Volume 2 Issue 5 - November 30, 2007
Effect of soil texture on the propagation and attenuation of acoustic wave at unsaturated conditions
Wei-Cheng Lo*, Chao-Lung Yeh, and Chang-Tai Tsai

Department of Hydraulic and Ocean Engineering
lowc@mail.ncku.edu.tw

Journal of Hydrology, 338, 273-284, 2007.

1. RESEARCH OBJECTIVES

Over the past decades, the increasing scientific focus on the mechanical behavior of a fluid-containing porous medium has generated great interest in connecting the velocity and dissipation of acoustic waves to the hydrological properties of the shallow subsurface, on which immense practical applications to soil science, agriculture engineering, geomechamics, resource engineering, and hydrogeology are being built.  An inverse issue is to use reflection and refraction seismology to infer the spatial distribution of permeability, porosity, and moisture content of sedimentary materials, as well as to detect the type of pore immiscible fluids, the information of which is crucial for improving the accurate description of water movement and contaminant transport through the vadose zone and groundwater aquifers.  A forward problem is to predict the propagation speed and attenuation coefficient of seismic waves, and induced strains from the knowledge of hydrological and elastic data, by which the dynamic response of partially-saturated porous materials to fluid-pumped and earthquake-triggered pressure changes is capable of being modeled.  The success in implementing these two problems requires a better understanding of the physics of elastic wave propagation and attenuation through unconsolidated soils containing two immiscible, compressible, viscous fluids.

As opposed to the existence of only a single compressional wave and a shear wave in a nonporous solid, three modes of acoustic (dilatational, compressional) wave has been demonstrated to occur in such a medium both theoretically and numerically.  That denoted by P1 has the highest magnitude of phase velocity, followed by P2 and P3 in descending order.  The P1 wave for which the displacements of fluid and solid are in-phase is similar to the ordinary dilatational wave in a nonporous medium.  The P2 and P3 waves exhibit a diffusion phenomenon and the latter (the P3 wave) is related to the relationship between capillary pressure and relative fluid saturation.  The P3 wave thus disappears when the pore space is saturated by a single fluid.

However, despite numerous numerical calculations that have been performed to study the dependence of elastic wave characteristics on water saturation with excitation frequency for consolidated rocks, few ones based on a physically-based theory were conducted for unconsolidated soils.  In addition, little is known about the influence of soil texture on acoustic wave motions at unsaturated conditions.  Thus, there remains a need to explore the effect in a systematic manner.  The present study systematically investigates changes in two important acoustic wave signatures (phase speed and attenuation coefficient) due to the influence of fluid saturation and soil texture including sand, loamy sand, sandy loam, loam, silt loam, sandy clay loam, clay loam, silty clay loam, sandy clay, silty clay, and clay.

2. APPROACH

The coupled partial differential equations describing dilatational wave propagation and attenuation through a partially-saturated porous medium were formulated by Lo et al. [2005] using continuum mechanics of mixtures.  Based on the dispersion equation derived in the Lo et al. [2005] model for depicting the relationship between excitation frequency and wave number, the phase velocity and attenuation coefficient of three different modes of acoustic wave were determined as a function of water saturation for eleven different types of unconsolidated soils.  A renowned constitutive equation (the van Genuchten - Mualem model) was applied to determine the capillary pressure – saturation and relative permeability – saturation relations.  Then, so as to provide a more general result, the calculated phase velocity and attenuation coefficient of the P1 and P2 waves for different soil textures were normalized into a dimensionless form using those obtained for sand.  By doing so, these dimensionless parameters were found to be independent of excitation frequency up to 500 Hz, a critical frequency assuring that each fluid flow is of the Darcy’s type under wave excitation for all soil texture classes examined.  The phase velocity and attenuation coefficient of the P3 wave were not shown in the present study since this wave has very high attenuation and low speed, thus making it very difficult to observe.

3. ACCOMPLISHMENTS

Our numerical studies show (Fig.1) that the normalized phase velocity of the P1 wave remains a nearly constant value at higher water saturations, but in the lower range it increases with water saturation, decreasing after approaching a maximum value, which occurs at water saturations between 0 and 0.2.  The magnitude of this value is dependent on soil types, which ranges from silty clay, clay, silty clay loam, sandy clay, clay loam, silt loam, loam, sandy clay loam, sandy loam, to loamy sand in descending order.  A dimensionless ratio of effective non-wetting fluid storativity factors associated with the capillary pressure curve is found to be the controlling parameter responsible for the occurrence of the maximum value.  It has been also seen that, when excitation frequency is lower, the phase velocity of the P1 wave is equal to a characteristic wave speed, defined as the square root of the ratio of the effective bulk modulus to the effective density of the fluid-filled porous medium.
Figure 1 The normalized phase velocity of the P1 wave for ten soil texture classes (loamy sand, sandy loam, loam, silt loam, sandy clay loam, clay loam, silty clay loam, sandy clay, silty clay, and clay)
Figure 2 The normalized attenuation coefficient of the P1 wave for ten soil texture classes (loamy sand, sandy loam, loam, silt loam, sandy clay loam, clay loam, silty clay loam, sandy clay, silty clay, and clay)


As far as the normalized attenuation coefficient of the P1 wave is concerned, we see in Figure 2 that not only does it reveal a distinct characteristic for soil textures, but it is also affected by the degree of saturation.  In addition, the normalized attenuation coefficient of the P1 wave is found to be less than unity no matter of water saturation for all soil textures, whereas that of the P2 wave is greater than unity (Fig. 4).  This physically indicates that sand has the highest attenuation coefficient for the P1 wave and the lowest one for the P2 wave among eleven different types of soils.  Loamy sand has the second highest attenuation coefficient for the P1 wave.  Lastly, it is noted in Figures 3 and 4 that the normalized phase velocity and attenuation coefficient of the P2 wave are closely related to the intrinsic permeability.  When a soil has a higher intrinsic permeability, the normalized phase velocity of the P2 wave is greater while the normalized attenuation coefficient of the P2 wave is smaller.  Two minor exceptions to this observation occur only at 0.99 water saturation between silt loam and sandy clay loam, and at 0.99 and 1 water saturations between silty clay loam and sandy clay.  This proportional relation provides an informative indicator in linking acoustic waves to the ability of soils in transporting fluid flows. 
Figure 3 The normalized phase velocity of the P2 wave for ten soil texture classes (loamy sand, sandy loam, loam, silt loam, sandy clay loam, clay loam, silty clay loam, sandy clay, silty clay, and clay)
Figure 4 The normalized attenuation coefficient of the P2 wave for ten soil texture classes (loamy sand, sandy loam, loam, silt loam, sandy clay loam, clay loam, silty clay loam, sandy clay, silty clay, and clay)


4. SIGNIFICANCE OF FINDINGS

The combination of acoustic wave characteristics obtained from our investigation concerning the normalized phase velocity and attenuation coefficient of the P1 and P2 waves should not only allow application of the results to other cases without having to perform an entire set of numerical simulations but can also serve as a basis for low-frequency acoustic wave methods used to measure the physical properties of soil, such as moisture content, porosity, permeability, and material density.  One of the practical applications for environmental remediation is to locate NAPL (nonaqueous phase liquid)-contaminated sites in groundwater aquifers by obtaining the distribution of seismic wave velocity and attenuation.

Reference
Lo, W. C., G. Sposito, and E. Majer, (2005), Wave propagation through elastic porous media containing two immiscible fluids, Water Resources Research, 41, W02025, (20 pp.)
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