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A mesh-independent finite element formulation for modeling crack growth in saturated porous media based on an enriched-FEM technique

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In this paper, a crack growth simulation is presented in saturated porous media using the extended finite element method, where the mass balance equation of fluid phase and the momentum balance of bulk and fluid phases are employed to obtain the fully coupled set of equations in the framework of $$u{-}p$$ formulation.
Abstract
In this paper, the crack growth simulation is presented in saturated porous media using the extended finite element method. The mass balance equation of fluid phase and the momentum balance of bulk and fluid phases are employed to obtain the fully coupled set of equations in the framework of $$u{-}p$$ formulation. The fluid flow within the fracture is modeled using the Darcy law, in which the fracture permeability is assumed according to the well-known cubic law. The spatial discritization is performed using the extended finite element method, the time domain discritization is performed based on the generalized Newmark scheme, and the non-linear system of equations is solved using the Newton–Raphson iterative procedure. In the context of the X-FEM, the discontinuity in the displacement field is modeled by enhancing the standard piecewise polynomial basis with the Heaviside and crack-tip asymptotic functions, and the discontinuity in the fluid flow normal to the fracture is modeled by enhancing the pressure approximation field with the modified level-set function, which is commonly used for weak discontinuities. Two alternative computational algorithms are employed to compute the interfacial forces due to fluid pressure exerted on the fracture faces based on a ‘partitioned solution algorithm’ and a ‘time-dependent constant pressure algorithm’ that are mostly applicable to impermeable media, and the results are compared with the coupling X-FEM model. Finally, several benchmark problems are solved numerically to illustrate the performance of the X-FEM method for hydraulic fracture propagation in saturated porous media.

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VALIDITY OF CUBIC LAW FOR FLUID FLOW IN A DEFORMABLE ROCK FRACTURE - eScholarship

Abstract: The validity of the cubic law for laminar flow of fluids through open fractures consisting of parallel planar plates has been established by others over a wide range of conditions with apertures ranging down to a minimum of 0.2 µm. The law may be given in simplified form by Q/Δh = C(2b)3, where Q is the flow rate, Δh is the difference in hydraulic head, C is a constant that depends on the flow geometry and fluid properties, and 2b is the fracture aperture. The validity of this law for flow in a closed fracture where the surfaces are in contact and the aperture is being decreased under stress has been investigated at room temperature by using homogeneous samples of granite, basalt, and marble. Tension fractures were artificially induced, and the laboratory setup used radial as well as straight flow geometries. Apertures ranged from 250 down to 4µm, which was the minimum size that could be attained under a normal stress of 20 MPa. The cubic law was found to be valid whether the fracture surfaces were held open or were being closed under stress, and the results are not dependent on rock type. Permeability was uniquely defined by fracture aperture and was independent of the stress history used in these investigations. The effects of deviations from the ideal parallel plate concept only cause an apparent reduction in flow and may be incorporated into the cubic law by replacing C by C/ƒ. The factor ƒ varied from 1.04 to 1.65 in these investigations. The model of a fracture that is being closed under normal stress is visualized as being controlled by the strength of the asperities that are in contact. These contact areas are able to withstand significant stresses while maintaining space for fluids to continue to flow as the fracture aperture decreases. The controlling factor is the magnitude of the aperture, and since flow depends on (2b)3, a slight change in aperture evidently can easily dominate any other change in the geometry of the flow field. Thus one does not see any noticeable shift in the correlations of our experimental results in passing from a condition where the fracture surfaces were held open to one where the surfaces were being closed under stress.
Journal ArticleDOI

Phase-field modeling of hydraulic fracture

TL;DR: In this paper, a theoretical framework implementing the phase-field approach to fracture is used to couple the physics of flow through porous media and cracks with the mechanics of fracture, which is a challenge for all diffuse crack representations, is on how to allow for the flow of fluid and the action of fluid pressure on the aggregate within the diffuse damage zone of the cracks.
Journal ArticleDOI

A three-phase XFEM model for hydraulic fracturing with cohesive crack propagation

TL;DR: In this paper, a three-phase hydro-mechanical model for hydraulic fracturing is proposed, where porous solid, fracturing fluid and host fluid are coupled through a so-called leak-off mass transfer term.
Journal ArticleDOI

Three-Dimensional poroelastic effects during hydraulic fracturing in permeable rocks

TL;DR: In this paper, a fully coupled three-dimensional finite-element model for hydraulic fractures in permeable rocks is presented, and used to investigate the ranges of applicability of the classical analytical solutions that are known to be valid in limiting cases.
Journal ArticleDOI

An XFEM-based method with reduction technique for modeling hydraulic fracture propagation in formations containing frictional natural fractures

TL;DR: In this paper, a reduction technique is proposed in a tightly coupled model in which the equilibrium and flow continuity equations are solved simultaneously by the Newton-Raphson method, which can significantly accelerate the simulation without worsening the convergence or losing the computational accuracy.
References
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TL;DR: In this article, the number of physical constants necessary to determine the properties of the soil is derived along with the general equations for the prediction of settlements and stresses in three-dimensional problems.
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Yielding of steel sheets containing slits

TL;DR: In this article, a relation between extent of plastic yielding and external load applied was investigated, and panels containing internal and edge slits were loaded in tension and lengths of plastic zones were measured.
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Theoretical Soil Mechanics

Karl Terzaghi
Journal ArticleDOI

A finite element method for crack growth without remeshing

TL;DR: In this article, a displacement-based approximation is enriched near a crack by incorporating both discontinuous elds and the near tip asymptotic elds through a partition of unity method.
Book ChapterDOI

The mathematical theory of equilibrium cracks in brittle fracture

TL;DR: In this paper, the authors present a unified view of the way basic problems in the theory of equilibrium cracks are formulated and discuss the results obtained thereby, and the object of the theory is the study of the equilibrium of solids in the presence of cracks.
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