Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids |
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Authors: | Wei-Cheng Lo Garrison Sposito Ernest Majer |
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Institution: | (1) Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan, 701, Taiwan;(2) Department of Geophysics, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA;(3) Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720-1710, USA |
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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. |
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Keywords: | Dilatational waves Immiscible fluid flow Poroelastic behavior |
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