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# Crack surface, Γc, showing local coordinate systems associated with finite elements of the mesh adopted for the solution of the fluid equation. The boundary of the crack surface, ∂Γc, is also shown in the figure.

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This paper presents a coupled hydro-mechanical formulation for the simulation of non-planar three-dimensional hydraulic fractures. Deformation in the rock is modeled using linear elasticity and the lubrication theory is adopted for the fluid flow in the fracture. The governing equations of the fluid flow and elasticity and the subsequent discretiza...

## Contexts in source publication

**Context 1**

... the fluid flow problem can be reduced to a two- dimensional problem. The fluid flow equation on a non-planar surface is derived below using local [coupled˙formulation˙paper -November 5, 2015] n n n − Ω ¯ t t t z ¯ t t t ¯ t t t coordinate systems associated with finite elements of the mesh adopted for the solution of the fluid equation, as illustrated in Figure 3. The fluid is assumed to be Newtonian and incompressible. ...

**Context 2**

... e e 2 (7) Figure 2 shows the schematic of the fluid flow inside a fracture Γ c with fracture opening w and fluid flux q q q. Figure 3 shows examples of local coordinate systems on Γ c and the boundary ∂ Γ c of the fracture surface. Each coordinate system with base vectors {¯ e e e 1 , ¯ e e e 2 } and position vector ¯ x x x = ( ¯ x 1 , ¯ x 2 ) is associated with a finite element of the mesh used for the solution of the fluid equation. ...

**Context 3**

... n n n c is the normal to the boundary of fracture surface, as shown in Figure 3. Flux ¯ q (s) is zero along the crack front. ...

**Context 4**

... matrix is assembled from fluid element stiffness matrices as usual. It is noted that since each fluid element has its own coordinate system, as illustrated in Figure 3, the gradient of the shape functions in Equation (70) is computed using these element systems on an element-by-element basis. However, there is no need to transform the fluid element stiffness matrix or load vector to, e.g., nodal coordinate systems, as typically done with shell finite elements, since we are dealing with a scalar quantity-the fluid pressure. ...

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Multistage hydraulic fracturing is a highly effective method for creating multiple transverse fractures to improve gas and oil reservoir production. It is critical to minimize the fracture spacing while also ensuring transverse propagation of fractures in multi-fractured horizontal wells. In this paper, a 3D fully coupled pore pressure-stress model...

## Citations

... This warrants the need to have a coupled hydromechanical framework that models the stated challenges. Other issues associated with modeling this complex phenomenon include the presence of different layers of rock formation; variation of the in situ confining stress; leakage of hydraulic fluid; effect of temperature and shear on the rheology of the fracture fluid; the movement of the suspended sorted sand within the fracture; etc 5 . As a result, several models for the simulation of hydraulic fracture propagation have been developed to date 6,7,8,9,5,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26 . ...

... Other issues associated with modeling this complex phenomenon include the presence of different layers of rock formation; variation of the in situ confining stress; leakage of hydraulic fluid; effect of temperature and shear on the rheology of the fracture fluid; the movement of the suspended sorted sand within the fracture; etc 5 . As a result, several models for the simulation of hydraulic fracture propagation have been developed to date 6,7,8,9,5,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26 . The injection rate of the fracturing fluid, rock properties, and rheology are the main input parameters for these models, while the geometry, path, and fluid pressure are some of the output of the simulations 27 . ...

Using a robust three-dimensional generalized/eXtended finite element method (G/XFEM) algorithm, this paper presents a comprehensive study on multiple hydraulic fracture propagation and their interactions under different treatment conditions. Aimed at capturing the complex multiphysics behavior, the resulting nonplanar fracture footprints under mixed-mode conditions are simulated using a formulation which couples the solid/rock domain equations and the fluid flow within the fractures/crevices. Due to the small length scale of the fracture process zone relative to the surrounding formation size, as typically encountered in tight shale reservoirs, linear elastic fracture mechanics based on a regularized Irwin criterion is adopted to describe the fractured solid/rock response while a power law is used for the fluid flow by assuming a Newtonian fluid behavior. Equipped with the capability for mesh adaptivity and automatic time step search algorithms, the G/XFEM utilized to discretize the resulting system of coupled nonlinear equations allows the adoption of independent meshes for the background solid domain and the fracture surfaces without any matching/compatibility requirement between the two. Fracture propagation directions are decided based on the modes I, II, and III stress intensity factors that are extracted using the Displacement Correlation Method. The presented model is first applied to a pair of misaligned fractures and then to an array of en échelon fractures for qualitative verification against a literature prediction by the boundary element method and an observed field behavior, respectively. Next, simulation of several sets and configurations of multiple hydraulic fractures, resulting in a total of 23 parametric studies, is carried out to investigate the influences of fracture spacing, injection fluid viscosity, number of fracture clusters, and the type of remote stress conditions.

... The XFEM method has recently been used in studying fracture propagation problems in the context of subsurface porous media, including the cohesive fracture growth [12,44,53], the hydraulic fracture propagation [17,19,[36][37][38]43], the frictional contact on geological faults [15,21,22,[30][31][32][33] and the dynamic rupture processes [10,34,35]. Recently, AES approach has been used in simulating the slip of pre-defined faults in the poroelastic media [11]. ...

... At the same time, the fracture may be closed, opened or extended, all driven by the nearby fluid pressure gradients. This driving mechanism for fracture deformation and propagation is slightly different from those reported in [21,22,[36][37][38]50] where the fracture is driven by the fluid pressure applied on the fracture face, instead of the pressure gradients around the fracture. ...

The assumed enhanced strain (AES) method is developed to simulate cohesive fracture propagation in the partially saturated porous media which includes the solid skeleton and the compressible pore water. The motivation of this research is to build a numerical framework allowing us to more physically investigate the complex coupled processes in the subsurface environment, such as the coupling between the solid skeleton deformation and fluid flow, fracture initiation and propagation driving by fluid flow and the evolution of water saturation and permeability. The detailed formulation describing the permeability enhancement due to fracture opening and volume increase is also presented. The numerical framework is based on the classical Biot’s mixture theory, where fractures can be naturally embedded into the framework with the AES method. The nonlinear discrete equations are derived by the consistent linearization technique and then solved with the Newton’s method. The AES method allows the fracture to propagate inside the elements and can be easily implemented in the standard nonlinear finite element codes. The implementation of the framework is fully verified with several numerical examples in comparison with the previous results published in the literature. The examples demonstrate that the proposed numerical framework is capable of capturing the interactions among the fracture, the bulk rock and the fluid flow. In the last example, we show the potential of the proposed framework in simulating the propagation of fluid-driven fractures.

... However, such model needs an explicit definition of fracture opening. While it can be straightforward to obtain the fracture opening with discrete fracture approaches, e.g., LEFM [12], XFEM [78] and GFEM [55], it is knotty to calculate the fracture-width-like quantities in continuum fracture approaches [129]. ...

We present a novel non-local integral-type damage formulation for hydraulic fracture of poro-viscoelastic media under the framework of irreversible thermodynamics. The poro-viscoelastic material is modeled by a generalized Maxwell model, whose shear modulus is described in terms of Prony series. A bilinear damage law is assumed, which is driven by three equivalent strain invariants. Darcy’s law is employed to describe the fluid flow in the entire domain including the fracture process zone, where the permeability is assumed to be nonlinear and anisotropic. A monolithic two-field (u-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{u}}-p$$\end{document}) mixed finite element method is employed to discretize the coupled hydromechanical system. A Newton–Raphson method is utilized to solve the nonlinear system, and a backward Euler scheme is applied to evolve the system in time. Several numerical examples are presented to investigate the time-dependent deformation response of saturated porous media. In particular, we study the effects of relaxation time and the ratios of anisotropic initial permeability on the strongly coupled processes of the solid deformation, fluid transport and damage evolution of geomaterials. In addition, the different modes of energy dissipation mechanisms including damage and solid viscous response are presented and discussed.

... In the past years, significant effort has been made to develop a robust, accurate, and efficient GFEM to model 3-D hydraulic fracture propagation near a wellbore [22][23][24][25][26]. The method can account for all three fracture modes in a mixed-mode propagation problem [22,27] and it has been optimized to be computationally efficient and robust [25]. ...

... The discretized system is presented in Section 4.2.4. In this article, the fully 3-D GFEM for hydraulic fracturing presented in [22][23][24][25][26] is extended to handle the simultaneous propagation of multiple hydraulic fractures and the coupling of hydraulic fracture and wellbore flows. A modification to the propagation criterion first introduced in [24] is proposed in Section 4.1.2 ...

... A brief summary of the method is presented in this section. Further details on the GFEM for hydraulic fracturing adopted here can be found in [22][23][24][25][26]. ...

In this article, a 3-D methodology for the simulation of hydraulic fracture propagation using the Generalized Finite Element Method (GFEM) is extended for the simultaneous propagation and interaction of multiple hydraulic fractures. A 3-D isotropic elastic material for the rock and Reynolds lubrication theory for the fluid flow in the fractures are assumed. The elastic solid governing equation is discretized in space with a quadratic GFEM and the equation for the flow in the fractures is discretized in space with a quadratic FEM. With the GFEM, the solid mesh does not need to fit the fracture, and accuracy is improved around the fracture front with the use of singular enrichment functions. Discretization error is further controlled by employing mesh adaptivity around the fracture front. The injected fluid partitioning among fractures is computed by modeling the wellbore, where the flow is assumed to be governed by the Hagen-Poiseuille relation. The pressure losses between the wellbore and hydraulic fractures are modeled with the sharp-edged orifice equation and with the use of 1-D connecting elements. A linear FEM is adopted for the spatial discretization of the equation governing the flow in the well-bore and the connection between wellbore and hydraulic fracture. A propagation criterion based on a regularization of Irwin's criterion is adopted and a methodology to automatically estimate the time step that leads to the propagation of fractures based on linear interpolation/extrapolation is presented. Three verification problems are solved and compared with results from the literature. A problem with initial fractures not aligned with the in situ stresses is analyzed. Complex final fracture geometries are observed due to this misalignment.

... Fracture modeling in poroelastic media has been attempted using different approaches. First attempts were based on Linear Elastic Fracture Mechanics (LEFM) [15,16], and later research efforts focused on generalized/Extended Finite Element (G/XFEM) [17][18][19] and cohesive zone elements [20,21]. These approaches define the domain as elastic or poroelastic, and use an explicit fluid-filled crack definition (Poiseuille's equation) to represent fluid-driven fractures. ...

... These approaches define the domain as elastic or poroelastic, and use an explicit fluid-filled crack definition (Poiseuille's equation) to represent fluid-driven fractures. However, extension of LEFM models to represent material non-linear response mechanisms is challenging and tracking crack propagation direction and capturing crack coalescence requires significant model development [18,[22][23][24]. Moreover, modeling the fluid inside a discrete crack leads to discontinuity in the characterization of fluid flow, or pressure-gradient discontinuity [25], which is a diffusive continuum process in nature. ...

Recent experiments and physical evidence show that fractured porous media feature cracks and fluid capillary networks at various scales. We present a multi-physics macro-scale model that can distinguish between the mechanics and transport interactions. The porous media is represented by a poroelastic domain incorporating non-local damage and non-local transport. The evolution of each of these processes is governed by a unique length scale and driving force, which allows for better flexibility in modeling hydraulic-deformation network systems. For consistency the governing equations of the non-local multi-physics problem are derived from thermodynamics principles. Hence, a four-field () mixed finite element formulation is developed. The non-linear system of equations is linearized and solved using Newton’s method and a backward Euler scheme is used to evolve the system in time, for which a consistent Jacobian matrix and residual vector are derived analytically. Two benchmark examples are investigated: hydraulic fracturing of rocks and soil consolidation. The numerical examples show the viability of this model, and how the variation of the two length scales and damage parameters can be used to describe different physical phenomena.

... The smaller the cluster spacing, the stronger the "stress shadow" effect between fractures, and the greater the influence on fracture propagation and fracture width [18][19][20][21][22][23][24][25][26][27]. In addition, some scholars established a finite element model for directional well fracturing based on the basic finite element theory and studied the propagation morphology of single fracture under uneven confining pressure [28,29]. Although many theoretical and experimental studies have been carried out on hydraulic fracture initiation and propagation in directional wells and horizontal wells, most of the physical simulation experiments on fracture initiation and propagation in directional wells were carried out under the condition of single-fracture or multifracture fracturing in a single stage, and there were few studies on the fracture propagation of multistage fracturing in directional wells. ...

Due to the limited space of offshore platform, it is unable to implement large-scale multistage hydraulic fracturing for the horizontal well in Lufeng offshore oilfield. Thus, multistage hydraulic fracturing technology in directional well was researched essentially to solve this problem. Modeling of fracture propagation during multistage fracturing in the directional and horizontal wells in artificial cores was carried out based on a true triaxial hydraulic fracturing simulation experiment system. The effects of horizontal stress difference, stage spacing, perforation depth, and well deviation angle on multifracture propagation were investigated in detail. Through the comparative analysis of the characteristics of postfrac rock and pressure curves, the following conclusions were obtained: (1) multistage fracturing in horizontal wells is conducive to create multiple transverse fractures. Under relatively high horizontal stress difference coefficient (1.0) and small stage spacing conditions, fractures tend to deflect and merge due to the strong stress interference among multiple stages. As a consequence, the initiation pressure for the subsequent stages increases by more than 8%, whereas in large stage spacing conditions, the interference is relatively lower, resulting in the relatively straight fractures. (2) Deepening perforation holes can reduce the initiation pressure and reduce the stress interference among stages. (3) When the projection trace of directional wellbore on horizontal plane is consistent with the direction of the minimum horizontal principal stress, fractures intersecting the wellbore obliquely are easily formed by multistage fracturing. With the decrease of well deviation angle, the angle between fracture surface and wellbore axis decreases, which is not conducive to the uniform distribution of multiple fractures. (4) When there is a certain angle between the projection trace of directional wellbore on horizontal plane and the direction of minimum horizontal principal stress, the growth of multiple fractures is extremely ununiform and the fracture paths are obviously tortuous.
1. Introduction
With the development of unconventional oil and gas reservoirs and the advancement of technology, directional wells, horizontal wells, and multistage fracturing technology are combined to increase the drainage area of the reservoir, hence to improve oil recovery and economic benefits. Currently, multistage fracturing is the main stimulation technology for unconventional resources; the principle of which is to enlarge the oil and gas discharge area by forming dense transverse fractures that are perpendicular to the wellbore [1–4]. However, due to the limited area of offshore platforms, high equipment operating costs, and high operational safety risks, it is difficult to apply mature onshore staged fracturing technologies to offshore oilfields. Meanwhile, the development of offshore horizontal staged fracturing technologies is far behind that of onshore oilfields [5]. In order to adapt to the characteristics of offshore platforms and treat more production zones at the same time, the research on multistage fracturing technology in directional wells is of great importance.
Series of theoretical studies on hydraulic fracture initiation and propagation in directional wells have been conducted [6–10]. The models of stress distribution near the wellbore of directional wells under different conditions have been established, and formulas of fracture initiation pressure and fracture initiation angle have been deduced. Zhou et al. [11] proposed the prediction model of fracture initiation by establishing the distribution model of stress field in the surrounding rock of directional wellbore and pointed out that the initiation mode of hydraulic fractures was affected by the azimuth of wellbore, in situ stress difference, and well deviation angle. Since the hypotheses of theoretical researches often somewhat differ from the actual conditions and studies of fracture propagation morphology are usually based on simplified two-dimensional or three-dimensional models, the results obtained have limitations to a certain degree. In addition to the theoretical model research, the physical simulation experiment is also an important means to study fracture initiation and propagation. Physical hydraulic fracturing simulation experiments of directional wells conducted by domestic and foreign scholars showed [12–17] that the controlling factors of hydraulic fracture initiation in directional wells mainly include well deviation angle, borehole azimuth, horizontal stress difference, and perforation parameters, and the fracture propagation is easy to deflect to produce complex forms. However, the earlier experiments mainly focused on the study of a single fracture in directional wells and did not take into account the interaction of simultaneous propagation of multiple fractures in directional wells. Many studies have shown that multifracture propagation tends to be unbalanced [4] due to the influence of (1) reservoir characteristics such as natural fractures, in situ stress distribution, and rock mechanical properties, (2) well completion factors such as stage spacing and cluster spacing, (3) perforation parameters, and (4) stress interference between fractures. The smaller the cluster spacing, the stronger the “stress shadow” effect between fractures, and the greater the influence on fracture propagation and fracture width [18–27]. In addition, some scholars established a finite element model for directional well fracturing based on the basic finite element theory and studied the propagation morphology of single fracture under uneven confining pressure [28, 29]. Although many theoretical and experimental studies have been carried out on hydraulic fracture initiation and propagation in directional wells and horizontal wells, most of the physical simulation experiments on fracture initiation and propagation in directional wells were carried out under the condition of single-fracture or multifracture fracturing in a single stage, and there were few studies on the fracture propagation of multistage fracturing in directional wells. Therefore, the propagation morphology of fractures formed in multistage fracturing under stress interference among stages in directional wells was not taken into account, and treating parameter optimization of multistage fracturing in directional wells still lacks direct experimental evidence.
To shed a light on the problem mentioned above, this paper presents the physical simulation research of staged fracturing and multifracture propagation in horizontal and directional wells using the true triaxial hydraulic fracturing physical simulation experiment system. Then, we compared and analyzed the influence factors that affect multifracture initiation and propagation of two types of wells. The effects of horizontal stress difference, perforation depth, stage spacing, well deviation angle, and wellbore azimuth (the angle between projection of wellbore axis in the horizontal plane and the direction of maximum horizontal principal stress) on fracture propagation morphology and pressure curve characteristics of multistage fracturing were considered.
2. Experimental Method
2.1. Sample Preparation
The research area is located in the south of Lufeng Sag of Zhu I Depression, Pearl River Mouth Basin, South China Sea. The facies of research formation are shallow shore lake and braided river delta, with buried depth of 3563-4272 m. The reservoir varies greatly in vertical and horizontal directions, including silty mudstone, siltstone, and fine sandstone, with strong heterogeneity. The reservoir rocks have elastic modulus of 21.3–34.4 GPa, Poisson’s ratio of 0.18–0.31, tensile strength of 2.1–4.6 MPa, maximum horizontal principal stress of 78.4–86.6 MPa, and minimum horizontal principal stress of 64.1–70.3 MPa.
The experimental samples were cement cubes (G grade cement, quartz sand, and water in a 3 : 1 : 1 ratio) with a side length of 30 cm (Figure 1(a)). The physical properties were similar to those of the reservoir lithology. The wellbores were prefabricated in the cement. Each wellbore was divided into three sections (one perforation cluster in each section and four perforations in each cluster) to facilitate staged fracturing in horizontal and directional wells [4]. Since the size of the core samples and wellbores was limited and the effect of stress interference among stages on fracture propagation morphology can be reflected by the model with single cluster in each stage, the design of single cluster in each stage was adopted. Each wellbore was composed of three parts: outer casing, inner wellbore, and fluid injection pipelines. The inner wellbore was a steel pipe with an outer diameter of 1.5 cm, an inner diameter of 0.8 cm, and a length of 20.0 cm. The outer casing was a steel pipe with an outer diameter of 2.0 cm, an inner diameter of 1.6 cm, and a length of 20.0 cm, and a certain number of thread grooves were processed on its outer surface to strengthen the bond between the casing and cement. Four drain holes with a diameter of 3 mm were drilled on the outer casing in each section. A steel tube with the same diameter as the holes was welded perpendicularly at the position of each hole to simulate the perforation process. The annulus of each stage was sealed off with a gasket to simulate stage packer. Each stage in the inner wellbore was sealed by a steel plate and linked with an injection line which connected to an individual intermediate vessel. A six-way valve was connected between the injection lines and three intermediate vessels to control the injection. The staged fracturing can be achieved by operating the valves to have one injection line and one intermediate vessel connected at one time, while keeping the others closed.
(a) Concrete core sample

... It is also noted that even though Discontinuous Galerkin (DG) methods have been shown to be effective for wave propagation problems [30,31], the authors are unaware of a class of DG methods that can adopt meshes that do not fit fracture surfaces. The proposed methodology leverages GFEMs and algorithms developed for the simulation of 3-D quasi-static propagation of hydraulic fractures [32][33][34][35][36]. The rock is assumed to be a 3-D isotropic linear elastic material governed by the elastic-wave equation, and the fluid is assumed to be Newtonian and governed by an approximation to the compressible Navier-Stokes equations with lubrication theory. ...

... Equation (2)). This term is discretized in time with the α-method as in [33] ...

... However, for simulating Krauklis waves, a 2-D mesh at the fracture location is required to discretize the domain of the fluid-filled fracture governing equations. Gupta and Duarte [33] present a robust and efficient procedure to automatically generate this 2-D mesh based on the intersection of the 3-D solid elements with the fracture surface in the context of hydraulic fracture propagation. Here is a brief description of the procedure to generate the 2-D fluid-filled fracture mesh (cf. ...

This paper proposes a novel methodology for 3-D simulations of pressure waves propagating in fluid-filled fractures in an elastic body -- the so-called Krauklis waves. The problem is governed by an approximation to the compressible Navier-Stokes equations for the viscous fluid in the fracture cavity coupled to the elastic-wave equation in the surrounding solid. The solid is assumed to be isotropic linear elastic and the fluid to be Newtonian. The elastic-wave equation is discretized in space with a quadratic Generalized Finite Element Method (GFEM) and in time by Newmark's method while the Krauklis waves equations are discretized in space by a quadratic standard FEM and in time by the α-method. A monolithic solver is adopted for the coupled problem and is numerically shown to be stable for the adopted approximation spaces. The GFEM is particularly appealing for the discretization of 3-D fractures since it does not require meshes fitting their geometry. Furthermore, analytical asymptotic solutions are used to enrich the GFEM approximation spaces, increasing the accuracy of the method. Mesh adaptivity around the fracture fronts is employed to further decrease the discretization error while controlling the computational time. The methodology is verified with analytical solutions and compared with experimental data. Different fracture geometries are investigated to demonstrate the complex 3-D effects of the physical phenomenon and the robustness of the proposed GFEM methodology.

... The extended finite element method (XFEM), proposed in [17] and [18] as a technique to allow arbitrary crack propagation without the need of re-meshing, has successfully been applied in the framework of fluid-driven fracture propagation in 2D benchmarks either in combination with CZM [19,20] or with classical mixed-mode propagation criteria based on the stress field at the crack tip [21]. The extremely challenging implementation of XFEM in the case of a pressurized crack propagating in 3D geometries was done in references [22][23][24]. XFEM methods have been verified against analytical solutions for plane-strain [19] and penny-shaped [24] impermeable cracks. The main shortcoming of the XFEM approach lies in cases where there is merging or branching of several cracks [25]. ...

The propagation of cracks driven by a pressurized fluid emerges in several areas of engineering, including structural, geotechnical, and petroleum engineering. In this paper, we present a robust numerical framework to simulate fluid-driven fracture propagation that addresses the challenges emerging in the simulation of this complex coupled nonlinear hydro-mechanical response. We observe that the numerical difficulties stem from the strong nonlinearities present in the fluid equations as well as those associated with crack propagation, from the quasi-static nature of the problem, and from the a priori unknown and potentially intricate crack geometries that may arise. An additional challenge is the need for large scale simulation owing to the mesh resolution requirements and the expected 3D character of the problem in practical applications. To address these challenges we model crack propagation with a high-order hybrid discontinuous Galerkin/cohesive zone model framework, which has proven massive scalability properties, and we model the lubrication flow inside the propagating cracks using continuous finite elements, furnishing a fully-coupled discretization of the solid and fluid equations. We find that a conventional Newton-Raphson solution algorithm is robust even in the presence of crack propagation. The parallel approach for solving the linearized coupled problem consists of standard iterative solvers based on domain decomposition. The resulting computational approach provides the ability to conduct highly-resolved and quasi-static simulations of fluid-driven fracture propagation with unspecified crack path. We conduct a series of numerical tests to verify the computational framework against known analytical solutions in the toughness and viscosity dominated regimes and we demonstrate its performance in terms of robustness and parallel scalability, enabling simulations of several million degrees of freedom on hundreds of processors.

... Several numerical algorithms have been designed to concurrently track two independent moving boundaries, following the evolution of the lag in time: the singular dislocation solutions, either for the case of semi-infinite crack propagating at constant velocity [30,39,35] or for the plane strain case [40]; finite elements [28,41,42]; XFEM [43,44,45,46,47,48]. The present work aims at establishing a set of governing equations that result in a novel numerical scheme, capable to accurately describe the evolution of a crack filled by a viscous Newtonian fluid in an infinite impermeable elastic medium. ...

Propagation of a fluid-driven crack in an impermeable linear elastic medium under axis-symmetric conditions is investigated in the present work. The fluid exerting the pressure inside the crack is an incompressible Newtonian one and its front is allowed to lag behind the propagating fracture tip. The tip cavity is considered as filled by fluid vapors under constant pressure having a negligible value with respect to the far field confining stress. A novel algorithm is here presented, which is capable of tracking the evolution of both the fluid and the fracture fronts. Particularly, the fracture tracking is grounded on a recent viscous regularization of the quasi-static crack propagation problem as a standard dissipative system. It allows a simple and effective approximation of the fracture front velocity by imposing Griffith's criterion at every propagation step. Furthermore, for each new fracture configuration, a non linear system of integro-differential equations has to be solved. It arises from the non local elastic relationship existing between the crack opening and the fluid pressure, together with the non linear lubrication equation governing the flow of the fluid inside the fracture.

... Many theoretical and computational methods have been developed in recent decades to model hydraulic fracture. Important work includes analytical methods (PKN model [6] and Palmer's model [7]), the displacement discontinuity method (DDM) [8][9][10][11], cohesive zone methods (CZM) [12][13][14][15], the extended finite element method (XFEM) [16][17][18][19] and the generalized finite element method (GFEM) [20][21][22], continuum damage mechanics (CDM) [23][24][25], and phase field methods (PFM) [5,[26][27][28][29]. ...

We present a novel phase field method for modeling hydraulic fracture propagation in poroelastic media. In this approach, a new phase field evolution equation is derived to account for damage dependent poro-elastic parameters (Biot’s coefficient, Biot’s modulus and porosity). The fluid flow obeys Darcy’s seepage law in the entire domain including the damage zone, where the rock permeability is assumed to be anisotropic, following the maximum principal strain. The fully coupled problem is solved by a staggered scheme in which the mechanical equilibrium and fluid flow equations are linearized and solved using a Newton–Raphson(NR) method. Several numerical results are presented to investigate the effectiveness of the proposed formulation. First, stability and convergence of the method are verified on a set of benchmark problems considering different time steps and mesh sizes. Second, it is shown that if the poroelastic parameters are kept constant and do not change with the phase field parameter, i.e. reducing to standard phase field approaches in the literature, the model will tend to underestimate the fracture length and overestimate the pore pressure. Finally, we study the interaction of a propagating hydraulic fracture in porous media with inclined natural fractures, and simulate the hydraulic fracture propagation with different perforation phase angle.