Simulation of Wave Propagation Along Fluid-Filled Cracks Using High-Order Summation-by-Parts Operators and Implicit-Explicit Time Stepping

We present an efficient, implicit-explicit numerical method for wave propagation in solids containing fluid-filled cracks, motivated by applications in geophysical imaging of fractured oil/gas reservoirs and aquifers, volcanology, and mechanical engineering. We couple the elastic wave equation in the solid to an approximation of the linearized, compressible Navier–Stokes equations in curved and possibly branching cracks. The approximate fluid model, similar to the widely used lubrication model but accounting for fluid inertia and compressibility, exploits the narrowness of the crack relative to wavelengths of interest. The governing equations are spatially discretized using high-order summation-by-parts finite difference operators and the fluid-solid coupling conditions are weakly enforced, leading to a provably stable scheme. Stiffness of the semidiscrete equations can arise from the enforcement of coupling conditions, fluid compressibility, and diffusion operators required to capture viscous boundary layers near the crack walls. An implicit-explicit Runge–Kutta scheme is used for time stepping, and the entire system of equations can be advanced in time with high-order accuracy using the maximum stable time step determined solely by the standard CFL restriction for wave propagation, irrespective of the crack geometry and fluid viscosity. The fluid approximation leads to a sparse block structure for the implicit system, such that the additional computational cost of the fluid is small relative to the explicit elastic update. Convergence tests verify highorder accuracy; additional simulations demonstrate applicability of the method to studies of wave propagation in and around branching hydraulic fractures.