Steadily moving semi-infinite fracture in plane poroelasticity
We present a boundary integral formulation for steadily propagating semi-infinite plane strain tensile and shear fractures in poroelastic media. By combining fundamental solutions of plane strain poroelasticity for an instantaneous fluid source and instantaneous edge dislocations (normal and slip modes) with temporal and spatial superposition principles, we derive boundary integral equations for steadily moving fractures under the adopted hydraulic boundary conditions. These equations relate the tractions (normal and shear stresses) and the pore fluid pressure on the fracture surfaces to the fracture opening, slip, and the fluid displacement function. Assuming prescribed traction and pore fluid pressure profiles, we develop a numerical methodology to solve the governing equations for fracture opening, slip, and the fluid displacement function. The formulation is systematically verified on several relevant problems, including a tensile fracture with exponential normal loading, a stress-free tensile fracture with an imposed exponential pore fluid pressure, and a shear fracture under uniform shear loading over a finite region, demonstrating excellent agreement with analytical and semi-analytical solutions. The resulting boundary integral framework provides an accurate and efficient tool for analyzing semi-infinite steadily propagating cracks in permeable poroelastic media. By supplementing the formulation with appropriate closure relations and additional physics, such as lubrication flow in hydraulic fractures or frictional strength evolution in shear fractures, it can be used to investigate a broad range of coupled fracture-fluid problems. The approach may also be adapted to other classes of elasto-diffusive problems by modifying the underlying physical parameters.