With the development of productivity and technology, increasingly open-pit coal mines have formed a series of high and steep slopes. The slope contains a rich network of fractures, and the stability of these slopes greatly affects the production and life of open-pit coal mines and the safety of personnel. At present, for low-strength fractured rock masses, such as mudstone, there are relatively few studies on the mechanical properties and microcrack propagation [1]. In an engineering project, such as open-pit coal mines and tunnels, rocks often contain more primary joints and flaws, which are more likely to cause damage to the rock masses. Therefore, it is necessary to conduct a more detailed study on the mechanical properties and microcrack propagation of mudstone containing a single preexisting flaw.
Because it is difficult to obtain and measure the internal fracture network of the rock masses, it has become a feasible research method to study the preexisting flaws with a small number. In terms of experiments, Yang and Jing conducted uniaxial compression tests of single-flaw sandstone and found that its mechanical properties have a certain correlation with the inclination of the fracture [2]. Wong and Einstein used gypsum prefabricated single-flaw specimens for uniaxial compression tests and obtained the propagation and evolution characteristics of wing cracks and secondary cracks [3]. Aliabadian et al. used 3D printing technology and digital image correlation technology, and the mechanical properties and strain field laws of rock-like materials with single and double flaws in Brazilian testing were studied [4]. Zhao et al. found that the evolution of the strain field was consistent with the initiation, propagation, and coalescence of microcracks [5]. Li et al. conducted uniaxial compression tests on a coal sample with a single preexisting flaw and conducted acoustic emission (AE) signal monitoring during the loading processes [6].
Crack mychat 4.14
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With the rapid development of computer technology, numerical simulation technology has become an essential technical method to study the mechanical properties of fractured rocks. As an effective supplement to laboratory tests, it plays an important role in the field of rock mechanics. Zhao et al. used an expanded distinct element method (EDEM) to simulate the crack propagation of a single fractured rock and found that with the change of the inclination of the fracture, the failure mode of the rock also changed accordingly [5]. Guo et al. used FLAC software to simulate the changes in the displacement field in tunnels and foundation pits [7, 8]. Bi et al. used the general particle dynamics (GPD) method to study the crack growth of rock-like materials containing multiple preexisting flaws under different confining pressures under three-dimensional conditions [9]. The discrete element method (DEM) proposed by Cundall and Strack [10] is a very widely used method in numerical simulation. particle flow code (PFC) [11] is commercial software developed based on DEM. To date, many researchers have used PFC to study the mechanical properties and the growth of microcracks in rock materials or rock-like materials with preexisting flaws. Zhang et al. used PFC to study the difference in AE characteristics of single-flaw rock-like materials under different loading rates [12]. Chen et al. used PFC to study the microcrack growth and mechanical properties of the rock containing nonpersistent joints [13].
The main objective of this work is to propose a parallel bond model with the Weibull distribution to model crack propagation in heterogeneous brittle rocks, such as mudstone. In the proposed framework, the micromechanical properties of brittle rocks are specifically defined through the Weibull distribution. This particular setting ensures that material heterogeneities at the grain scale and allows for the crack propagation more random at the microscopic level. In this paper, PFC2D 5.0 is adopted to investigate the mechanical properties, cracking behavior, and AE characteristics of mudstone under uniaxial compression. The main structure of this paper is as follows: in Section 2, the parallel bond model, Weibull distribution, microparameter calibration, and acoustic emission based on moment tensor are introduced. In Section 3, mechanical properties, cracking behavior, and damage constitutive model of mudstone under uniaxial compression are studied. And conclusions are given in Section 4.
To study the relationship between deformation and strength of rock materials, constructing a damage constitutive model is one of the more basic and effective methods. Therefore, based on the results of laboratory tests or numerical simulations, the introduction of damage variables is to reflect the difference in the degree of internal damage of the rock during the loading processes. There are many methods to define damage variables, such as based on elastic modulus [30], number of cracks [31], and AE characteristics [32]. In our research, the damage variable D is defined by the number of AE events simulated by the theory of moment tensor:where Cn is the total number of AE events of the rock from the original loading point to a specific time and C is the total number of AE events of the rock.
In hydraulic fracturing, fluid is pumped at high pressure into a crack in a rock mass in order to open the crack. Hydraulic fracturing has many applications. In mining, high pressure water is used to break rocks instead of explosives which leave small rock fragments in the air that could damage the lungs of miners [1]. In geothermal reservoirs, in their natural state, the cracks allow only a small flow of water. In order to increase the flow, high pressure water is pumped into the crack network at one borehole and extracted from another borehole [2]. Hydraulic fracturing is also used to enhance the extraction of oil and gas in large underground shale deposits [3].
Spence and Sharp [4] were the first to show that the equations of hydraulic fracture admit a similarity solution. These authors considered the enlargement of a lens shaped crack and a two-dimensional crack by a viscous fluid modelled by lubrication theory. The fluid pressure and the crack shape were connected by a singular integral equation from linear elasticity. The theory was applied to magma-driven propagation of cracks in geophysics [5] [6] [7].
The Cautchy principal value integral in the singular integral equation relating pressure to the crack shape is difficult to analyze. An important simplification was the Perkins-Kern-Nordgren (PKN) approximation in which the normal stress at the fluid-rock interface is proportional to the half-width of the fracture [8] [9]. The PKN approximation puts the differential equations of hydraulic fracture in a form which can be analyzed using the theory of Lie point symmentries and conservation laws.
The scientific literature on hydraulic fracturing is now very large. We will only comment briefly on the application of Lie point symmetries to hydraulic fracturing, because that approach will be used in this paper. Lie point symmetry analysis was first applied to hydraulic fracturing by Fitt, Mason and Moss who considered the propagation of a fracture in impermeable rock [1]. Fareo and Mason analyzed the propagation of a hydraulic fracture in permeable rock with fluid leak-off into the rock mass and investigated the effect of leak-off on the rate of propagation of a fracture [10]. Fareo and Mason also considered a fracture driven by a power law fluid in an impermeable rock mass [11]. Anthonyrajah, Mason and Fareo [12] compared laminar and turbulent fluid driven fractures using the wall shear stress model of Emerman, Turcotte and Spence [6] for the fluid and the PKN approximation instead of the Cautchy principal value integral model to relate the fluid pressure to the crack shape.
The effects of tortuosity on hydraulic fracturing due to asperities or surface roughness at the fluid rock interface and contact regions caused by touching asperities will be investigated in this paper. The hydraulic fracture with tortuosity will be replaced by a symmetric two-dimensional hydraulic fracture without asperities but with a modified Reynolds flow law, which accounts for the effect that the presence of asperities at the fluid rock interface has on the fluid flow, and with a modified crack law, which models the effect of the contact regions on the stress at the fluid-rock interface [2].
In this paper, we model the contact regions by the hyperbolic crack law which was introduced by Goodman [13]. This is motivated by the discussion by Fitt et al. [2] that the hyperbolic crack law is generally considered to be a more realistic model to describe the presence of contact regions (deformations formed by touching asperities) in a fracture than the linear crack law. Although the concept of tortuosity was investigated in [14] [15], the contact regions in these papers were modelled by the linear crack law which was proposed by Pine and Cundall [16] and further discussed by Fitt et al. [2].
In this paper, we aim to solve the problem of the evolution of a hydraulic fracture with tortuosity described by the hyperbolic crack law. The fluid flow in the fracture is described by a modified Reynolds flow law and the fluid pressure and crack shape are related by the PKN approximation. The aim of this work is to derive numerical and approximate analytical solutions for the evolution of the half-width and length of the model symmetric fracture which replaces the fracture with tortuosity. The problem is formulated mathematically and in dimensionless form in Section 2.6.
In Section 2, we present the derivation of the model describing fluid flow in a tortuous hydraulic fracture with contact regions modelled by the hyperbolic crack law. Section 3 outlines the derivation of the group invariant solution of the problem. In Section 4 we consider possible operating conditions at the fracture entry and investigate the existence of analytical solutions. Section 5 describes a numerical solution for the problem. In Section 6, the investigation of the width averaged fluid velocity leads to the derivation of an approximate analytical solution for the problem. In this Section, a comparison of the numerical and the approximate analytical solutions is made. Finally in Section 7 we summarize important findings and conclude the paper. 2ff7e9595c
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