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.

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

A 3-D conjugated bond-pair-based peridynamic formulation for initiation and propagation of cracks in brittle solids

By introducing the concept of mathematical cover and physical cover, the numerical manifold method (NMM) is able to solve continuous and discontinuous problems in a unified way. In this paper, the NMM is developed to analyze three dimensional (3D) fracture propagation. The maximum tensile stress criterion is implemented to determine whether the fracture will propagate and the direction of fracture propagation. Three benchmark problems are analyzed to validate the present algorithm and program. The numerical results replicate available experimental results and existing numerical results. The present algorithm and 3D NMM code are promising for 3D fracture propagation. They deserve to be further developed for the analysis of rock mechanic problems in which the initiation and propagation of multiple fractures, tensile and shear fractures, and fracture propagation under compressive loading are taken into account.

Three-dimensional fracture propagation with numerical manifold method

Phase-field modeling of fluid-driven dynamic cracking in porous media

Hydraulic fracturing modeling using the enriched numerical manifold method

Crack coalescence between two non-parallel flaws in rock-like material under uniaxial compression

In this paper, a new numerical simulation tool named TOUGH-FEMM is presented and applied to model three-dimensional (3D) hydraulic fracturing in porous rock. The fluid flow in both fractures and porous rock is modeled using TOUGH2, which is a well-established code for analysis of multiphase and multi-component fluid flow. Rock deformations associated with fracture propagation are modeled using finite element-meshfree method (FEMM). FEMM is an approach to simulate fracture propagation without remeshing, in which the fracture path does not need to be predetermined. Fracture mechanics with mixed-mode stress intensity factors are employed to detect fracture instability and determine the direction of fracture propagation. TOUGH-FEMM is verified for modeling fluid-driven fracture propagation in 3D through a number of simulation examples, including modeling of hydraulic fracturing laboratory experiments and by comparison to independent numerical simulation results for multiple interacting hydraulic fractures at ten to hundred meter scale.

/pdf/modeling-three-dimensional-fluid-driven-propagation-of-26t8q1yawj.pdf

Xuhai Tang

Papers

Three-dimensional fracture propagation with numerical manifold method

Hydraulic fracturing modeling using the enriched numerical manifold method

Crack coalescence between two non-parallel flaws in rock-like material under uniaxial compression

Modeling Three-Dimensional Fluid-Driven Propagation of Multiple Fractures using TOUGH-FEMM

A three-node triangular element with continuous nodal stress