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Journal articles on the topic 'Phase Field Fracture'

Author: Grafiati

Published: 30 June 2021

Last updated: 20 February 2023

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1

Zhao, Jinzhou, Qing Yin, John McLennan, Yongming Li, Yu Peng, Xiyu Chen, Cheng Chang, Weiyang Xie, and Zhongyi Zhu. "Iteratively Coupled Flow and Geomechanics in Fractured Poroelastic Reservoirs: A Phase Field Fracture Model." Geofluids 2021 (December 20, 2021): 1–13. http://dx.doi.org/10.1155/2021/6235441.

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Fluid-solid coupling in fractured reservoirs plays a critical role for optimizing and managing in energy and geophysical engineering. Computational difficulties associated with sharp fracture models motivate phase field fracture modeling. However, for geomechanical problems, the fully coupled hydromechanical modeling with the phase field framework is still under development. In this work, we propose a fluid-solid fully coupled model, in which discrete fractures are regularized by the phase field. Specifically, this model takes into account the complex coupled interaction of Darcy-Biot-type fluid flow in poroelastic media, Reynolds lubrication governing flow inside fractures, mass exchange between fractures and matrix, and the subsequent geomechanical response of the solid. An iterative coupling method is developed to solve this multifield problem efficiently. We present numerical studies that demonstrate the performance of our model.

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2

Ni, Lin, Xue Zhang, Liangchao Zou, and Jinsong Huang. "Phase-field modeling of hydraulic fracture network propagation in poroelastic rocks." Computational Geosciences 24, no. 5 (April 19, 2020): 1767–82. http://dx.doi.org/10.1007/s10596-020-09955-4.

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Abstract Modeling of hydraulic fracturing processes is of great importance in computational geosciences. In this paper, a phase-field model is developed and applied for investigating the hydraulic fracturing propagation in saturated poroelastic rocks with pre-existing fractures. The phase-field model replaces discrete, discontinuous fractures by continuous diffused damage field, and thus is capable of simulating complex cracking phenomena such as crack branching and coalescence. Specifically, hydraulic fracturing propagation in a rock sample of a single pre-existing natural fracture or natural fracture networks is simulated using the proposed model. It is shown that distance between fractures plays a significant role in the determination of propagation direction of hydraulic fracture. While the rock permeability has a limited influence on the final crack topology induced by hydraulic fracturing, it considerably impacts the distribution of the fluid pressure in rocks. The propagation of hydraulic fractures driven by the injected fluid increases the connectivity of the natural fracture networks, which consequently enhances the effective permeability of the rocks.

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3

Berry, M. D., D. W. Stearns, and M. Friedman. "THE DEVELOPMENT OF A FRACTURED RESERVOIR MODEL FOR THE PALM VALLEY GAS FIELD." APPEA Journal 36, no. 1 (1996): 82. http://dx.doi.org/10.1071/aj95005.

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A fractured reservoir model has been developed for the Palm Valley gas field, located WSW of Alice Springs, in the Amadeus Basin, NT. Definition of this complex, naturally fractured, Ordovician gas reservoir has required an integrated approach involving multiple studies to develop the geological model that has formed the basis for reservoir simulation and the rationale for the location of new wells. In addition, new seismic data provided fundamental input to the structure/fracture model of the field. Results suggest a primary, northsouth compression for the origin of structures in the basin and that Palm Valley resulted from a single, balanced folding phase. The seismic data show that Palm Valley is not an arcuate anticline as previously mapped, but is an elongate WSW to ENE trending, doubly plunging anticline with an offset crest and minor reverse faults at reservoir level. Investigations have shown that the majority of fractures in the reservoir outcrop are extremely ordered and there is definite structural control of the fracture spacing in the brittle reservoir rocks. Fracture trajectory and fracture intensity maps have been constructed, the latter providing the mechanism for distribution of fracture parameters around the field. The orientation of fractures measured at depth in the reservoir match exactly the fractures predicted from the structure/fracture model. This is the first time a fractured reservoir model that has been developed for Palm Valley, and it will form the basis for the further study and future development of the field.

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4

Tsoflias, Georgios P., Jean‐Paul Van Gestel, Paul L. Stoffa, Donald D. Blankenship, and Mrinal Sen. "Vertical fracture detection by exploiting the polarization properties of ground‐penetrating radar signals." GEOPHYSICS 69, no. 3 (May 2004): 803–10. http://dx.doi.org/10.1190/1.1759466.

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Vertically oriented thin fractures are not always detected by conventional single‐polarization reflection profiling ground‐penetrating radar (GPR) techniques. We study the polarization properties of EM wavefields and suggest multipolarization acquisition surveying to detect the location and azimuth of vertically oriented fractures. We employ analytical solutions, 3D finite‐difference time‐domain modeling, and field measurements of multipolarization GPR data to investigate EM wave transmission through fractured geologic formations. For surface‐based multipolarization GPR measurements across vertical fractures, we observe a phase lead when the incident electric‐field component is oriented perpendicular to the plane of the fracture. This observation is consistent for nonmagnetic geologic environments and allows the determination of vertical fracture location and azimuth based on the presence of a phase difference and a phase lead relationship between varying polarization GPR data.

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5

Choo, Jinhyun, and Fan Fei. "Phase-field modeling of geologic fracture incorporating pressure-dependence and frictional contact." E3S Web of Conferences 205 (2020): 03004. http://dx.doi.org/10.1051/e3sconf/202020503004.

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Geologic fractures such as joints and faults are central to many problems in energy geotechnics. Notable examples include hydraulic fracturing, injection-induced earthquakes, and geologic carbon storage. Nevertheless, our current capabilities for simulating the development and evolution of geologic fractures in these problems are still insufficient in terms of efficiency and accuracy. Recently, phase-field modeling has emerged as an efficient numerical method for fracture simulation which does not require any algorithm for tracking the geometry of fracture. However, existing phase-field models of fracture neglected two distinct characteristics of geologic fractures, namely, the pressure-dependence and frictional contact. To overcome these limitations, new phase-field models have been developed and described in this paper. The new phase-field models are demonstrably capable of simulating pressure-dependent, frictional fractures propagating in arbitrary directions, which is a notoriously challenging task.

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6

Santillan Sanchez, David, Hichem Mazighi, and Mustapha Kamel Mihoubi. "Hybrid phase-field modeling of multi-level concrete gravity dam notched cracks." Frattura ed Integrità Strutturale 16, no. 61 (June 19, 2022): 154–75. http://dx.doi.org/10.3221/igf-esis.61.11.

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Phase-field models have become a powerful tool to simulate crack propagation. They regularize the fracture discontinuity and smooth the transition between the intact and the damaged regions. Based on the thermodynamic function and a diffusive field, they regularize the variational approach to fracture that generalizes Grifﬁth’s theory for brittle fracture. Phase-field models are capable to simulate complex fracture patterns efficiently and straightforwardly. In this paper, we introduce a hybrid phase-ﬁeld approach to simulate the crack propagation in laboratory-scale and life-scale structures. First, we apply our methodology to the three-point bending test on notched laboratory beams. Second, we simulate the fracture propagation in a life-size structure: the Koyna gravity dam. We account for the pressure load inside the fracture, and we study the effect of the position and number of initial fractures in the upstream face and the value of the Grifﬁth critical energy release, on the fracture propagation under a flood event. The position of the fracture plays an important role in the final fracture pattern and crest displacements, whereas the value of the Grifﬁth critical energy release alters the onset of the fracture propagation. We conclude that phase-ﬁeld models are a promising computational tool that may be applied to real engineering problems.

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7

Bourne, Stephen J., Lex Rijkels, Ben J. Stephenson, and Emanuel J. M. Willemse. "Predictive Modelling of Naturally Fractured Reservoirs Using Geomechanics and Flow Simulation." GeoArabia 6, no. 1 (January 1, 2001): 27–42. http://dx.doi.org/10.2113/geoarabia060127.

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ABSTRACT To optimise recovery in naturally fractured reservoirs, the field-scale distribution of fracture properties must be understood and quantified. We present a method to systematically predict the spatial distribution of natural fractures related to faulting and their effect on flow simulations. This approach yields field-scale models for the geometry and permeability of connected fracture networks. These are calibrated by geological, well test and field production data to constrain the distributions of fractures within the inter-well space. First, we calculate the stress distribution at the time of fracturing using the present-day structural reservoir geometry. This calculation is based on a geomechanical model of rock deformation that represents faults as frictionless surfaces within an isotropic homogeneous linear elastic medium. Second, the calculated stress field is used to govern the simulated growth of fracture networks. Finally, the fractures are upscaled dynamically by simulating flow through the discrete fracture network per grid block, enabling field-scale multi-phase reservoir simulation. Uncertainties associated with these predictions are considerably reduced as the model is constrained and validated by seismic, borehole, well test and production data. This approach is able to predict physically and geologically realistic fracture networks. Its successful application to outcrops and reservoirs demonstrates that there is a high degree of predictability in the properties of natural fracture networks. In cases of limited data, field-wide heterogeneity in fracture permeability can be modelled without the need for field-wide well coverage.

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8

Wang, Huimin, J. G. Wang, Feng Gao, and Xiaolin Wang. "A Two-Phase Flowback Model for Multiscale Diffusion and Flow in Fractured Shale Gas Reservoirs." Geofluids 2018 (2018): 1–15. http://dx.doi.org/10.1155/2018/5910437.

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A shale gas reservoir is usually hydraulically fractured to enhance its gas production. When the injection of water-based fracturing fluid is stopped, a two-phase flowback is observed at the wellbore of the shale gas reservoir. So far, how this water production affects the long-term gas recovery of this fractured shale gas reservoir has not been clear. In this paper, a two-phase flowback model is developed with multiscale diffusion mechanisms. First, a fractured gas reservoir is divided into three zones: naturally fractured zone or matrix (zone 1), stimulated reservoir volume (SRV) or fractured zone (zone 2), and hydraulic fractures (zone 3). Second, a dual-porosity model is applied to zones 1 and 2, and the macroscale two-phase flow flowback is formulated in the fracture network in zones 2 and 3. Third, the gas exchange between fractures (fracture network) and matrix in zones 1 and 2 is described by a diffusion process. The interactions between microscale gas diffusion in matrix and macroscale flow in fracture network are incorporated in zones 1 and 2. This model is validated by two sets of field data. Finally, parametric study is conducted to explore key parameters which affect the short-term and long-term gas productions. It is found that the two-phase flowback and the flow consistency between matrix and fracture network have significant influences on cumulative gas production. The multiscale diffusion mechanisms in different zones should be carefully considered in the flowback model.

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9

Bharali, Ritukesh, Fredrik Larsson, and Ralf Jänicke. "Computational homogenisation of phase-field fracture." European Journal of Mechanics - A/Solids 88 (July 2021): 104247. http://dx.doi.org/10.1016/j.euromechsol.2021.104247.

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10

Chen, Lin, and René de Borst. "Phase-field modelling of cohesive fracture." European Journal of Mechanics - A/Solids 90 (November 2021): 104343. http://dx.doi.org/10.1016/j.euromechsol.2021.104343.

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11

Freddi, Francesco. "Fracture energy in phase field models." Mechanics Research Communications 96 (March 2019): 29–36. http://dx.doi.org/10.1016/j.mechrescom.2019.01.009.

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12

Wilson, Zachary A., and Chad M. Landis. "Phase-field modeling of hydraulic fracture." Journal of the Mechanics and Physics of Solids 96 (November 2016): 264–90. http://dx.doi.org/10.1016/j.jmps.2016.07.019.

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13

Ambati, M., T. Gerasimov, and L. De Lorenzis. "Phase-field modeling of ductile fracture." Computational Mechanics 55, no. 5 (April 10, 2015): 1017–40. http://dx.doi.org/10.1007/s00466-015-1151-4.

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14

Kuhn, C., and R. Müller. "A phase field model for fracture." PAMM 8, no. 1 (December 2008): 10223–24. http://dx.doi.org/10.1002/pamm.200810223.

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15

Kuhn, Charlotte, and Ralf Müller. "Phase field simulation of thermomechanical fracture." PAMM 9, no. 1 (December 2009): 191–92. http://dx.doi.org/10.1002/pamm.200910070.

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16

Mauthe, Steffen, and Christian Miehe. "Phase-Field Modeling of Hydraulic Fracture." PAMM 15, no. 1 (October 2015): 141–42. http://dx.doi.org/10.1002/pamm.201510061.

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17

Kueper, Bernard H., C. Stephan Haase, and Helen L. King. "Leakage of dense, nonaqueous phase liquids from waste impoundments constructed in fractured rock and clay: theory and case history." Canadian Geotechnical Journal 29, no. 2 (April 1, 1992): 234–44. http://dx.doi.org/10.1139/t92-027.

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This paper examines the behaviour of dense, nonaqueous phase liquids (DNAPLs) in fractured media, with an emphasis on waste-disposal ponds constructed in fractured clay and rock. Calculations are presented to estimate the height of DNAPL that may accumulate at the base of a disposal pond prior to initial entry into a water-saturated fracture. This height is found to be a function of the fluid densities, the DNAPL–water interfacial tension, the fracture aperture, and the position of the water table. A numerical model is applied to estimate the steady-state rate of DNAPL leakage from a disposal pond underlain by vertical fractures. This rate of leakage is found to be a function of the spacing of fractures, the fracture aperture, the DNAPL density, and the height of the water table in the formation outside of the impoundment. It is demonstrated that a wide range of leakage rates can occur over a relatively narrow range of parameters. A conceptual analysis of two-phase flow examines the conditions that lead to both uniform and sparse DNAPL migration pathways beneath a disposal pond in fracture networks. A case history is presented as field evidence of the concepts discussed. In particular, the case history demonstrates that sparse DNAPL migration pathways can occur in fractured rock, and that relatively low dissolved phase concentrations can exist in the immediate vicinity of DNAPL source zones. Key words : nonaqueous phase liquids, ground water, fractures, disposal ponds.

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18

Meadows, Mark A., and Don F. Winterstein. "Seismic detection of a hydraulic fracture from shear‐wave VSP data at Lost Hills Field, California." GEOPHYSICS 59, no. 1 (January 1994): 11–26. http://dx.doi.org/10.1190/1.1443523.

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A shear‐wave (S‐wave) VSP experiment was performed at Lost Hills Field, California, in an attempt to detect hydraulic fractures induced in a nearby well. The hydrofrac well was located between an impulsive, S‐wave source on the surface and a receiver well containing a clamped, three‐component geophone. Both direct and scattered waves were detected immediately after shut‐in, when the hydraulic pumps were shut off and recording started. The scattered energy disappeared within about an hour, which is consistent with other measurements that indicate some degree of fracture closure and leak‐off within that period. Although S‐wave splitting was evident, no change was detected in the fast wave (polarized parallel to the fracture). However, the slow wave (polarized perpendicular to the fracture) did change over a period of about an hour, after which the prehydrofrac wavelet shape was recovered. The fact that only the wave polarized perpendicular to the fracture was affected is a dramatic confirmation of both theoretical predictions and laboratory observations of S‐wave behavior in a fractured medium. Subtracting the prehydrofrac wavelet from the wavelets recorded within the first hour after shut‐in revealed scattered wavelets that were diminished and phase‐rotated versions of the incident (prehydrofrac) wavelet. Arrival times of the direct and scattered waves were matched by ray tracing. We accounted for the scattered‐wave amplitudes by using numerical solutions of S‐wave diffractions off of ribbon‐shaped fractures. Amplitudes derived from full‐wavefield Born scattering, however, did not match recorded amplitudes. The phase of the scattered wavelets was matched very well by Born scattering when the incident wavelet was input, but only for fracture lengths no larger than half those predicted from fracture‐simulator models. These results show that a carefully controlled experiment, combined with accurate modeling, can provide important information about the geometry of induced fractures.

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19

Zhang, Yan, Xiaobing Lu, Xuhui Zhang, and Peng Li. "Proppant Transportation in Cross Fractures: Some Findings and Suggestions for Field Engineering." Energies 13, no. 18 (September 19, 2020): 4912. http://dx.doi.org/10.3390/en13184912.

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The proppant transportation is a typical two-phase flow process in a complex cross fracture network during hydraulic fracturing. In this paper, the proppant transportation in cross fractures is investigated by the computational fluid dynamics (CFD) method. The Euler–Euler two-phase flow model and the kinetic theory of granular flow (KTGF) are adopted. The dimensionless controlling parameters are derived by dimensional analysis. The equilibrium proppant height (EPH) and the ratio of the proppant mass (RPM) in the secondary fracture to that in the whole cross fracture network are used to describe the movement and settlement of proppants in the cross fractures. The main features of the proppant transportation in the cross fractures are given, and several relative suggestions are presented for engineering application in the field. The main controlling dimensionless parameters for relative EPH are the proppant Reynolds number and the inlet proppant volume fraction. The dominating dimensionless parameters for RPM are the relative width of the primary and the secondary fracture. Transportation of the proppants with a certain particle size grading into the cross fractures may be a good way for supporting the hydraulic fractures.

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20

Lerner, D. N., G. P. Wealthall, and A. Steele. "Assessing Risk from DNAPLs in Fractured Aquifers." Journal of Agricultural and Marine Sciences [JAMS] 7, no. 2 (June 1, 2002): 47. http://dx.doi.org/10.24200/jams.vol7iss2pp47-52.

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Chlorinated solvents are among the most widespread pollutants of groundwater. As DNAPLs (dense nonaqueous phase liquids), they can move rapidly and in complex patterns through fractures to reach and contaminate large volumes of aquifer, and then dissolve to cause significant pollution of groundwater. However, clean-up of DNAPLs in fractured rocks is virtually impossible and certainly expensive. Risk assessment should be used to decide whether the pollution is serious enough to justify major expenditure on clean-up or containment. A key aspect of risk assessment for DNAPLs in fractured aquifers is to understand how deep they are likely to have penetrated through the fracture network. This paper addresses two aspects of such predictions: measuring fracture apertures in situ and the connectivity of fracture networks with respect to DNAPLs. Fracture aperture is an in-situ field technique that has been developed and implemented to measure aperture variability and NAPL entry pressure in an undisturbed, water-saturated rock fracture. The field experiment also provided the opportunity to measure the wetting phase relative permeability at residual non-wetting phase saturation. The RADIO (Radial Aperture Determination by the Injection of Oil) method employs a constant rate injection of a non-toxic NAPL into a fracture isolated by a double packer array. The method was applied at the field site in Scotland, and measured apertures out to ~5m from the borehole. It showed that hydraulic aperture (from packer tests) was a poor estimator of the controlling aperture for DNAPL movement. This is the first time such large-scale aperture measurements have been made, and the technique is the first which can provide useful aperture estimates for risk analysis of DNAPL movement.Network connectivity is a fundamental property of the fracture system. DNAPL connectivity extends the concept to take account of the fluid properties.

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Li, Haifeng, Wei Wang, Yajun Cao, and Shifan Liu. "Phase-Field Modeling Fracture in Anisotropic Materials." Advances in Civil Engineering 2021 (July 30, 2021): 1–13. http://dx.doi.org/10.1155/2021/4313755.

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The phase-field method is a widely used technique to simulate crack initiation, propagation, and coalescence without the need to trace the fracture surface. In the phase-field theory, the energy to create a fracture surface per unit area is equal to the critical energy release rate. Therefore, the precise definition of the crack-driving part is the key to simulate crack propagation. In this work, we propose a modified phase-field model to capture the complex crack propagation, in which the elastic strain energy is decomposed into volumetric-deviatoric energy parts. Because of the volumetric-deviatoric energy split, we introduce a novel form of the crack-driving energy to simulate mixed-mode fracture. Furthermore, a new degradation function is proposed to simulate crack processes in brittle materials with different degradation rates. The proposed model is implemented by a staggered algorithm and to validate the performance of the phase-field modelling, and several numerical examples are constructed under plane strain condition. All the presented examples demonstrate the capability of the proposed approach in solving problems of brittle fracture propagation.

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22

Dou, Zhi, Zhifang Zhou, Yefei Tan, and Yanzhang Zhou. "Numerical Study of the Influence of Cavity on Immiscible Liquid Transport in Varied-Wettability Fractures." Journal of Chemistry 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/961256.

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Field evidence indicates that cavities often occur in fractured rocks, especially in a Karst region. Once the immiscible liquid flows into the cavity, the cavity has the immiscible liquid entrapped and results in a low recovery ratio. In this paper, the immiscible liquid transport in cavity-fractures was simulated by Lattice Boltzmann Method (LBM). The interfacial and surface tensions were incorporated by Multicomponent Shan-Chen (MCSC) model. Three various fracture positions were generated to investigate the influence on the irreducible nonwetting phase saturation and displacement time. The influences of fracture aperture and wettability on the immiscible liquid transport were discussed and analyzed. It was found that the cavity resulted in a long displacement time. Increasing the fracture aperture with the corresponding decrease in displacement pressure led to the long displacement time. This consequently decreased the irreducible nonwetting phase saturation. The fracture positions had a significant effect on the displacement time and irreducible saturation. The distribution of the irreducible nonwetting phase was strongly dependent on wettability and fracture position. Furthermore, this study demonstrated that the LBM was very effective in simulating the immiscible two-phase flow in the cavity-fracture.

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23

Li, Liyong, and Seong H. Lee. "Efficient Field-Scale Simulation of Black Oil in a Naturally Fractured Reservoir Through Discrete Fracture Networks and Homogenized Media." SPE Reservoir Evaluation & Engineering 11, no. 04 (August 1, 2008): 750–58. http://dx.doi.org/10.2118/103901-pa.

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Summary This paper describes a hybrid finite volume method, designed to simulate multiphase flow in a field-scale naturally fractured reservoir. Lee et al. (WRR 37:443-455, 2001) developed a hierarchical approach in which the permeability contribution from short fractures is derived in an analytical expression that from medium fractures is numerically solved using a boundary element method. The long fractures are modeled explicitly as major fluid conduits. Reservoirs with well-developed natural fractures include many complex fracture networks that cannot be easily modeled by simple long fracture formulation and/or homogenized single continuity model. We thus propose a numerically efficient hybrid method in which small and medium fractures are modeled by effective permeability, and large fractures are modeled by discrete fracture networks. A simple, systematic way is devised to calculate transport parameters between fracture networks and discretized, homogenized media. An efficient numerical algorithm is also devised to solve the dual system of fracture network and finite volume grid. Black oil formulation is implemented in the simulator to demonstrate practical applications of this hybrid finite volume method. Introduction Many reservoirs are highly fractured due to the complex tectonic movement and sedimentation process the formation has experienced. The permeability of a fracture is usually much larger than that of the rock matrix; as a result, the fluid will flow mostly through the fracture network, if the fractures are connected. This implies that the fracture connectivities and their distribution will determine fluid transport in a naturally fractured reservoir (Long and Witherspoon 1985). Because of statistically complex distribution of geological heterogeneity and multiple length and time scales in natural porous media, three approaches (Smith and Schwartz 1993) are commonly used in describing fluid flow and solute transport in naturally fractured formations:discrete fracture models;continuum models using effective properties for discrete grids; andhybrid models that combine discrete, large features and equivalent continuum. Currently, most reservoir simulators use dual continuum formulations (i.e., dual porosity/permeability) for naturally fractured reservoirs in which matrix blocks are divided by very regular fracture patterns (Kazemi et al. 1976, Van Golf-Racht 1982). Part of the primary input into these simulation models is the permeability of the fracture system assigned to the individual grid-blocks. This value can only be reasonably calculated if the fracture systems are regular and well connected. Field characterization studies have shown, however, that fracture systems are very irregular, often disconnected, and occur in swarms (Laubach 1991, Lorenz and Hill 1991, Narr et al. 2003). Most naturally fractured reservoirs include fractures of multiple- length scales. The effective grid-block permeability calculated by the boundary element method becomes expensive as the number of fractures increases. The calculated effective properties for grid-blocks also underestimates the properties for long fractures whose length scale is much larger than the grid-block size. Lee et al. (2001) proposed a hierarchical method to model fluid flow in a reservoir with multiple-length scaled fractures. They assumed that short fractures are randomly distributed and contribute to increasing the effective matrix permeability. An asymptotic solution representing the permeability contribution from short fractures was derived. With the short fracture contribution to permeability, the effective matrix permeability can be expressed in a general tensor form. Thus, they also developed a boundary element method for Darcy's equation with tensor permeability. For medium-length fractures in a grid-block, a coupled system of Poisson equations with tensor permeability was solved numerically using a boundary element method. The grid-block effective permeabilities were used with a finite difference simulator to compute flow through the fracture system. The simulator was enhanced to use a control-volume finite difference formulation (Lee et al. 1998, 2002) for general tensor permeability input (i.e., 9-point stencil for 2-D and 27-point stencil for 3-D). In addition, long fractures were explicitly modeled by using the transport index between fracture and matrix in a gridblock. In this paper we adopt their transport index concept and extend the hierarchical method:to include networks of long fractures;to model fracture as a two-dimensional plane; andto allow fractures to intersect with well bore. This generalization allows us to model a more realistic and complex fracture network that can be found in naturally fractured reservoirs. To demonstrate this new method for practical reservoir applications, we furthermore implement a black oil formulation in the simulator. We explicitly model long fractures as a two-dimensional plane that can penetrate several layers. The method, presented in this paper, allows a general description of fracture orientation in space. For simplicity of computational implementation however, both the medium-length and long fractures considered in this paper are assumed to be perpendicular to bedding boundaries. In addition, we derive a source/sink term to model the flux between matrix and long fracture networks. This source/sink allows for coupling multiphase flow equations in long fractures and matrix. The paper is organized as follows. In Section 2 black oil formulation is briefly summarized and the transport equations for three phase flow are presented. The fracture characterization and hierarchical modeling approach, based on fracture length, are discussed in Section 3. In Section 4 we review homogenization of short and medium fractures, which is part of our hierarchical approach to modeling flow in porous media with multiple length-scale fractures. In Section 5 we discuss a discrete network model of long fractures. We also derive transfer indices between fracture and effective matrix blocks. In Section 6 we present numerical examples for tracer transport in a model with simple fracture network, interactions of fractures and wells, and black oil production in a reservoir with a complex fracture network system. Finally, the summary of our main results and conclusion follows.

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Jammoul, M., and M. F. Wheeler. "A Phase-Field-Based Approach for Modeling Flow and Geomechanics in Fractured Reservoirs." SPE Journal 27, no. 02 (December 21, 2021): 1195–208. http://dx.doi.org/10.2118/203906-pa.

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Summary Modeling the geomechanical deformations of fracture networks has become an integral part of designing enhanced geothermal systems and recovery mechanisms for unconventional reservoirs. Stress changes in the reservoir can cause variations in the apertures of fractures resulting in large changes in their transmissivities. At the same time, sustained high-injection pressures can induce shear slipping along existing fractures and faults and trigger seismic activity. In this work, we extend the phase-field method to solve for flow and geomechanical deformations in naturally fractured reservoirs. The framework can predict the opening/closure of fractures as well as their shear slipping because of induced stresses and poromechanical effects. The flow through fractures is modeled on spatially nonconforming grids using the enhanced velocity mixed finite element method. The geomechanics equations are discretized using the continuous Galerkin (CG) finite element method. The flow and mechanics equations are decoupled using the fixed stress iterative scheme. The implementation is validated against the analytical solutions of Mandel’s problem and Sneddon’s benchmark test. Two synthetic examples are presented to illustrate the impact of poroelastic deformations and the accompanying dynamic behavior of fractures on the safety and productivity of subsurface projects.

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25

Zhang, Hao, Hui Peng, Xiao-yang Pei, Ping Li, Tie-gang Tang, and Ling-cang Cai. "A phase-field model for spall fracture." Journal of Applied Physics 129, no. 12 (March 28, 2021): 125903. http://dx.doi.org/10.1063/5.0043675.

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26

Xue, Tianju, Sigrid Adriaenssens, and Sheng Mao. "Mapped phase field method for brittle fracture." Computer Methods in Applied Mechanics and Engineering 385 (November 2021): 114046. http://dx.doi.org/10.1016/j.cma.2021.114046.

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27

Yoshioka, Keita, Mostafa Mollaali, and Olaf Kolditz. "Variational phase-field fracture modeling with interfaces." Computer Methods in Applied Mechanics and Engineering 384 (October 2021): 113951. http://dx.doi.org/10.1016/j.cma.2021.113951.

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Strobl, M., and Th Seelig. "Phase field modeling of Hertzian indentation fracture." Journal of the Mechanics and Physics of Solids 143 (October 2020): 104026. http://dx.doi.org/10.1016/j.jmps.2020.104026.

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29

Levitas, Valery I., Alexander V. Idesman, and Ameeth K. Palakala. "Phase-field modeling of fracture in liquid." Journal of Applied Physics 110, no. 3 (August 2011): 033531. http://dx.doi.org/10.1063/1.3619807.

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Miehe, C., F. Welschinger, and M. Hofacker. "A phase field model of electromechanical fracture." Journal of the Mechanics and Physics of Solids 58, no. 10 (October 2010): 1716–40. http://dx.doi.org/10.1016/j.jmps.2010.06.013.

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Kuhn, Charlotte, and Ralf Müller. "A continuum phase field model for fracture." Engineering Fracture Mechanics 77, no. 18 (December 2010): 3625–34. http://dx.doi.org/10.1016/j.engfracmech.2010.08.009.

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32

Bilgen, Carola, Alena Kopaničáková, Rolf Krause, and Kerstin Weinberg. "A phase-field approach to conchoidal fracture." Meccanica 53, no. 6 (August 28, 2017): 1203–19. http://dx.doi.org/10.1007/s11012-017-0740-z.

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33

Conti, S., M. Focardi, and F. Iurlano. "Phase field approximation of cohesive fracture models." Annales de l'Institut Henri Poincaré C, Analyse non linéaire 33, no. 4 (July 2016): 1033–67. http://dx.doi.org/10.1016/j.anihpc.2015.02.001.

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Schlüter, Alexander, Adrian Willenbücher, Charlotte Kuhn, and Ralf Müller. "Phase field approximation of dynamic brittle fracture." Computational Mechanics 54, no. 5 (June 21, 2014): 1141–61. http://dx.doi.org/10.1007/s00466-014-1045-x.

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Kuhn, Charlotte, Timo Noll, and Ralf Müller. "On phase field modeling of ductile fracture." GAMM-Mitteilungen 39, no. 1 (May 30, 2016): 35–54. http://dx.doi.org/10.1002/gamm.201610003.

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36

Schillinger, Dominik, Michael J. Borden, and Henryk K. Stolarski. "Isogeometric collocation for phase-field fracture models." Computer Methods in Applied Mechanics and Engineering 284 (February 2015): 583–610. http://dx.doi.org/10.1016/j.cma.2014.09.032.

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Kuhn, Charlotte, Alexander Schlüter, and Ralf Müller. "A Phase Field Approach for Dynamic Fracture." PAMM 13, no. 1 (November 29, 2013): 87–88. http://dx.doi.org/10.1002/pamm.201310039.

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Schlüter, Alexander, Charlotte Kuhn, and Ralf Müller. "Phase Field Approximation of Dynamic Brittle Fracture." PAMM 14, no. 1 (December 2014): 143–44. http://dx.doi.org/10.1002/pamm.201410059.

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Bilgen, Carola, Alena Kopaničáková, Rolf Krause, and Kerstin Weinberg. "A phase-field approach to pneumatic fracture." PAMM 17, no. 1 (December 2017): 71–74. http://dx.doi.org/10.1002/pamm.201710022.

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40

Verhoosel, Clemens V., and René de Borst. "A phase-field model for cohesive fracture." International Journal for Numerical Methods in Engineering 96, no. 1 (July 24, 2013): 43–62. http://dx.doi.org/10.1002/nme.4553.

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41

Sidharth, P. C., and B. N. Rao. "A Review on phase-field modeling of fracture." Proceedings of the 12th Structural Engineering Convention, SEC 2022: Themes 1-2 1, no. 1 (December 19, 2022): 449–56. http://dx.doi.org/10.38208/acp.v1.534.

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Abstract:

In cases with complicated crack topologies, the computational modeling of failure processes in materials owing to fracture based on sharp crack discontinuities fails. Diffusive crack modeling based on the insertion of a crack phase-field can overcome this. The phase-field model (PFM) portrays the fracture geometry in a diffusive manner, with no abrupt discontinuities. Unlike discrete fracture descriptions, phase-field descriptions do not need numerical monitoring of discontinuities in the displacement field. This considerably decreases the complexity of implementation. These qualities enable PFM to describe fracture propagation more successfully than numerical approaches based on the discrete crack model, especially for complicated crack patterns. These models have also demonstrated the ability to forecast fracture initiation and propagation in two and three dimensions without the need for any ad hoc criteria. The phase-field model, among numerous options, is promising in the computer modeling of fracture in solids due to its ability to cope with complicated crack patterns such as branching, merging, and even fragmentation. A brief history of the application of the phase-field model in predicting solid fracture has been attempted. An effort has been made to keep the conversation focused on recent research findings on the subject. Finally, some key findings and recommendations for future research areas in this field are discussed.

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Seleš, Karlo, Tomislav Lesičar, Zdenko Tonković, and Jurica Sorić. "A Phase Field Staggered Algorithm for Fracture Modeling in Heterogeneous Microstructure." Key Engineering Materials 774 (August 2018): 632–37. http://dx.doi.org/10.4028/www.scientific.net/kem.774.632.

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The phase field approach to fracture modelling is based on a variational principle of the energy minimization as an extension of the Griffith’s brittle fracture theory. It introduces a scalar damage field, to differentiate between the fractured and intact material state. That way, it regularizes the sharp crack discontinuities and eliminates the need for the explicit tracking of the fracture surfaces. Moreover, the numerical implementation complexity is thus vastly reduced. In this contribution, the staggered phase field algorithm for the modelling of brittle fracture is implemented within the finite element program Abaqus. A common issue of the existing Abaqus implementations of the staggered phase field schemes is the computationally demanding fine incrementation of the loading applied, required to obtain an accurate solution. The computational time is reduced by imposing an appropriate convergence control paired with the Abaqus automatic time incrementation. Therefore, by taking advantage of the Abaqus computational efficiency, an accurate solution can be obtained for a moderate time step. The proposed model is verified on the symmetrically double notched tensile benchmark test. Compared to the existing implementations, it demonstrates an improvement in accuracy and the computational performance. Furthermore, a heterogeneous steel microstructure is analyzed displaying the model’s ability to solve crack nucleation and curvilinear crack paths.

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Tsoflias, Georgios P., and Matthew W. Becker. "Ground-penetrating-radar response to fracture-fluid salinity: Why lower frequencies are favorable for resolving salinity changes." GEOPHYSICS 73, no. 5 (September 2008): J25—J30. http://dx.doi.org/10.1190/1.2957893.

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Time-lapse ground-penetrating-radar (GPR) surveys exploit signal-amplitude changes to monitor saline tracers in fractures and to identify groundwater flow paths. However, the relationships between GPR signal amplitude, phase, and frequency with fracture aperture and fluid electrical conductivity are not well understood. We used analytical modeling, numerical simulations, and field experiments of multifrequency GPR to investigate these relationships for a millimeter-scale-aperture fracture saturated with water of varying salinity. We found that the response of lower-frequency radar signals detects changes in fluid salinity better than the response of higher-frequency signals. Increasing fluid electrical conductivity decreases low-frequency GPR signal wavelength, which improves its thin-layer resolution capability. We concluded that lower signal frequencies, such as [Formula: see text], and saline tracers of up to [Formula: see text] conductivity are preferable when using GPR to monitor flow in fractured rock. Furthermore, we found that GPR amplitude and phase responses are detectable in the field and predictable by EM theory and modeling; therefore, they can be related to fracture aperture and fluid salinity for hydrologic investigations of fractured-rock flow and transport properties.

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44

Kristensen, Philip K., Christian F. Niordson, and Emilio Martínez-Pañeda. "An assessment of phase field fracture: crack initiation and growth." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 379, no. 2203 (June 21, 2021): 20210021. http://dx.doi.org/10.1098/rsta.2021.0021.

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The phase field paradigm, in combination with a suitable variational structure, has opened a path for using Griffith’s energy balance to predict the fracture of solids. These so-called phase field fracture methods have gained significant popularity over the past decade, and are now part of commercial finite element packages and engineering fitness- for-service assessments. Crack paths can be predicted, in arbitrary geometries and dimensions, based on a global energy minimization—without the need for ad hoc criteria. In this work, we review the fundamentals of phase field fracture methods and examine their capabilities in delivering predictions in agreement with the classical fracture mechanics theory pioneered by Griffith. The two most widely used phase field fracture models are implemented in the context of the finite element method, and several paradigmatic boundary value problems are addressed to gain insight into their predictive abilities across all cracking stages; both the initiation of growth and stable crack propagation are investigated. In addition, we examine the effectiveness of phase field models with an internal material length scale in capturing size effects and the transition flaw size concept. Our results show that phase field fracture methods satisfactorily approximate classical fracture mechanics predictions and can also reconcile stress and toughness criteria for fracture. The accuracy of the approximation is however dependent on modelling and constitutive choices; we provide a rationale for these differences and identify suitable approaches for delivering phase field fracture predictions that are in good agreement with well-established fracture mechanics paradigms. This article is part of a discussion meeting issue ‘A cracking approach to inventing new tough materials: fracture stranger than friction’.

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Leggett, Smith Edward, Ding Zhu, and Alfred Daniel Hill. "Thermal Effects on Far-Field Distributed Acoustic Strain-Rate Sensors." SPE Journal 27, no. 02 (November 23, 2021): 1036–48. http://dx.doi.org/10.2118/205178-pa.

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Summary Fiber-optic cables cemented outside of the casing of an unconventional well measure crosswell strain changes during fracturing of neighboring wells with low-frequency distributed acoustic sensing (LF-DAS). As a hydraulic fracture intersects an observation well instrumented with fiber-optic cables, the fracture fluid injected at ambient temperatures can cool a section of the sensing fiber. Often, LF-DAS and distributed temperature sensing (DTS) cables are run in tandem, enabling the detection of such cooling events. The increasing use of LF-DAS for characterizing unconventional hydraulic fracture completions demands an investigation of the effects of temperature on the measured strain response by LF-DAS. Researchers have demonstrated that LF-DAS can be used to extract the temporal derivative of temperature for use as a differential-temperature-gradient sensor. However, differential-temperature-gradient sensing is predicated on the ability to filter strain components out of the optical signal. In this work, beginning with an equation for optical phase shift of LF-DAS signals, a model relating strain, temperature, and optical phase shift is explicitly developed. The formula provides insights into the relative strength of strain and temperature effects on the phase shift. The uncertainty in the strain-rate measurements due to thermal effects is estimated. The relationship can also be used to quantify uncertainties in differential-temperature-gradient sensors due to strain perturbations. Additionally, a workflow is presented to simulate the LF-DAS response accounting for both strain and temperature effects. Hydraulic fracture geometries are generated with a 3D fracture simulator for a multistage unconventional completion. The fracture width distributions are imported by a displacement discontinuity method (DDM) program to compute the strain rates along an observation well. An analytic model is used to approximate the temperature in the fracture. Using the derived formulae for optical phase shift, the model outputs are then used to compute the LF-DAS response at a fiber-optic cable, enabling the generation of waterfall plots including both strain and thermal effects. The model results suggest that before, during, and immediately following a fracture intersecting a well instrumented with fiber, the strain on the fiber drives the LF-DAS signal. However, at later times, as completion fluid cools the observation well, the temperature component of the LF-DAS signal can be equal to or exceed the strain component. The modeled results are compared to a published field case in an attempt to enhance the interpretation of LF-DAS waterfall plots. Finally, we propose a sensing configuration to identify the events when “wet fractures” (fractures with fluids) intersect the observation well.

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46

Wick, Thomas, Gurpreet Singh, and Mary F. Wheeler. "Fluid-Filled Fracture Propagation With a Phase-Field Approach and Coupling to a Reservoir Simulator." SPE Journal 21, no. 03 (June 15, 2016): 0981–99. http://dx.doi.org/10.2118/168597-pa.

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Summary A quantitative assessment of hydraulic-fracturing jobs relies on accurate predictions of fracture growth during slickwater injection for single and multistage fracturing scenarios. This requires consistent modeling of underlying physical processes, from hydraulic fracturing to long-term production. In this work, we use a recently introduced phase-field approach to model fracture propagation in a porous medium. This approach is thermodynamically consistent and captures several characteristic features of crack propagation such as joining, branching, and nonplanar propagation as a result of heterogeneous material properties. We describe two different phase-field fracture-propagation models and then present a technique for coupling these to a fractured-poroelastic-reservoir simulator. The proposed coupling approach can be adapted to existing reservoir simulators. We present 2D and 3D numerical tests to benchmark, compare, and demonstrate the predictive capabilities of the fracture-propagation model as well as the proposed coupling scheme.

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47

Noii, Nima, Amirreza Khodadadian, Jacinto Ulloa, Fadi Aldakheel, Thomas Wick, Stijn François, and Peter Wriggers. "Bayesian inversion for unified ductile phase-field fracture." Computational Mechanics 68, no. 4 (August 26, 2021): 943–80. http://dx.doi.org/10.1007/s00466-021-02054-w.

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AbstractThe prediction of crack initiation and propagation in ductile failure processes are challenging tasks for the design and fabrication of metallic materials and structures on a large scale. Numerical aspects of ductile failure dictate a sub-optimal calibration of plasticity- and fracture-related parameters for a large number of material properties. These parameters enter the system of partial differential equations as a forward model. Thus, an accurate estimation of the material parameters enables the precise determination of the material response in different stages, particularly for the post-yielding regime, where crack initiation and propagation take place. In this work, we develop a Bayesian inversion framework for ductile fracture to provide accurate knowledge regarding the effective mechanical parameters. To this end, synthetic and experimental observations are used to estimate the posterior density of the unknowns. To model the ductile failure behavior of solid materials, we rely on the phase-field approach to fracture, for which we present a unified formulation that allows recovering different models on a variational basis. In the variational framework, incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. The overall formulation is revisited and extended to the case of anisotropic ductile fracture. Three different models are subsequently recovered by certain choices of parameters and constitutive functions, which are later assessed through Bayesian inversion techniques. A step-wise Bayesian inversion method is proposed to determine the posterior density of the material unknowns for a ductile phase-field fracture process. To estimate the posterior density function of ductile material parameters, three common Markov chain Monte Carlo (MCMC) techniques are employed: (i) the Metropolis–Hastings algorithm, (ii) delayed-rejection adaptive Metropolis, and (iii) ensemble Kalman filter combined with MCMC. To examine the computational efficiency of the MCMC methods, we employ the $$\hat{R}{-}convergence$$ R ^ - c o n v e r g e n c e tool. The resulting framework is algorithmically described in detail and substantiated with numerical examples.

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Schmidt, Jaroslav, Alena Zemanová, Jan Zeman, and Michal Šejnoha. "Phase-Field Fracture Modelling of Thin Monolithic and Laminated Glass Plates under Quasi-Static Bending." Materials 13, no. 22 (November 16, 2020): 5153. http://dx.doi.org/10.3390/ma13225153.

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A phase-field description of brittle fracture is employed in the reported four-point bending analyses of monolithic and laminated glass plates. Our aims are: (i) to compare different phase-field fracture formulations applied to thin glass plates, (ii) to assess the consequences of the dimensional reduction of the problem and mesh density and refinement, and (iii) to validate for quasi-static loading the time-/temperature-dependent material properties we derived recently for two commonly used polymer foils made of polyvinyl butyral or ethylene-vinyl acetate. As the nonlinear response prior to fracture, typical of the widely used Bourdin–Francfort–Marigo model, can lead to a significant overestimation of the response of thin plates under bending, the numerical study investigates two additional phase-field fracture models providing the linear elastic phase of the stress-strain diagram. The typical values of the critical fracture energy and tensile strength of glass lead to a phase-field length-scale parameter that is challenging to resolve in the numerical simulations. Therefore, we show how to determine the fracture energy concerning the applied dimensional reduction and the value of the length-scale parameter relative to the thickness of the plate. The comparison shows that the phase-field models provide very good agreement with the measured stresses and resistance of laminated glass, despite the fact that only one/two cracks are localised using the quasi-static analysis, whereas multiple cracks evolve during the experiment. It was also observed that the stiffness and resistance of the partially fractured laminated glass can be well approximated using a 2D plane-stress model with initially predefined cracks, which provides a better estimation than the one-glass-layer limit.

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Phansalkar, Dhananjay, Kerstin Weinberg, Michael Ortiz, and Sigrid Leyendecker. "A spatially adaptive phase-field model of fracture." Computer Methods in Applied Mechanics and Engineering 395 (May 2022): 114880. http://dx.doi.org/10.1016/j.cma.2022.114880.

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50

Noii, Nima, Amirreza Khodadadian, and Thomas Wick. "Bayesian inversion for anisotropic hydraulic phase-field fracture." Computer Methods in Applied Mechanics and Engineering 386 (December 2021): 114118. http://dx.doi.org/10.1016/j.cma.2021.114118.

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