Variational phase-field model based on lower-dimensional interfacial element in FEM framework for investigating fracture behavior in layered rocks

Highlights

• A variational phase-field model is proposed for interfacial fracture in layered rocks.

• The lower-dimensional interfacial element and its tangential derivative describe the phase-field evolution at the interface.

• A quantitative analysis of interfacial fracture is conducted for perfectly-bonded and weakly-bonded interfaces.

• The model successfully captures cracks penetrating the interface or deflecting at the interface.

Abstract

The fracture characteristics of layered shale rock determine the efficiency of shale gas production through hydraulic fracturing. An effective and robust numerical method may play a significant role in predicting the crack propagation path in layered shale. In this study, a lower-dimensional interfacial element and tangential derivative variable are developed to describe the evolution of the phase-field at the interface within the framework of the phase-field approximation. The variational phase-field model is derived based on a new energetic framework. The lower-dimensional interface is discretized together with the rock matrix. A separated coupling strategy is adopted to solve the coupled system, and the equations are solved sequentially during each time step. The model is validated by conducting two 2D classical benchmark tests and a 3D three-point-bending experiment. Further, the presented approach is applied to modeling layered shale failure in 2D notched square shale specimens subjected to tension. The stiff-to-stiff, soft-to-stiff, and stiff-to-soft configurations are designed to investigate the fracture behavior in layered rocks with perfectly-bonded and weakly-bonded interfaces. The results indicate that the proposed method exhibits extreme robustness and can be applied to evaluate the interfacial fracture and crack-interface interaction.

Introduction

Interfacial fractures in layered rocks require special considerable attention in underground rock engineering [1], [2]. For example, in shale gas extraction, shale contains numerous bedding structures owing to its unique composition, geological structure, and diagenetic history. The lower cementation of the bedding plane (interface) is weak and tends to fracture before the shale matrix is damaged. Therefore, predicting the fracture patterns in layered shale is crucial for understanding the formation mechanism of fracture networks when hydraulic fracturing is performed [3]. This requires an effective and robust numerical method that can predict and describe the complex fracture behaviors of the layered rock.

Several numerical methods have been proposed to address the crack propagation problem in rock engineering, such as the extended finite element method [4], cracking particle methods [5], [6], phase-field method [7], and peridynamics [8]. These are commonly discrete or continuous methods, depending on whether the crack surface displacement is continuous. The phase-field method, as a popular method based on continuous assumptions, has gradually attracted the attention of researchers. However, the theory needs to be expanded further to consider the interfacial fracture of layered rocks. The phase-field method originated from the fracture variational model proposed by Francfort and Marigo [9] based on the Griffith fracture theory for quasi-static brittle cracks (i.e., energy-based fracture theory). Subsequently, Bourdin et al. presented a numerical implementation of the fracture variational model [10]. In the past two decades, many researchers have developed the aforementioned fracture variational method, which has gradually evolved into the current mainstream phase-field method. Among these works, the contributions of Miehe et al. [7], [11] resulted in the acceptance of the phase-field method by many researchers. Miehe et al. [7], [11] proposed a thermodynamically consistent framework for the phase-field model of crack propagation in elastic solids, developed the incremental variational principle, and presented a numerical implementation of the multi-field finite element method. Borden et al. developed a phase-field method for dynamic problems based on its variational structure [12]. In the phase-field method, the elastic energy is split into different parts to model different types of cracking mechanisms. The two main energy splitting methods were presented by Amor et al. [13] and Miehe et al. [7]; in these, the compressive fractures are inhibited and only tensile cracks are captured. Additionally, the compressive-shear fractures in rocks cannot be ignored and can be captured by using a new driving force considering cohesion and internal friction in the phased-field framework proposed by Zhou et al. [14].

The phase-field method can be implemented by solving the phase-field diffusion equation based on the traditional finite element method (FEM). Additionally, a physics-informed neural network [15] and deep neural networks [16] have been proposed to solve the phase-field fracture problem. The phase-filed method has significant advantages in simulating crack initiation, propagation, coalescence, and branching as it avoids tedious crack surface tracking. The implementations in FEM-based software, such as ABAQUS [17], [18], [19], [20] and COMSOL [21], accelerate the application of the phase-field method. However, algorithms based on software are often inherently implicit, and an explicit phase-field method has been presented recently to improve convergence [22].

Owing to its variational structure, the phase-field method can be conveniently extended to fractures under multiphysics coupling conditions, such as hydraulic fracturing in porous media [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], thermally-induced cracking [33], [34], [35], [36], fracturing under thermo-hydro-mechanical-chemo coupling [37], [38], [39], and cracking in multi-phase media [40], [41]. Additionally, phase-field models for elasto-plastic-ductile solids and their implementations have been presented [19], [34], [42], [43], [44], [45]. However, the phase-field method has some limitations, such as high computational cost, unclear length scale parameter, and inaccurate determination of the crack tip. Reviewers on the phase-field method can be found in the literature [46], [47].

Recently, phase-field models for modeling interfacial fractures in layered materials have been proposed by some researchers. Verhoosel and de Borst [48] introduced an additional field to consider the displacement jump at the interface. Nguyen et al. [49] proposed a novel phase-field method to simulate interfacial fracture, and the method does not require an additional field, and both matrix and interface fractures are described by smeared discontinuities. Subsequently, they extended their work to simulate an interfacial fracture in the phase-field framework, including a regularized interfacial transition zone[50] and a dimensional-reduced model based on a rigorous asymptotic analysis[51]. Li et al. [52] presented a phase-field method to model the interfacial damage in elastoplastic composites. Zhang et al.[53], [54] implemented a phase-field model for fiber-reinforced composites based on ABAQUS. The fiber/matrix interface debonding and kinking, matrix crack propagation, and delamination between adjacent plies were investigated[54]. Xia et al. [55] presented a phase-field framework with a regularized description of both matrix and interface cracks and extended it to a fully coupled hydro-mechanical framework, in which the jumps at the interfaces between the porous matrix and inclusions were described by using a regularized approximation. Msekh et al.[56] predicted the fracture properties of clay/epoxy nanocomposites with interphase zones using a phase-field model, where the interphase zones are treated as real regions. Hansen-Dörr et al. [57], [58] proposed a phase-field method for adhesive interfacial failure, in which the discrete interface is regularized by a finite length using the finite interface regularization length scale. Zhuang et al.[59] studied the characteristics of hydraulic fracture propagation in naturally-layered porous media using phase-field method. They also mentioned that their research only considered a perfectly bonded interface, which is not always the case in geological settings. In our previous work, we characterized the difference between the parameters of the shale bedding plane and matrix by employing the interpolation function within the framework of the phase field. Although this method can reveal the influence of the bedding plane on rock fracture to a certain extent, it lacks theoretical support and is also smeared about the given width of the bedding plane[32], [60].

To conveniently predict the crack propagation path in layered shale within the framework of the phase-field theory, in this study, we extend the phase-field method based on the discrete fracture seepage theory in porous media to simulate interfacial fracture in layered rocks. A lower-dimensional interfacial element is developed. The interfacial fractures are modeled separately. The energy functional is revised by incorporating the interface-related terms and work performed external loads and then deriving the governing equations. A variational phase-field model considering the subdomain interfaces is proposed to investigate the interfacial fracture in layered rocks. The remainder of this paper is organized as follows. The derivation of the variational phase-field theory for interfacial fracture is provided in Section 2. The numerical implementation based on the FEM in COMSOL Multiphysics is described in Section 3. The model verification is presented in Section 4. Numerical simulations to investigate the sensitivity of the model parameters are detailed in Section 5. Finally, the conclusions drawn are explained in Section 5.