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SUMMARYThis paper describes a fully coupled finite element/finite volume approach for simulating field-scale hydraulically driven fractures in three dimensions, using massively parallel computing platforms. The proposed method is capable of capturing realistic representations of local heterogeneities, layering and natural fracture networks in a reservoir. A detailed description of the numerical implementation is provided, along with numerical studies comparing the model with both analytical solutions and experimental results. The results demonstrate the effectiveness of the proposed method for modeling large-scale problems involving hydraulically driven fractures in three dimensions.

SUMMARYWe develop both stable and stabilized methods for imposing Dirichlet constraints on embedded, threedimensional surfaces in finite elements. The stable method makes use of the vital vertex algorithm to develop a stable space for the Lagrange multipliers together with a novel discontinuous set of basis functions for the multiplier field. The stabilized method, on the other hand, follows a Nitsche type variational approach for three-dimensional surfaces. Algorithmic and implementational details of both methods are provided. Several three-dimensional benchmark problems are studied to compare and contrast the accuracy of the two approaches. The results indicate that both methods yield optimal rates of convergence in various quantities of interest, with the primary differences being in the surface flux. The utility of the domain integral for extracting accurate surface fluxes is demonstrated for both techniques.

The extended finite element method (X-FEM) has proven to be an accurate, robust method for solving embedded interface problems. With a few exceptions, the X-FEM has mostly been used in conjunction with piecewise-linear shape functions and an associated piecewise-linear geometrical representation of interfaces. In the current work, the use of spline-based finite elements is examined along with a Nitsche technique for enforcing constraints on an embedded interface. To obtain optimal rates of convergence, we employ a hierarchical local refinement approach to improve the geometrical representation of curved interfaces. We further propose a novel weighting for the interfacial consistency terms arising in the Nitsche variational form with B-splines. A qualitative dependence between the weights and the stabilization parameters is established with additional element level eigenvalue calculations. An important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of large heterogeneities as well as elements with arbitrarily small volume fractions. We demonstrate the accuracy and robustness of the proposed method through several numerical examples. 677 of accuracy in the geometric representation of the interface becomes important when higher-order rates of convergence are sought with these methods. To improve the geometrical representation of the curved interface, the approximation can be classified as falling into one of two categories, p-refinement and h-refinement. The p-refinement approach is to use higher-order interpolation for the interface. Cheng and Fries [4] used a higher-order representation of the level set to describe the interface with higher-order accuracy. Kästner et al.[5] employed a similar idea and further improved the compatibility between the level set representation and the integration sub-domains by computing the level set values at local elements. Higher-order splines were used by Benowitz and Waisman [6] to interpolate the arbitrary shape of inclusions. The development of NURBS-based sub-elements for the representation of curved material interfaces can be found in Haasemann et al. [7]. These issues have also been examined recently in the context of discontinuous Galerkin method for interface problems. In particular, Huynh et al. [8] proposed the use of superparametric elements at the interface to recover optimal rates of convergence.The h-refinement approach is to locally refine the mesh in the vicinity of the interface. The level set function is interpolated by means of finite element shape functions, which need not necessarily be the same with the ones used for the finite element approximation. This allows one to introduce an adapted mesh to describe the geometry while keeping a higher-order approximation of field quantities on a uniform coarse mesh. Thanks to this approach, Dréau et al. [9] and Legrainet al. [10] obtained optimal rates of convergence with higher-order X-FEM for the case of free surfaces and material interfaces with c...

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In this work, we present the application of a fully coupled hydro‐mechanical method to investigate the effect of fracture heterogeneity on fluid flow through fractures at the laboratory scale. Experimental and numerical studies of fracture closure behavior in the presence of heterogeneous mechanical and hydraulic properties are presented. We compare the results of two sets of laboratory experiments on granodiorite specimens against numerical simulations in order to investigate the mechanical fracture closure and the hydro‐mechanical effects, respectively. The model captures fracture closure behavior and predicts a nonlinear increase in fluid injection pressure with loading. Results from this study indicate that the heterogeneous aperture distributions measured for experiment specimens can be used as model input for a local cubic law model in a heterogeneous fracture to capture fracture closure behavior and corresponding fluid pressure response.

We investigate various strategies to enforce the kinematics at an embedded interface for transient problems within the extended finite element method. In particular, we focus on explicit time integration of the semi-discrete equations of motion and extend both dual and primal variational frameworks for constraint enforcement to a transient regime. We reiterate the incompatibility of the dual formulation with purely explicit time integration and the severe restrictions placed by the Courant-Friedrichs-Levy condition on primal formulations. We propose an alternate, consistent formulation for the primal method and derive an estimate for the stabilization parameter, which is more amenable in an explicit dynamics framework. Importantly, the use of the new estimate circumvents the need for any tolerances as an interface approaches an element boundary. We also show that with interfacial constraints, existing mass lumping schemes can lead to prohibitively small critical time steps. Accordingly, we propose a mass lumping procedure, which provides a more favorable estimate. These techniques are then demonstrated on several benchmark numerical examples, where we compare and contrast the accuracy of the primal methods against the dual methods in enforcing the constraints. EMBEDDED CONSTRAINTS IN EXPLICIT DYNAMICS 207 simulating perfect or frictional contact, as well as in phase transformation and solidification problems [4,5]. The key challenge lies in developing a method that is easy to implement, and which preserves optimal rates of convergence in both bulk and interfacial fields.A commonly used approach is to enforce the constraints weakly by building them into the variational statement of the problem through the use of Lagrange multipliers or penalty methods. Unfortunately, as observed in the classical mixed finite element literature, choosing a suitable basis for interpolating the Lagrange multipliers is a non-trivial task [6,7]. The most convenient choice of the basis functions for interpolating the Lagrange multipliers violates the Ladyzhenskaya-Babuska-Brezzi (LBB) condition, resulting in an overly constrained system, which manifests as artificial oscillations in the multiplier field [8]. Penalty methods, on the other hand, result in suboptimal convergence rates because they break the variational consistency of the method [9].An inf-sup stable approximation can be constructed by methodically coarsening the approximation space for the interfacial field with respect to the bulk field. This approach was pursued by researchers working on the fictitious domain methods (see Glowinski et al. [10], Burman and Hansbo [11]) as well as the finite difference immersed interface methods of Bedrossian et al. [12]. In a similar vein, for enriched approximations with embedded interfaces, Moës et al. [13] developed a coarsening strategy for the X-FEM and Kim et al. [14] used a mortaring approach with a distinctly coarser discretization for the interfacial multipliers. It is noteworthy that the extension of these algorithms to three d...

We develop a local, implicit crack tracking approach to propagate embedded failure surfaces in three-dimensions. We build on the global crack-tracking strategy of Oliver et al. (Int. J. Numer. Anal. Meth. Geomech., 2004; 28: 609-632) that tracks all potential failure surfaces in a problem at once by solving a Laplace equation with anisotropic conductivity. We discuss important modifications to this algorithm with a particular emphasis on the effect of the Dirichlet boundary conditions for the Laplace equation on the resultant crack path. Algorithmic and implementational details of the proposed method are provided. Finally, several three-dimensional benchmark problems are studied and results are compared with available literature. The results indicate that the proposed method addresses pathological cases, exhibits better behavior in the presence of closely interacting fractures, and provides a viable strategy to robustly evolve embedded failure surfaces in 3D.