Hydraulic fracture diagnostics from Krauklis-wave resonance and tube-wave reflections

Abstract

Fluid-filled fractures support guided waves known as Krauklis waves. The resonance of Krauklis waves within fractures occurs at specific frequencies; these frequencies, and the associated attenuation of the resonant modes, can be used to constrain the fracture geometry. We use numerical simulations of wave propagation along fluid-filled fractures to quantify fracture resonance. The simulations involve solution of an approximation to the compressible Navier-Stokes equation for the viscous fluid in the fracture coupled to the elastic-wave equation in the surrounding solid. Variable fracture aperture, narrow viscous boundary layers near the fracture walls, and additional attenuation from seismic radiation are accounted for in the simulations. We then determine how tube waves within a wellbore can be used to excite Krauklis waves within fractures that are hydraulically connected to the wellbore. The simulations provide the frequency-dependent hydraulic impedance of the fracture, which can then be used in a frequency-domain tube-wave code to model tube-wave reflection/ transmission from fractures from a source in the wellbore or at the wellhead (e.g., water hammer from an abrupt shut-in). Tube waves at the resonance frequencies of the fracture can be selectively amplified by proper tuning of the length of a sealed section of the wellbore containing the fracture. The overall methodology presented here provides a framework for determining hydraulic fracture properties via interpretation of tube-wave data.

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

“…Without free-stream preservation, the discrete conservative form can act as an erroneous source term causing spurious oscillations. By constructing the metric coefficients using the exact same difference operators D q and D r in (27), the discrete metric identities are exactly satisfied. (See [34] for a proof in the context of moving grids, and geometric conservation laws.…”

“…Note that the strong form of (27) can be directly advanced in time using explicit time stepping since the mass matrices are diagonal and the inverse of S is known.…”

“…Since the variational formulation holds for all nontrivial test functions, we obtain the strong formulation of the semidiscrete approximation of the solid by eliminating the test functions in (27) and inverting the matrices on the left-hand side. Note that the strong form of (27) can be directly advanced in time using explicit time stepping since the mass matrices are diagonal and the inverse of S is known.…”

“…For the discrete approximation of the conservative form in (27) to be a consistent approximation of the continuous conservative form in (3), the metric identities (5) must be exactly satisfied when discretized. This condition is known as free-stream preservation and guarantees that the discrete conservative form differentiates a constant solution exactly [21].…”

“…In (27), all of the material properties, Jacobian, and metric coefficients, evaluated at each grid point, are stored in diagonal matrices. The matrix I 2 is a 2 × 2 identity Downloaded 08/27/17 to 130.236.83.247.…”