Efficient co‐solution of time step size and independent state in simulations of fluid‐driven fracture propagation with embedded meshes

Abstract

We present an efficient time-continuation scheme for fluid-driven fracture propagation problems in the extended finite element method framework. The approach applies a monolithic solution strategy to a fully coupled and implicit approximation of hydro-mechanical systems in conjunction with simultaneous linear elastic propagation of multiple fractures. At the end of each time step, the process ensures that the weakest fracture tip is in an equilibrium propagation regime. Furthermore, the solution process provides an initialization procedure for the newly created fracture spaces and an a priori estimate of the stress intensity factor growth rate, improving simulation robustness, and efficiency.The solution process is validated using the Kristianovich-Geertsma-de Klerk analytical solution under the toughness-and viscosity-dominated regimes. It is also extended to and demonstrated on problems with multiple fractures undergoing simultaneous propagation with stress shadow interactions. Numerical examples demonstrate that the solution process can reduce the required computational cost by one order of magnitude compared to other existing methods.

Spatial and temporal constraints of the cohesive modeling: A unified criterion for fluid‐driven fracture

“…And for $$$ \kappa \gg 1 $$$, the fracture propagating under the toughness‐dominated regime and the opening of the fracture can be written as 78 : $$$$ {w}_{\mathrm{K}}\left(x,t\right)=0.6828{\left(\frac{K^{\prime 2}{Q}_0t}{E^{\prime 2}}\right)}^{\frac{1}{3}}{\left(1+\frac{x}{l_{\mathrm{K}}}\right)}^{\frac{1}{2}}{\left(1-\frac{x}{l_{\mathrm{K}}}\right)}^{\frac{1}{2}}, $$$$ $$$$ {l}_{\mathrm{K}}(t)=0.9324{\left(\frac{E^{\prime }{Q}_0t}{K^{\prime }}\right)}^{\frac{2}{3}}. $$$$ …”

“…And for 𝜅 ≫ 1, the fracture propagating under the toughness-dominated regime and the opening of the fracture can be written as 78 :…”

“…In this regard, to overcome this challenge, Hageman and Borst 77 proposed an arc‐length‐like method to determine the time step size during fracture propagation. Ren and Younis 78 presented a time‐continuation scheme for the XFEM to reduce the computational cost. Both of them determined the time step size by the iteration process during the solving procedure, where the fracture propagation distance is fixed, and the constraint for fracture propagation is based on the LEFM.…”

Automatic Time Step Control to Resolve Hydromechanically Driven Fault Reactivation, Spontaneous Nucleation, and Seismic Arrest

“…This choice of mixed discretization facilitates the resolution of sharp fluid saturation fronts while incorporating their effects on poromechanical behavior in a stable manner. While our base model accommodates fracture segment intersections and fracture propagation , 2022, these effects are disabled in the present work. The first extension to the base model implements a numerically exact treatment of fracture contact conditions by incorporating the normal and tangential traction and stick-slip behavior.…”

“…The quasi‐static model couples an extended finite element method (XFEM) (Khoei, 2014; Khoei & Nikbakht, 2007) for mechanics with an embedded discrete fracture and matrix (EDFM) (Jiang & Younis, 2016, 2017; Li & Lee, 2008; Moinfar et al., 2013) approximation for multiphase fluid flow within the matrix and fractures. Although the original quasi‐static model resolves fluid‐driven fracture propagation under the LEFM hypothesis (Ren & Younis, 2021, 2022), this work disregards these effects. The model is extended to incorporate inertial mechanics using a stable and second‐order fully implicit Newmark temporal discretization approximation.…”