Properties of low‐frequency trapped mode in viscous‐fluid waveguides

Abstract

A B S T R A C TWe derived the velocity and attenuation of a generalized Stoneley wave being a symmetric trapped mode of a layer filled with a Newtonian fluid and embedded into either a poroelastic or a purely elastic rock. The dispersion relation corresponding to a linearized Navier-Stokes equation in a fracture coupling to either Biot or elasticity equations in the rock via proper boundary conditions was rigorously derived. A cubic equation for wavenumber was found that provides a rather precise analytical approximation of the full dispersion relation, in the frequency range of 10 −3 Hz to 10 3 Hz and for layer width of less than 10 cm and fluid viscosity below 0.1 Pa·s [100 cP]. We compared our results to earlier results addressing viscous fluid in either porous rocks with a rigid matrix or in a purely elastic rock, and our formulae are found to better match the numerical solution, especially regarding attenuation. The computed attenuation was used to demonstrate detectability of fracture tip reflections at wellbore, for a range of fracture lengths and apertures, pulse frequencies, and fluid viscosity.

Laboratory Measurements of the Impact of Fracture and Fluid Properties on the Propagation of Krauklis Waves

“…Because the theory of the crack wave was first introduced by Krauklis in 1962, the crack wave is also called the "Krauklis wave" in the literature (Korneev, 2008;Krauklis, 1962;Staecker & Wang, 1973). Following Krauklis (1962), many researchers have studied the Krauklis wave analytically and numerically (Dunham & Ogden, 2012;Ferrazzini & Aki, 1987;Frehner, 2014;Korneev, 2008;Nikitin et al, 2016). Korneev et al (2014) analytically examined the phase velocity for a laboratory fracture model with a finite background medium thickness (a trilayer model).…”

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

“…The key concept here is a particular type of guided wave that propagates along fluid-filled cracks. These waves, known as crack waves or Krauklis waves, have been studied extensively in the context of the oil and gas industry, volcano seismology, and other fields (Krauklis, 1962;Paillet and White, 1982;Chouet, 1986;Ferrazzini and Aki, 1987;Korneev, 2008Korneev, , 2010Yamamoto and Kawakatsu, 2008;Dunham and Ogden, 2012;Lipovsky and Dunham, 2015;Nikitin et al, 2016). At the frequencies of interest here (approximately 1-1000 Hz), they are anomalously dispersed waves of opening and closing that propagate along fractures at speeds approximately 10-1000 m∕s.…”