Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids

Abstract

An analytical theory is presented for the low-frequency behavior of dilatational waves propagating through a homogeneous elastic porous medium containing two immiscible fluids.The theory is based on the Berryman-Thigpen-Chin (BTC) model, in which capillary pressure effects are neglected. We show that the BTC model equations in the frequency domain can be transformed, at sufficiently low frequencies, into a dissipative wave equation (telegraph equation) and a propagating wave equation in the time domain. These partial differential equations describe two independent modes of dilatational wave motion that are analogous to the Biot fast and slow compressional waves in a single-fluid system. The equations can be solved analytically under a variety of initial and boundary conditions. The stipulation of "low frequency" underlying the derivation of our equations in the time domain is shown to require that the excitation frequency of wave motions be much smaller than a critical frequency. This frequency is shown to be the inverse of an intrinsic time scale that depends on an effective kinematic shear viscosity of the interstitial fluids and the intrinsic permeability of the porous medium. Numerical calculations indicate that the critical frequency in both unconsolidated and consolidated materials containing water and a nonaqueous phase liquid ranges typically from kHz to MHz. Thus engineering problems involving the dynamic response of an unsaturated porous medium to low excitation frequencies (e.g. seismic wave stimulation) should be accurately modeled by our equations after suitable initial and boundary conditions are imposed.

Propagation of plane P-waves at interface between elastic solid and unsaturated poroelastic medium

“…Thus, the frequency correction function should Fig. 2 Effects of frequency on amplitude ratios with Sr = 0.8 be introduced in the high frequency range [26] . Therefore, the present work is restricted to a relatively low frequency, i.e., below 10 kHz, which covers the common frequencies used in the seismic and acoustic fields.…”

Motional modes of dilatational waves in elastic porous media containing two immiscible fluids

“…doi:10.1016/j.advwatres.2009.12.007 move in phase, whereas the P2 wave involves out-of-phase solidfluid motions leading to attenuation driven by viscous drag. When the pore space contains two immiscible fluids and the excitation frequency is well below a critical value equal to the inverse of the sum of viscous damping time scales for the pore fluids, the P1 and P2 waves can be described analytically by a propagating wave equation and a dissipative wave equation (telegraph equation), respectively [14], in complete analogy with Biot theory. This critical value is shown to vary typically from kHz to MHz in both unconsolidated and consolidated materials containing water and a NAPL [14].…”

“…When the pore space contains two immiscible fluids and the excitation frequency is well below a critical value equal to the inverse of the sum of viscous damping time scales for the pore fluids, the P1 and P2 waves can be described analytically by a propagating wave equation and a dissipative wave equation (telegraph equation), respectively [14], in complete analogy with Biot theory. This critical value is shown to vary typically from kHz to MHz in both unconsolidated and consolidated materials containing water and a NAPL [14]. Independently of fluid saturation and wave excitation frequency (in the seismic range), the speed of the P1 wave is equal to the square root of the ratio of the effective bulk modulus of the system to its effective mass density [2,13,20].…”

Analytical decoupling of poroelasticity equations for acoustic-wave propagation and attenuation in a porous medium containing two immiscible fluids

“…[ 32] Not available One-fluid system Two-fluid system Models without inertial coupling Time Chandler and Johnson [28] L o [ 31] domain, yielding a propagating-wave equation for the Biot fast wave and a dissipative wave equation for the Biot slow wave [30]. Decoupling of the Lo et al [13] poroelasticity equations for dilatational waves in a porous medium containing two immiscible fluids also can be accomplished in the time domain if inertial coupling terms are dropped [31].…”

“…Berryman et al [24] decoupled their model equations, which neglect capillary pressure changes but retain inertial coupling terms, following the frequency-domain method used by Berryman [27] for the single-fluid case ( Table 1). In a generalization of the result for a single-fluid system [13], conversion of the decoupled frequency-domain equations of Berryman et al [24] for dilatational waves into the time domain can be accomplished [32] if the wave excitation frequency is well below a critical frequency, equal to the ratio of an effective kinematic shear viscosity for the two interstitial fluids [24] to the intrinsic permeability of the porous medium (Table 1).…”