The Extended Finite Element Method (XFEM) approach is applied to the coupled problem of fluid flow, solid deformation, and fracture propagation. The XFEM model description of hydraulic fracture propagation is part of a joint project in which the developed numerical model will be verified against large-scale laboratory experiments. XFEM forms an important basis towards future combination with heat and mass transport simulators and extension to more complex fracture systems. The crack is described implicitly using three level-sets to evaluate enrichment functions. Additionally, an explicit crack representation is used to update the crack during propagation. The level-set functions are computed exactly from the explicit representation. This explicit/implicit representation is applied to a fluid-filled crack in an impermeable, elastic solid and compared to the early-time solution of a plane-strain hydraulic fracture problem with a fluid lag.

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The Extended Finite Element Method (XFEM) with an implicit-explicit crack description is used to solve a plane-strain hydraulic fracture problem. In its typical form, a hydraulic fracture problem combines the physical processes of fluid flow, solid deformation, and fracture propagation. The crack is described implicitly using three level-sets to evaluate enrichment functions and to locate the enriched region. Additionally, an explicit crack description is used to update the crack during propagation. This approach is applied to a partially fluid-filled crack in an semi-infinite elastic solid without leak off and compared to the reference solution given in the literature. The results for the crack opening, pressure and crack length for various values of the fluid fraction are in good agreement with the analytical solutions. Problem FormulationHydraulic fracture propagation is based on at least three physical processes. A fluid flow within the crack imposes a pressure load on its surfaces and causes the crack to grow. The following assumptions are usually made for the hydraulic fracture model [1]: i) the fluid flow is governed by the lubrication theory, ii) solid deformation is modeled using the theory of linear elasticity, and iii) the propagation criterion is given by the conventional energy-release-rate approach of linear elastic fracture mechanics (LEFM). The cracked medium is modeled as a linear elastic solid. The fracturing fluid with dynamic viscosity µ is modeled as laminar flow between two parallel plates with a constant injection rate Q 0 . With the fluid flux q = − w 3 12µ ∇p, the Reynolds (lubrication) equation is then given byand describes the conservation of the fluid mass for an incompressible fluid. For a fracture propagating in an impermeable solid, the leak-off q l is negligible and, therefore, set to zero. It is assumed that a fluid lag develops between the fluid front and the crack tip. However, for reasons of simplicity the lag size is not part of the solution. The XFEM formulation with an explicit-implicit crack description used in this work is based on the work done by [6]. The enriched approximation of the displacements is stated as follows:The first term on the right hand side describes the classical FEM-approximation with continuous shape functions N i (x) and nodal unknowns u i . The discontinuity in the displacement field across the crack path and the singular behavior at the tip is accounted for by step-functions Ψ step and a set of crack-tip enrichment functions Ψ m tip (r, θ), respectively. Numerical resultsThe numerical results for the crack opening and pressure distribution at the wellbore of a plane-strain hydraulic fracture problem are compared to the similarity solution with a fluid lag.The results are scaled to dimensionless quantities. For a detailed description of the scaling for the pressure Π, the opening Ω and the crack length γ see the original publication [13]. Figures 1(a)-(c) show good agreement of the similarity solution for various values of the fluid fraction ξ f = ...

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