The far-field deformation caused by a hydraulic fracture in an inhomogeneous elastic half-space

Chen, Zuorong; Jeffrey, Robert G.; Pandurangan, Venkataraman

doi:10.1016/j.ijsolstr.2017.09.034

Cited by 8 publications

(6 citation statements)

References 33 publications

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“…The dielectric case only requires one of the two linked functions C2, S2 = 1 À C2. Calculating C2 and C4 say, 6 using cos 2 u = t 2 and cos 4 u = t 4 , one obtains for z-oriented spheroids (p = 1,2)…”

Section: Spheroid Shape Function From the Rt-irt Methods And The Relatmentioning

confidence: 99%

“…The Eshelby (1957) seminal work on ellipsoidal inclusions [1], which proved uniformity for the interior strain localization tensor of ellipsoids in infinite media, allowed tremendous progress in the fields of effective property and stress-strain field estimates in composite materials for the many reinforcedmatrix structure types with all kinds of reinforcing (or weakening) included phases and also for aggregates. With inclusions ranging from nano to mega scales, applications cover major research domains, such as bio-, geo-, space-, engineering and mechanics, sometimes in an unexpected manner [2][3][4][5][6]. At the limit of infinitely long (prolate) or flat (oblate) ellipsoids, this ''Eshelby uniformity property'' still holds as long as the embedding medium is infinite too, which means infinite cylinders and laminates share this property.…”

Section: Introductionmentioning

confidence: 99%

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Uniformity of the Green operator and Eshelby tensor for hyperboloidal domains in infinite media

Franciosi

2020

Mathematics and Mechanics of Solids

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Since Eshelby’s (1957) result (Eshelby, JD. The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc R Soc London A 1957; 421: 379–396) that ellipsoids in an infinite matrix have uniform localization tensors, all attempts to find other finite domain shapes sharing that same property have failed and valuable proofs were provided that none could exist. Since that “uniformity property” also applies to infinite cylinders and layers as limits of prolate and oblate spheroids, we examine the cases of hyperboloidal domains of which infinite cylinders and platelets also are the limits. As members of the quadric surface family, hyperboloids expectably also have uniform Eshelby tensor and Green operator when embedded in infinite media, with specific features expectable too from unboundedness and not convex curvatures. Using the Radon transform method applied by the author to various inclusion shapes, as well as to finite and infinite patterns, since the uniformity of a shape function (inverse Radon transform of the domain indicator function) implies the uniformity of the related Green operator and Eshelby tensor, we examine the shape functions of axially symmetric hyperboloids. We establish that those of the two-sheet types are uniform and those of the one-sheet types are not, an additional neck-related term carrying the non-uniformity. The Green operators are next examined in considering an isotropic embedding medium with elastic- (including dielectric-) like properties. The results regarding the operator (non) uniformity correspond to those concerning the shape functions. The established operator uniformity characteristics imply validity of all Eshelby-derived ellipsoid properties. Yet, determining the Green operator solution calls for overcoming the issue of the infinite hyperbolic planar sections (the operator finiteness), with also attention being paid to positive definiteness. Options are compared from which an obtained satisfying solution with regard to both issues raises questioning theoretical and practical points on mathematical and mechanical grounds. While further studies are in progress, some application tracks are indicated.

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Section: Spheroid Shape Function From the Rt-irt Methods And The Relatmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Uniformity of the Green operator and Eshelby tensor for hyperboloidal domains in infinite media

Franciosi

2020

Mathematics and Mechanics of Solids

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“…An inclusion/inhomogeneity can be related either to an equivalent dislocation (Eshelby 1957, Mura 1987 or to an equivalent force (Pan 2004a. With the known fundamental solutions of the equivalent dislocation or equivalent force, the inclusion/inhomogeneity approach can be applied to hydraulic fracture analysis (Chen et al 2018a), to finite 3D inhomogeneity in both elastic and viscoelastic halfspaces (Wu and Wang 1988, Bonafede 1990, Wu et al 1991, Bonafede and Ferrari 2009, Zhong et al 2019 and even to the current quantum-wire and quantum-dot nanotechnology (Pan 2004a, Pan et al 2008, Zou and Pan 2012, Lee et al 2015. Furthermore, the dislocation solutions can be directly or indirectly applied to solve various crack/ fracture problems, as in Bonafede andRivalta (1999a, 1999b) for the tensile dislocation/crack problems in a half-plane or a bimaterial plane.…”

Section: Gfs In a Half-space By Forces And Dislocationsmentioning

confidence: 99%

“…the FEM). Recently, the analytical code ELLNs in MATLAB was published by Chen et al (2018a). It is based on the analytical solution by Pan et al (2015b) combined with the SMM (i.e.…”

Section: Summary Of Sectionmentioning

confidence: 99%

Green’s functions for geophysics: a review

Pan

2019

Rep. Prog. Phys.

118

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The Green's function (GF) method, which makes use of GFs, is an important and elegant tool for solving a given boundary-value problem for the differential equation from a real engineering or physical field. Under a concentrated source, the solution of a differential equation is called a GF, which is singular at the source location, yet is very fundamental and powerful. When looking at the GFs from different physical and/or engineering fields, i.e. assigning the involved functions to real physical/engineering quantities, the GFs can be scaled and applied to large-scale problems such as those involved in Earth sciences as well as to nano-scale problems associated with quantum nanostructures. GFs are ubiquitous and everywhere: they can describe heat, water pressure, fluid flow potential, electromagnetic (EM) and gravitational potentials, and the surface tension of soap film. In the undergraduate courses Mechanics of Solids and Structural Analysis, a GF is the simple influence line or singular function. Dropping a pebble in the pond, it is the circular ripple traveling on and on. It is the wave generated by a moving ship in the opening ocean or the atom vibrating on a nanoscale sheet induced by the atomic force microscopy. In Earth science, while various GFs have been derived, a comprehensive review is missing. Thus, this article provides a relatively complete review on GFs for geophysics. In section 1, the George Green's potential functions, GF definition, as well as related theorems and basic relations are briefly presented. In section 2, the boundary-value problems for elastic and viscoelastic materials are provided. Section 3 is on the GFs in full-and half-spaces (planes). The GFs of concentrated forces and dislocations in horizontally layered half-spaces (planes) are derived in section 4 in terms of both Cartesian and cylindrical systems of vector functions. The corresponding GFs in a self-gravitating and layered spherical Earth are presented in section 5 in terms of the spherical system of vector functions. The singularity and infinity associated with GFs in layered systems are analyzed in section 6 along with a brief review of various layer matrix methods. Various associated mathematical preliminaries are listed in appendix, along with the three sets of vector function systems. It should be further emphasized that, while this review is targeted at geophysics, most of the GFs and solution methods can be equally applied to other engineering and science fields. Actually, many GFs and solutions methods reviewed in this article are derived by engineers and scientists from allied fields besides geophysics. As such, the updated approaches of constructing and deriving the GFs reviewed here should be very beneficial to any reader.

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“…The problem of deformation and stress disturbance around a reservoir due to the heat transport and fluid migration may be solved on the basis of the Eshelby model, which involves the concept of eigenstrain to handle the inelastic issue [27][28][29]. In this study, the stress fields around a penny-shaped reservoir with thermo-porous eigenstrains in a full space are investigated via the numerical equivalent inclusion method.…”

Section: Introductionmentioning

confidence: 99%

A Computational Scheme for Evaluating the Stress Field of Thermally and Pressure Induced Unconventional Reservoir

Zhang

Zhou³

2021

Lithosphere

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The fluid flow connecting the hydraulic fracture and associated unconventional gas or oil reservoir is of great importance to explore such unconventional resource. The deformation of unconventional reservoir caused by heat transport and pore pressure fluctuation may change the stress field of surrounding layer. In this paper, the stress distribution around a penny-shaped reservoir, whose shape is more versatile to cover a wide variety of special case, is investigated via the numerical equivalent inclusion method. Fluid production or hydraulic injection in a subsurface resource caused by the change of pore pressure and temperature within the reservoir may be simulated with the help of the Eshelby inclusion model. By employing the approach of classical eigenstrain, a computational scheme for solving the disturbance produced by the thermally and pressure induced unconventional reservoir is coded to study the effect of Biot coefficient and some other important factors. Moreover, thermo-poro transformation strain and arbitrarily orientated reservoir existing within the surrounding layer are also considered.

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The far-field deformation caused by a hydraulic fracture in an inhomogeneous elastic half-space

Cited by 8 publications

References 33 publications

Uniformity of the Green operator and Eshelby tensor for hyperboloidal domains in infinite media

Uniformity of the Green operator and Eshelby tensor for hyperboloidal domains in infinite media

Green’s functions for geophysics: a review

A Computational Scheme for Evaluating the Stress Field of Thermally and Pressure Induced Unconventional Reservoir

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