Discrete Fracture Approach Research Articles

A combined ALE-cohesive fracture approach for the arbitrary crack growth analysis

This work introduces an advanced numerical model, based on a cohesive zone approach and the moving mesh technique, for simulating fracture propagation in quasi-brittle materials. The proposed procedure involves two stages: first, a mesh boundary representing the crack is selected and aligned with the crack growth direction by using the Arbitrary Lagrangian-Eulerian (ALE) methodology; next, a zero-thickness interface cohesive element, equipped with a traction-separation law, is adaptively inserted along the previously selected mesh boundary, in order to describe the nonlinear fracture process. The proposed model allows for multiple crack onset and propagation without requiring mesh-updated procedures and sensibly reduces the well-known mesh dependency issues of the standard discrete fracture approaches. Numerical analyses of mixed-mode fracture in concrete specimens are performed and suitable comparisons with experiments show the effectiveness and reliability of the proposed model in predicting arbitrary crack growth.

Modeling gas breakthrough and flow phenomena through engineered barrier systems using a discrete fracture approach

Compacted clays are being considered to build engineered barrier systems (EBS) intended for the safe isolation of high-level nuclear waste (HLW) and spent nuclear fuel (SNF). The corrosion of the metallic canister containing the HLW/SNF will lead to the generation and buildup of the gas pressure in the more internal part of the clay buffer. This phenomenon would eventually trigger the formation and propagation of fractures in the clay barrier, jeopardizing its safety functions. In this work we propose to use the fragmentation technique (MFT) to model evolving fractures in clays triggered by gas pressurization. The MFT has been successfully used to model the formation of fractures in concrete, drying cracks in soil, hydraulic and thermo-fractures in rocks. In this work, we extend the MFT to deal with multiphase fluid flow in deformable porous media, and we upgraded a fully coupled computer finite element code using the extended technique. The proposed approach is first verified against analytical solutions and is then applied to model gas breakthrough experiments in clays. A very satisfactory performance of the method is observed in all the analyses, showing the potential of the MFT to tackle multiphase flow problems in deformable porous media with evolving discontinuities.

3D hybrid coupled dual continuum and discrete fracture model for simulation of CO2 injection into stimulated coal reservoirs with parallel implementation

This work presents the development of a 3D hybrid coupled dual continuum and discrete fracture model for simulating coupled flow, reaction and deformation processes relevant to fractured reservoirs with multiscale fracture system, e.g., coal and shale, efficiently and accurately. In this hybrid model, the natural fracture network and coal matrix are described together by a dual continuum approach and the large fractures are represented explicitly by the discrete fracture approach. A combination of different types of elements is used for spatial discretization. Large fractures are discretised with lower-dimensional interface elements and continuum domains with higher-dimensional elements. The coupling between the two models is achieved via the principle of superposition. To reduce computational time of simulations for complex and large-scale problems, a hybrid MPI/OpenMP parallel scheme is implemented in this work. The developed model is applied to investigate coupled thermal, hydraulic, and mechanical processes associated with CO2 sequestration and enhanced coalbed methane recovery. The results demonstrate capabilities of the model to adequately capture the effects of multiscale fracture system and their coupled behaviour during CO2 injection and methane recovery from coal reservoirs. Performance of the proposed parallelisation scheme was tested by comparing computation times of serial and parallel implementations. A good performance improvement was achieved, the speedup using parallelized scheme reaches up to about 10 times along with satisfactory scalability for considered application example. The findings of this work support developments and improvements of efficient advanced numerical models to study coupled THCM behaviour in fractured porous geomaterials.

Cracking behavior analysis of reinforced concrete structures by using a cohesive fracture model

The simulation of cracking behavior in reinforced concrete structures is one of the most challenging tasks of engineering practice due to the highly complex phenomena which arise in the structural element under both ultimate and service loads such as the interaction between concrete and steel reinforcement, crack coalescence and branching, and multiple crack propagation. In the literature, the most known and well-established constitutive models for concrete are based on a continuum representation for this material, in which both kinematic and static variables (i.e. strains and stresses) possess regular spatial distributions. However, due to the discrete nature of the cracks, this hypothesis should be removed for a concrete cracking analysis to be correct. In this context, the present work aims to investigate the cracking behavior, in terms of crack width and crack spacing, in reinforced concrete structures by means of a discrete fracture approach implemented in a finite element framework. In particular, such a model relies on an inter-element cohesive model to simulate crack onset and propagation used in combination with an embedded truss model to reproduce the mechanical behavior of the steel rebars as well as their interaction with the surrounding concrete. The adopted integrated model has been employed to perform numerical simulations for predicting the load-carrying capacity and the related crack pattern of real-scale reinforced concrete structural elements. Comparisons with experimental outcomes, in terms of loading curve, crack width and crack spacing, have demonstrated the capabilities and effectiveness of the proposed model to investigate the cracking behavior in reinforced concrete structures providing more accurate crack patterns, than existing models.

A new discrete fracture approach based on the use of coupling finite elements for modeling fluid transport in naturally fractured porous media

The presence of natural fractures in a porous medium strongly influences the fluid flow path. The complex geometric characteristics of the fracture network create preferential channels that change the permeability of the medium. In this context, this paper presents a novel discrete fracture model to address the hydrodynamic behavior of naturally fractured porous media by using a coupling scheme in which the initially independent standard finite element meshes of porous rock and fractures are coupled via coupling finite elements to enforce the continuity of the pressure field between the meshes through a penalty parameter. The main feature of the technique is the capacity to treat problems with non-matching, overlapping and interfacial meshes with no need to add extra degrees of freedom to describe the compatibility between the meshes. In addition, the proposed coupling finite elements also connect non-conforming meshes of fractures, so that their interaction emerges naturally from the coupling scheme. The new theoretical framework and finite element equations are described in detail and 2D and 3D numerical experiments are conducted to validate and verify the technique against results reported in the literature. The obtained results show that the proposed coupling scheme is efficient and able to capture the main phenomena associated with the fluid transport in fractured porous media.

Investigation of concrete cracking phenomena by using cohesive fracture-based techniques: A comparison between an embedded crack model and a refined diffuse interface model

Cracking simulation in concrete-like structures is the subject of numerous investigations of computational kind, in which different modeling techniques based on either smeared or discrete fracture approaches have been proposed. In this work, two different finite element-based cohesive fracture models, proposed by some of the authors and belonging to the discrete approaches, have been compared: (i) a novel refined diffuse interface model and (ii) a well-established embedded crack model, which are based on an inter- and intra- element fracture approach, respectively. In particular, the first one is based on an intrinsic cohesive zone formulation, according to which nonlinear interface elements, equipped with a softening constitutive law, are inserted among all the finite elements of the computational domain. The second one, which has been successfully used to simulate concrete cracking along nonprescribed paths and is here used a reference model, combines the cohesive crack concept with the embedded strong discontinuity approach, allowing the cracks to be inserted across finite elements as discontinuities in the corresponding displacement field once a certain stress criterion is satisfied. These two models have been compared by analyzing the cracking behavior in concrete specimens subjected to general loading conditions. A critical discussion of the obtained computational outcomes, in terms of computational efficiency and numerical accuracy for both predicted loading curve and crack pattern, shows the strengths and limits of the investigated discrete fracture models.

Finite element analysis of concrete cracking: a comparative study between a diffuse interface model and an embedded crack model

The accurate analysis of cracking phenomena in the concrete structures has been a research topic of growing interest over the past few decades, with notable developments in the modeling techniques based on either smeared or discrete fracture approaches. The well-known cohesive zone models, belonging to discrete fracture approaches, are commonly judged as very effective for accurately representing the real crack pattern in quasi-brittle materials. The present work deals with comparing two different finite element-based cohesive fracture models: (i) a novel diffuse interface model, and (ii) an existing embedded crack model, based on an inter- and intra- element fracture approach, respectively. The first one relies on an intrinsic cohesive formulation by which the damage process inside the material is represented as a collection of imperfect interfaces randomly placed at the internal boundaries of a finite element mesh. The second one is based on a strong discontinuity approach according to which the crack is modeled as a discontinuity embedded into the displacement field of a standard continuum, allowing concrete cracking along nonprescribed paths to be correctly simulated. Cracking behavior in concrete specimens subjected to general loading conditions has been simulated by these two models and a detailed comparison between the numerical results is provided. Finally, a critical discussion regarding of computational efficiency and numerical accuracy highlights the efficacy of the newly proposed diffuse interface model.

A Discrete Fracture Modeling Approach for Analysis of Coalbed Methane and Water Flow in a Fractured Coal Reservoir

As a complex two-phase flow in naturally fractured coal formations, the prediction and analysis of CBM production remain challenging. This study presents a discrete fracture approach to modeling coalbed methane (CBM) and water flow in fractured coal reservoirs, particularly the influence of fracture orientation, fracture density, gravity, and fracture skeleton on fluid transport. The discrete fracture model is first verified by two water-flooding cases with multi- and single-fracture configurations. The verified model is then used to simulate CBM production from a discrete fractured reservoir using four different fracture patterns. The results indicate that fluid behavior is significantly affected by orientation, density, and fracture connectivity. Finally, several cases are performed to investigate the influence of gravity and fracture skeleton. The simulation results show that gas migrates upwards to the top reservoir during fluid extraction owing to buoyancy and the connected fracture skeleton plays a dominant role in fluid transport and methane production efficiency. Overall, the developed discrete fracture model provides a powerful tool to study two-phase flow in fractured coal reservoirs.

Pressure performance of multi-stage fractured horizontal well with stimulated reservoir volume and irregular fractures distribution in shale gas reservoirs

With the development of hydraulic fracturing technology of horizontal well, most unconventional gas reservoirs are developed by horizontal well multi-stage fracturing technology to improve gas well production. In this paper, we present a semi-analytical model of multi-stage fractured horizontal well (MFHWs) with consideration of adsorbed gas desorption and the irregular hydraulic fractures distribution in shale gas with the stimulated reservoir volume (SRV). On the bases of the point source function, Langmuir isotherm adsorption and Fick's law, wellbore pseudo-pressure solution of finite-conductivity MFHWs model is obtained by employing Laplace transform, finite cosine Fourier transform. and the discrete fracture approach in shale gas. The pressure transient responses log-log curves of MFHWs are drawn by Gauss elimination method and Stehfest numerical algorithm. In the light of the characteristics of the pressure derivative curve, the characteristics of each flow stage are identified and analyzed in detail. With the comparison of the commercial well test software numerical solution and the previous literature, the comparison results reach a good agreement under condition of same parameter. The effects of vital parameters, such as hydraulic fractures number and size of the SRV region, hydraulic fractures distribution styles and crossflow coefficient, on pressure transient curve are analyzed in detail. The presented model in this paper not only provides some insights for pressure dynamic analysis of shale gas reservoir development, but can help to engineers understand the pressure dynamics in shale gas reservoirs with SRV and irregular fracture distribution.

Two-Phase Compositional Flow Simulation in Complex Fractured Media by 3D Unstructured Gridding with Horizontal and Deviated Wells

Summary A higher-order numerical model for compositional two-phase flow in fractured media is presented in this paper. The simulation of horizontal and deviated wells is incorporated in the formation using unstructured grids. All commonly used types of finite elements are accounted for in the algorithm: quadrangular and triangular elements in 2D, and hexahedra, prisms and tetrahedra elements in 3D. The fracture crossflow equilibrium (FCFE) approach is applied to model the flow exchange between the fractures and the matrix. FCFE is combined with the hybridized form of the mass conservative mixed finite element (MHFE) and the higher-order discontinuous Galerkin (DG) method. A computer-aided design (CAD) interface is developed that connects the mesh generator to the CAD software. The interface allows to design, mesh, and incorporate horizontal and deviated wells into the higher-order simulator. The algorithm allows flow simulation in fractures in all ranges of permeability values as opposed to the embedded discrete fracture matrix (EDFM) approach where low permeable fractures affect the accuracy of the results. The efficiency, accuracy, and strengths of the model are demonstrated in comparison to alternatives including the embedded discrete fracture approach in different examples. Detailed incorporation of complex wells is presented in this work.

Comparison of solute/heat transport in fractured formations using discrete fracture and equivalent porous media modeling at the reservoir scale

In deep reservoirs, the formations consist of naturally or artificially fractured rocks. The fluid flow coupled with heat transfer through fractures is commonly modeled by discrete fracture approach. In this study, fluid flow and solute/heat transport phenomena were investigated using the equivalent porous media approach and compared to discrete fracture modeling. In the equivalent porous media approach, the fractures were replaced with a porous medium. Its effective porosity and permeability were adjusted to get the simulation results similar to discrete fracture modeling results. In addition, the solid deformation, in equivalent porous media approach, was coupled with heat transfer, while it was de-coupled with fluid flow to increase the simulations’ efficiency. The equivalent porous media approach was applied to two benchmark problems; geothermal doublet, and thermohaline aquifer-aquitard-aquifer model. The comparisons between the equivalent porous media and discrete fracture modeling approaches were in a good agreement and implied the feasibility and efficiency of the equivalent porous media relative to the fully coupled discrete fracture modeling approach.

Powell–Sabin B-splines for smeared and discrete approaches to fracture in quasi-brittle materials

Non-Uniform Rational B-splines (NURBS) and T-splines can have some drawbacks when modelling damage and fracture. The use of Powell–Sabin B-splines, which are based on triangles, can by-pass these drawbacks. Herein, smeared as well as discrete approaches to fracture in quasi-brittle materials using Powell–Sabin B-splines are considered.For the smeared formulation, an implicit fourth-order gradient damage model is adopted. Since quadratic Powell–Sabin B-splines employ C1-continuous basis functions throughout the domain, they are well-suited for solving the fourth order partial differential equation that emerges in this higher order damage model. Moreover, they can be generated from an arbitrary triangulation without user intervention. Since Powell–Sabin B-splines are generated from a classical triangulation, they are not necessarily boundary-fitting and in that case they are not isogeometric in the strict sense.For discrete fracture approaches, the degree of continuity of T-splines is reduced to C0 at the crack tip. Hence, stresses need to be evaluated and weighted at the integration points in the vicinity of the crack tip in order to decide when the critical stress is reached. In practice, stress fields are highly irregular around crack tips. Furthermore, aligning a T-spline mesh with the new crack segment can be difficult. Powell–Sabin B-splines also remedy these drawbacks as they are C1-continuous at the crack tip and stresses can be directly computed, which vastly increases the accuracy and simplifies the implementation. Moreover, re-meshing is more straightforward using Powell–Sabin B-splines. A current limitation is that, in three dimensions, there is no procedure (yet) for constructing Powell–Sabin B-splines on arbitrary tetrahedral meshes.

Evaluation of recoverable energy potential from enhanced geothermal systems: a sensitivity analysis in a poro‐thermo‐elastic framework

AbstractThis study presents application of an efficient approach to simulate fluid flow and heat transfer in naturally fractured geothermal reservoirs. Fluid flow is simulated by combining single continuum and discrete fracture approaches. The local thermal nonequilibrium approach is used to simulate heat transfer by conduction in the rock matrix and convection (including conduction) in the fluid. Fluid flow and heat transfer models are integrated within a coupled poro‐thermo‐elastic framework. The developed model is used to evaluate the long‐term response of a geothermal reservoir with specific boundary conditions and injection/production schedule. A comparative study and a sensitivity analysis are carried out to evaluate the capability of the integrated approach and understand the degree by which different reservoir parameters affect thermal depletion of Soultz geothermal reservoir, respectively. Also observed, there exists an optimum fracture permeability after which the reservoir stimulation does not change the recovery factor significantly. Estimation of fluid temperature by the assumption of local thermal nonequilibrium heat transfer between the fracture fluid and the rock matrix gives fluid temperature of about 3°C less than that of estimated by thermal equilibrium heat transfer at early stage of hot water production.

Evaluation of water coning phenomenon in naturally fractured oil reservoirs

Water coning is a complex phenomenon observed in conventional and unconventional reservoirs. This phenomenon takes place due to the imbalance between viscous and gravitational forces during simultaneous production of oil and water. In a fractured reservoir, controlling of water coning is challenging due to the complexity originates from large number of uncertain variables associated with such reservoirs system. This paper presents a fully coupled poroelastic multiphase fluid-flow numerical model to provide a new insight and understanding of water coning phenomenon in naturally fractured reservoir under effect of various rock and fluid properties. These properties include anisotropy ratio, fracture permeability, mobility ratio, and production rate. The simulation workflow of the developed numerical model is based on upstream flux weighted finite element discretization method and a new hybrid methodology, which combines single-continuum and discrete fracture approach. Moreover, the capillary pressure effect is included during the discretization of the partial differential equations of multiphase fluid flow. The numerical system is decoupled using implicit pressure and explicit saturation (IMPES) approach. Discretization of water saturation equation using standard finite element method produces solution with spatial oscillations due to its hyperbolic nature. To overcome this, Galerkin’s least square technique (GLS) is employed to stabilize the equation solutions. The developed numerical scheme is validated successfully against Eclipse-100 and then applied to a case study of fractured reservoir taken from Southern Vietnam. The results showed that the break through time is very sensitive to the distributions of fracture network, anisotropy ratio between fracture horizontal, vertical permeability, and mobility ratio. Furthermore, it has been concluded that aquifer strength has a little effect on coning behavior during oil production process.

An efficient numerical model for multicomponent compressible flow in fractured porous media

An efficient and accurate numerical model for multicomponent compressible single-phase flow in fractured media is presented. The discrete-fracture approach is used to model the fractures where the fracture entities are described explicitly in the computational domain. We use the concept of cross flow equilibrium in the fractures. This will allow large matrix elements in the neighborhood of the fractures and considerable speed up of the algorithm. We use an implicit finite volume (FV) scheme to solve the species mass balance equation in the fractures. This step avoids the use of Courant–Freidricks–Levy (CFL) condition and contributes to significant speed up of the code. The hybrid mixed finite element method (MFE) is used to solve for the velocity in both the matrix and the fractures coupled with the discontinuous Galerkin (DG) method to solve the species transport equations in the matrix. Four numerical examples are presented to demonstrate the robustness and efficiency of the proposed model. We show that the combination of the fracture cross-flow equilibrium and the implicit composition calculation in the fractures increase the computational speed 20–130 times in 2D. In 3D, one may expect even a higher computational efficiency.

Saltwater intrusion estimation in a karstified coastal system using density-dependent modelling and comparison with the sharp-interface approach

Saltwater intrusion is a naturally occurring phenomenon that is exacerbated significantly by excessive groundwater exploitation in coastal aquifers. In order to determine the extent of saltwater intrusion in a karstified coastal aquifer in Crete, Greece, a three-dimensional, density-dependent groundwater flow and transport model was developed and compared to the more traditional sharp-interface approach. The karstified medium was modelled using a combination of the equivalent porous medium approach (for lower-order fractures) and a discrete fracture approach (for the main fractures/faults). The model takes into consideration the geomorphologic characteristics of the karstic system, such as the depth and orientation of the fault network, and the diffusion phenomena associated with the variable densities of freshwater and saltwater—parameters that create a complex system, inducing uncertainty in the model. The model results showed that the orientation of the fractures, the pumping activity and the fluid density effects drive the seawater intrusion front asymmetrically inland. Editor Z.W. Kundzewicz Citation Dokou, Z. and Karatzas, G.P., 2012. Saltwater intrusion estimation in a karstified coastal system using density-dependent modelling and comparison with the sharp-interface approach. Hydrological Sciences Journal, 57 (5), 985–999.

A risk assessment tool for contaminated sites in low-permeability fractured media

A risk assessment tool for contaminated sites in low-permeability fractured media is developed, based on simple transient and steady-state analytical solutions. The discrete fracture (DF) tool, which explicitly accounts for the transport along fractures, covers different source geometries and history (including secondary sources) and can be applied to a wide range of compounds. The tool successfully simulates published data from short duration column and field experiments. The use for risk assessment is illustrated by three typical risk assessment case studies, involving pesticides, chlorinated solvents, benzene and MTBE. The model is compared with field data and with results from a simpler approach based on an Equivalent Porous Media (EPM). Risk assessment conclusions of the DF and EPM approaches are very different due to the early breakthrough, long term tailing, and lower attenuation due to degradation associated with fractured media. While the DF tool simulates the field data, it is difficult to conclude that the DF model is superior to an EPM model because of a lack of long term monitoring data. However, better agreement with existing field data by the DF model using observed physical fracture parameters favors the use of this model over the EPM model for risk assessments.

Physical and numerical modelling of a dual-porosity fractured rock surrounding an in-pit uranium tailings management facility

In-pit tailings management facilities (TMFs) have been used to dispose of uranium mine tailings in northern Saskatchewan. The pervious surround method can be employed to reduce groundwater flow through the tailings, and thereby moderate the rate of flux of radioactive and metal contamination that could enter the local groundwater via advection. A laboratory-scale analogue of an in-pit TMF was constructed to evaluate the impacts of a dual-porosity representation of the host rock on the effectiveness of the pervious surround concept in reducing peak concentrations and mass flux at a downgradient receptor. This work complemented the work of West et al. (2003) in which similar laboratory-scale experiments were conducted, simulating the host rock using the equivalent porous media and discrete fracture approaches. The experiments were adequately simulated using SWIFT II, a dual-porosity flow and transport model. Finally, to illustrate the impact of a dual-porosity representation of a fractured host rock on a contaminant plume from a TMF in the field, a field-scale scenario was modelled. The simulations illustrate the impact of diffusion into the host rock matrix on the simulated peak concentrations and the time for the peak concentrations to reach a downgradient receptor.

Terrain Discontinuity Effects in the Regional Flow of a Complex Karstified Aquifer

Karst aquifers are characterized by spatial heterogeneity due to the presence of highly permeable channels and conduits in low-permeable fractured rocks (matrix block). Recent studies have reported a close relationship between surface and subsurface water in karstic regions due to the water flow through a complicated network of paths formed by fracture intersections. Subsurface flow in karstified aquifers ranges between conduit flow, in large passages with relatively high flow velocities, and diffuse flow, in the matrix block where Darcy’s law is still valid. In this paper, we present the simulation of a complex karstified aquifer system in Crete, Greece, where the presence of main faults drastically affects the regional flow. A discrete fracture approach in conjunction with an equivalent porous medium approach was adopted to simulate the mixed flow in the area of interest. The simulation results have shown that the length and the orientation of the dominant faults, primarily during the rainy season, affect the flow field.

Site Characterization in Densely Fractured Dolomite: Comparison of Methods

One of the challenges in characterizing fractured-rock aquifers is determining whether the equivalent porous medium approximation is valid at the problem scale. Detailed hydrogeologic characterization completed at a small study site in a densely fractured dolomite has yielded an extensive data set that was used to evaluate the utility of the continuum and discrete-fracture approaches to aquifer characterization. There are two near-vertical sets of fractures at the site; near-horizontal bedding-plane partings constitute a third fracture set. Eighteen boreholes, including five coreholes, were drilled to a depth of approximately 10.6 m. Borehole geophysical logs revealed several laterally extensive horizontal fractures and dissolution zones. Flowmeter and short-interval packer testing identified which of these features were hydraulically important. A monitoring system, consisting of short-interval piezometers and multilevel samplers, was designed to monitor four horizontal fractures and two dissolution zones. The resulting network consisted of >70 sampling points and allowed detailed monitoring of head distributions in three dimensions. Comparison of distributions of hydraulic head and hydraulic conductivity determined by these two approaches suggests that even in a densely fractured-carbonate aquifer, a characterization approach using traditional long-interval monitoring wells is inadequate to characterize ground water movement for the purposes of regulatory monitoring or site remediation. In addition, traditional multiwell pumping tests yield an average or bulk hydraulic conductivity that is not adequate for predicting rapid ground water travel times through the fracture network, and the pumping test response does not appear to be an adequate tool for assessing whether the porous medium approximation is valid.