Blast-Induced Liquefaction and Determination of Soil-Density Changes with Ground-Penetrating Radar, Treasure Island, CA
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RELATION OF SOIL VOID RATIO TO RADAR-WAVE VELOCITY
Through the laboratory technique of time-domain reflectometry (TDR), a method essentially identical to our field field approach with crosshole GPR, Topp, et al. (1980) found that a singular relationship exists between the volumetric soil moisture  v,(volume of water to the volume of the total soil mass) and the dielectric constant. The complex dielectric constant of soil,
is composed of real ( ) and imaginary ( ) parts. The radar velocity is dependent only on the real part of the dielectric constant, where v = c/()1/2 and c is the velocity of light in air. The real part of the dielectric constant varies from 1 in air (v = c = 0.3 m/ns) to in excess of 30 for fine grained soils (v ~ c/6 ~ 0.05 m/ns). The radar velocity in soil varies from 0.15 m/ns in dry sand and 0.06 m/ns for saturated sand, to below 0.05 m/ns for soft cohesive soil. For a suite of soils types and water contents, Topp et al. (1980) found the following relation between and v for v between 0 and 0.6,
= 3.03 + 9.3kv + 146kv2 - 76.7kv3 [2].
Saturated soils have all of the void space filled with water. Under such conditions = n, the soil porosity. Resolving equation [2] for T and assuming full saturation, the porosity n can be determined from the real part of the dielectric constant as follows:
n = - 0.080607 + 0.037649k -; 0.0011413k2 - 1.5789E-5k3 [3].
In the field we measure radar velocity, rather than dielectric constant. To estimate porosity directly from radar velocity, the relationship = (c/Vr)2 (where c = 0.3 m/ns) is used to modify [3], as follows:
n = 2.5025 - 75.54kV + 920.1kV2 - 4094.8kV3 [4].
For equation [4], and [6] below, velocity is presented in units of m/ns. The geotechnical characterization of density state is typically done in terms of void ratio (e). Void ratio is the void volume normalized by the volume of the dry sediment grains. We substituted void ratio for porosity in equations [1] and solved for e in terms of dielectric constant and velocity:
e = - 0.035129 + 0.030695k - 3.5531E-4k2+ 9.6159E-6k3 [5]
and
e = 13.482 - 533.47kV + 7526.4kV2 - 36615kV3 [6].
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RESULTS: DENSITY CHANGE DURING LIQUEFACTION
The equations [4] and [6], above, were used to map porosity and void ratio in the soil prior to, and following, blast-induced soil liquefaction. Prior to liquefaction, radar velocity in the central portion of the soil column ranged between 0.054 and 0.060 m/ns, and an average velocity of 0.057 m/ns (see Box 1 at right). These velocities translate into void ratios ranging from 0.846 to 0.647, and an average void ratio of 0.738. Generally, there is a zone of low to intermediate void ratios in the central portion of the image and a low void ratio zone in the upper left region (see Box 2). A locally high void ratio on the right side of the plane is seen at depths of 2.5-2.75 meters, and also in the lower left corner.
After blasting, GPR velocities had risen considerably throughout the tomogram. The velocity range for the post-liquefaction soil ranged from 0.056 to 0.064 m/ns, and an average velocity of 0.060 m/ns. The void ratios associated with these velocities range from 0.554 to 0.770, and an average of 0.664. By comparing the tomograms, it can be seen that almost the entire tomographic plane underwent some level of densification and reduction of void ratio during liquefaction.
Subtraction of the post-liquefaction void ratios from the initial values produces a difference tomogram (Box 3). On average, the soil experienced a densification (reduction in void space) of  e = 0.074. The range of void ratio change spans from -0.066 to 0.172. That is, the entirety of the tomographic plane densified, with the exception of a narrow zone on the left side. That zone apparently loosened during the liquefaction event, or formed a void when sand was ejected to the surface. The average volumetric strain due to void ratio reduction was 4.2%. Given that the estimated thickness of the liquefied layer at the site was 4 meters, this strain would result in 17.0 cm of settlement.
The observed surface settlement at the test site is an independent measure of the volumetric strain. Maximum settlements of 16.8, 17.8, and 20.7 cm were recorded along three transects across the site, and are remarkably similar to the estimated 17-cm settlement derived from radar tomograms.
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