--> Numerical Simulations of Hydraulic Fracture Propagation — A Coupled Eulerian-Lagrangian Approach

# Numerical Simulations of Hydraulic Fracture Propagation — A Coupled Eulerian-Lagrangian Approach

## Abstract

Current numerical simulations of hydraulic fracturing do a poor job of predicting how fracture networks propagate during hydrofrac operations. These simulations can propagate fluid filled cracks in 3D domains using poroelastic governing equations, a realistic, anisotropic, distribution of material properties, initial ambient stress and fluid pressure conditions that include geostatic and tectonic loads, and time dependent fluid pressure loading. They are incapable of modeling branching fluid filled cracks. The relative magnitude of the differential stress controls the direction and morphology of fracture propagation-as the differential stress magnitude diminishes, fracture orientations become random and favor a branching morphology. This study utilizes a Coupled Eulerian-Lagrangian (CEL) approach to simulate hydrofrac propagation using a method that allows branching fractures. A CEL formulation is a Finite Element Method (FEM) technique that has three fundamental components, an Eulerian FEM (EFEM), which models the fluid, a Lagrangian FEM (LFEM), which models the solid, and general contact specifications to couple the two FEMs. In the EFEM, fluid, driven by fluid pressure gradients resulting from an injection source, is allowed to move through a fixed mesh. The LFEM has a deformable mesh and can track relatively small deformation and stress in elastic domains. Distributions of material properties can be propagated throughout domain if available. The general contact specifications govern the coupling between the two FEMs, satisfying quasi-static equilibrium over linear piece-wise surfaces normal to the fluid and bound by Lagrangian elements collocated with Eulerian elements having partial saturation. Both FEMs occupy the same space and contain appropriate material properties and initial, boundary, and loading conditions. The LFEM contains a cavity to simulate the injection point, and fluids within the EFEM are initially restricted to the cavity. The system is loaded to achieve the desired geostatic equilibrium and fluid flux is applied to the saturated zone of the EFEM. Fluid pressure increases until it exceeds the strength of some point of the chamber wall, which ruptures, introducing a fracture. The CEL analysis remeshes the LFEM to account for the crack, fluid flows into the fracture, and new coupling interfaces are created. With continuing pressurization the fracture propagates according to the time dependent stress field and specified rock strength.