore, the effects of mesh size, fracture and slurry parameters on the grouting performance are systematically investigated using a random fracture network example. The numerical results indicate that the grouted zone and propagation depth decrease as the mesh size of numerical model and yield strength increases, while it increases as initial fracture aperture and grouting pressure increases.

Experimental and theoretical analysis of hydraulic fracturing and gas fracturing of rock under true triaxial compressions

A test system was developed for true triaxial hydraulic fracturing (HF) and gas fracturing (GF) combined with acoustic emission (AE) monitoring and 3D computerized tomography (CT) scanning reconstruction to carry out a series of HF and GF tests under different true triaxial loading conditions. The critical breakdown pressure characteristics, the AE activities, and the fracture propagation and spatial morphologies were obtained during the fracturing process. The test results showed that the GF breakdown pressure was approximately 29.0–31.3% less than the HF breakdown pressure. Moreover, for the GF specimen, the gas pore pressure induced a large number of new microfractures before critical fracture. However, both HF and GF under true axial compression can only produce simple main longitudinal fractures (LFs, along the borehole) and transverse fractures (TFs, across the borehole) that are strictly dependent on the initial true triaxial stress field. Based on the experimental observations, a hypothesis for the initiation of the HF and GF under true triaxial compression was proposed by using the maximum tangential stress on the borehole wall and the maximum axial stress along the borehole. Accordingly, analytical solutions of the critical breakdown pressure for the HF (impermeable) and GF (permeable) specimens were derived. The theoretical predictions agree well with the observations from the 3D numerical simulations and tests. Lastly, a novel pulsed GF method (dynamic loading fracturing) was proposed to overcome the dependence of the in-situ crustal stress field associated with traditional HF and GF methods (static loading fracturing). Laboratory test results showed that the dynamic pulsed GF at moderate pressure can generate a complex and random fracture network independent of the in-situ stress field in rock.

On the effective stress coefficient of saturated fractured rocks

The applicability of the classic Terzaghi or Biot effective stress laws to fractured rocks is not clear. Based on discrete element models, bulk modulus method and equivalent strain method are presented to determine the effective stress coefficient of saturated fractured rocks, and were examined with the regular fracture network I. Two types of discrete fracture networks, including regular and random network models, were built, and a total of over 300 numerical simulations were performed using the equivalent strain method for sensitivity analysis of the effective stress coefficient of fractured rocks to the fracture geometries characteristics, and the mechanical properties of both the intact rock and the fracture. The classic effective stress laws for porous media cannot be extended to fractured rocks, mainly due to the different volumetric deformation behaviors between porous and fractured materials. The effective stress coefficient is the largest in the regular fracture network I and the smallest in the random fracture network. As the normal and shear stiffness of the fracture increase, the effective stress coefficient decreases, while it increases with the increasing elastic modulus and Poisson’s ratio of the intact rock. An empirical model was proposed to predict the effective stress coefficient of saturated fractured rocks.

Emergence of Stable Laws for First Passage Times in Three-Dimensional Random Fracture Networks.

We study first passage behaviors in the flow through three-dimensional random fracture networks. Network and flow heterogeneity lead to the emergence of heavy-tailed first passage time distributions that evolve with increasing distance between the start and target planes, and transition toward stable laws. Analysis of the spatial memory of the first passage process shows that particle motion can be quantified stochastically by a time domain random walk conditioned on the initial velocity data. This approach identifies advective tortuosity, the velocity point distribution and the average fracture link length as key quantities for the prediction of first passage times. Using this approach, we develop a theory for the evolution of first passage times with increasing distance between the start and target planes and the convergence towards stable laws.

3D Numerical Simulation of Elastic Wave Propagation in Discrete Fracture Network Rocks

Fractures play an important role in controlling rock block stability and the hydraulic properties of fractured rock formations. Understanding elastic wave propagation in fractured media can result in significant advances for the geophysical prediction of fracture parameters from seismic data. However, most natural fracture characteristics, such as fracture length, aperture, angle and location are random; therefore, fracture models must be built discretely and follow some stochastic principles. We construct stochastic models of fractured rock samples using a random fracture network rather than a single fracture. Three-dimensional (3D) wave field computation is a computationally complex problem. Here, the 3D fourth-order in space, second-order in time, displacement-stress staggered-grid finite-difference scheme is used for accurate simulations. Our numerical examples demonstrate the effects of varying fracture number, aperture, and length distribution of the fracture network on the seismic response. The wave field scattering caused by the contrast between fractures and background media is one of the key features, and the resulting scattering is more obvious for S-waves than for P-waves. Such an approach can be applied to any fracture network model that provides a link between fracture parameters and seismic attributes.

Investigation on heat extraction characteristics in randomly fractured geothermal reservoirs considering thermo‐poroelastic effects

AbstractA fully coupled thermal‐hydraulic‐mechanical (THM) model was developed to investigate the underlying response mechanisms during heat extraction in fractured geothermal reservoirs. The random fracture network in the stimulated zone was reproduced based on the fractal theory. The coupled model accounts for the dominant physical phenomena including (a) fluid flow, heat transport, and solid deformation in porous media and fractures; (b) local thermal nonequilibrium (LTNE) between rock matrix and flowing fluid; and (c) temperature‐dependent fluid thermodynamic properties and stress‐dependent pore and fracture permeability. The proposed model was validated by several analytic solutions. Sequentially, the evolution of pore pressure, equivalent temperature, effective stress, and reservoir permeability was analyzed. The sensitivity of the heat extraction performance to fracture network morphology was discussed. Results show that interconnected large‐scale fractures dominate mass and heat transport. Widely distributed small‐scale fractures contribute to the cooling of the heat rock mass along many flow paths in parallel. The change in effective stress associated with fully coupled thermo‐poroelastic effects may induce fracture shear dilation and pore expansion, resulting in an enhancement of the overall reservoir permeability. There is a significant temperature difference between the solid phase and the fluid phase in fractures, but not in porous media. It is more reasonable to use the LTNE theory to analyze the heat extraction of fractured geothermal reservoirs. The fracture network morphology has a profound effect on heat extraction efficiency and injectivity. Undeveloped fracture networks will result in a higher injection pressure, earlier thermal breakthrough, and shorter lifetime, but higher heat extraction ratio. Properly increasing the fracture density can delay the thermal breakthrough time, prolong the service life, and improve the injectivity. Fully connected fracture networks may result in thermal short‐circuiting and earlier thermal breakthrough. Generating a complex and scattered fracture network but without preferential channels is conducive to extracting more heat from geothermal reservoirs.

Coupled Hydro-Mechanical Analysis of Gas Production in Fractured Shale Reservoir by Random Fracture Network Modeling

Accurate construction of a shale-reservoir fracture network is a prerequisite for optimizing the fracturing methods and determining shale-gas extraction schemes. Considering the influence of geological conditions, stress levels, desorption–adsorption, and fissure characteristics and distribution, establishing a shale-gas reservoir fracture-network model based on a random fracture network is significant. According to the discrete network model and Monte Carlo stochastic theory, the random fracture network of natural and artificial fractures in a shale-gas reservoir stimulation zone was constructed in this study. The porosity and permeability of the stimulation zone were calculated according to the network geometric properties. The fracture network was reconstructed, and the fissure connectivity was characterized. Numerical simulation of the seepage flow was performed for shale-gas reservoirs with different fracking-fracture combinations. The results showed that the local permeability dominated by the main fracture was the main factor that affected the initial shale-gas production efficiency. The total shale-gas productivity was mainly controlled by the effective stimulated volume. The evenly distributed secondary fracture network could effectively improve the effective stimulated volume of the stimulation zone. A 4% increase in the effective stimulated volume could improve the accumulated gas production by approximately 12%. Moreover, when the ratio of the main fracture to the secondary fracture was approximately 1:14, the accumulated gas production was optimized.

Influence of Grain Size Heterogeneity and In-Situ Stress on the Hydraulic Fracturing Process by PFC2D Modeling

A modified fluid-mechanically coupled algorithm in PFC2D was adopted in this article to study the influence of grain size heterogeneity and in-situ stress on hydraulic fracturing behavior. Simulated results showed that the in-situ stress and grain size heterogeneity significantly affect the initiation, growth, and spatial distribution of the hydraulic fractures: (1) the initiation and breakdown pressure are gradually reduced with the increase of the grain size heterogeneity; (2) with increased in-situ stress, the initiation and breakdown pressure increase, and the reduction effect of grain size heterogeneity on the breakdown pressure becomes more obvious; (3) in grain size homogeneous rock, the initiation pressure decreases with increasing in-situ stress ratio, however, the initiation pressure of grain size heterogeneous rock is almost unaffected by the in-situ stress ratio; (4) The in-situ stress ratio and grain size heterogeneity affect the spatial distribution of hydraulic fractures simultaneously. When the in-situ stress ratio is larger than 1, the hydraulic fractures propagate substantially along the direction of the maximum principal stress. When the in-situ stress ratio is 1, the initiation position and extension direction of hydraulic fractures are random and complex fracture networks can easily develop in a grain size homogeneous model.

Connectivity, formation factor and permeability of 2D fracture network

The purpose of this paper is to investigate the effects of fracture connectivity and length distributions on the electrical formation factor, F, of random fracture network using percolation theory. We assumed that the matrix was homogeneous and low-permeable, but the connectivity and length distributions of fracture system were randomly variable. F of fracture network is analyzed via finite element method. The main result is that: different from the classical percolation “universal” power law for porous-type rocks, F of fracture network obeys a normalized “universal” scaling relation using the length-scale 〈l〉/L (〈l〉 is fracture mean length, and L is the domain size). Our proposed formation factor model, derived from the normalized “universal” scaling relationship, is valid in fracture network with constant fracture length and length distributions, showing that the normalized “universal” scaling law is independent of fracture patterns. The normalized scaling relation is also successfully used to derive the permeability model of 2D random fracture network using the previously published dataset, which obtained better fitting results than before.

Anomalous transport in disordered fracture networks: Spatial Markov model for dispersion with variable injection modes

We investigate tracer transport on random discrete fracture networks that are characterized by the statistics of the fracture geometry and hydraulic conductivity. While it is well known that tracer transport through fractured media can be anomalous and particle injection modes can have major impact on dispersion, the incorporation of injection modes into effective transport modeling has remained an open issue. The fundamental reason behind this challenge is that—even if the Eulerian fluid velocity is steady—the Lagrangian velocity distribution experienced by tracer particles evolves with time from its initial distribution, which is dictated by the injection mode, to a stationary velocity distribution. We quantify this evolution by a Markov model for particle velocities that are equidistantly sampled along trajectories. This stochastic approach allows for the systematic incorporation of the initial velocity distribution and quantifies the interplay between velocity distribution and spatial and temporal correlation. The proposed spatial Markov model is characterized by the initial velocity distribution, which is determined by the particle injection mode, the stationary Lagrangian velocity distribution, which is derived from the Eulerian velocity distribution, and the spatial velocity correlation length, which is related to the characteristic fracture length. This effective model leads to a time-domain random walk for the evolution of particle positions and velocities, whose joint distribution follows a Boltzmann equation. Finally, we demonstrate that the proposed model can successfully predict anomalous transport through discrete fracture networks with different levels of heterogeneity and arbitrary tracer injection modes.