# 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 liquidmore »

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- Earth Sciences Division

- OSTI Identifier:
- 950854

- Report Number(s):
- LBNL-1656E

Journal ID: TPMEEI; TRN: US200911%%98

- DOE Contract Number:
- DE-AC02-05CH11231

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Transport in Porous Media; Journal Volume: 68; Related Information: Journal Publication Date: 2007

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 54; 58; BOUNDARY CONDITIONS; CRITICAL FREQUENCY; EXCITATION; INTERSTITIALS; PARTIAL DIFFERENTIAL EQUATIONS; PERMEABILITY; PRESSURE DEPENDENCE; SEISMIC WAVES; SHEAR; STIMULATION; VISCOSITY; WATER; WAVE EQUATIONS; WAVE PROPAGATION

### Citation Formats

```
Lo, W.-C., Sposito, G., and Majer, E.
```*Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids*. United States: N. p., 2007.
Web. doi:10.1007/s11242-006-9059-2.

```
Lo, W.-C., Sposito, G., & Majer, E.
```*Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids*. United States. doi:10.1007/s11242-006-9059-2.

```
Lo, W.-C., Sposito, G., and Majer, E. Thu .
"Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids". United States.
doi:10.1007/s11242-006-9059-2. https://www.osti.gov/servlets/purl/950854.
```

```
@article{osti_950854,
```

title = {Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids},

author = {Lo, W.-C. and Sposito, G. and Majer, E.},

abstractNote = {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.},

doi = {10.1007/s11242-006-9059-2},

journal = {Transport in Porous Media},

number = ,

volume = 68,

place = {United States},

year = {Thu Feb 01 00:00:00 EST 2007},

month = {Thu Feb 01 00:00:00 EST 2007}

}