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DIVISION S-1-SOIL PHYSICS

Elastic Wave Velocities in Partially Saturated Ottawa Sand

Experimental Results and Modeling

National Center for Physical Acoustics, Univ. of Mississippi, Coliseum Drive, University, MS 38677 USA

dvelea{at}olemiss.edu

	ABSTRACT

TOP ABSTRACT INTRODUCTION NOTES Materials and methods Results and discussion Summary and conclusions REFERENCES

 
A theoretical model is needed to predict the macroscopic mechanical properties of soil from the size, shape, and elastic properties of its constituent particles. To test one such model, we compared measured and calculated values of compressional and shear wave velocities in Ottawa sand. The sand was packed in a cylindrical tank 0.9 m in diameter and 0.9 m deep. The velocities were measured in the horizontal direction as a function of depth as the zero tension level of the water in the sand was slowly raised. In the air-dry sand the velocities varied nonuniformly with depth, reaching a maximum value about two-thirds of the way to the bottom of the tank. When water was introduced into the bottom of the sand, the nonuniform depth dependence was removed. At higher saturations, the velocities gradually decreased until the zero tension level was at the top of the sand. The nonuniform depth dependence in the dry sand has been attributed to the tank wall supporting part of the gravitational stress in the material. A modified Digby (1981) model was found to adequately account for the results in the wet material. A lumped parameter combining the contacts per grain, size, and the grain roughness was used to fit the data. In terms of the model, it is concluded that the water in the contacts between the grains had little effect on the normal contact stiffness, but reduced the tangential contact stiffness to zero.

Abbreviations: TDR, time domain reflectometry

	INTRODUCTION

TOP ABSTRACT INTRODUCTION NOTES Materials and methods Results and discussion Summary and conclusions REFERENCES

 
MOST PREVIOUS STUDIES of compressional and shear wave velocities in unconsolidated granular materials used samples confined by relatively high hydrostatic or uniaxial stresses. Confining the granular material increases the grain-to-grain contact forces and the reproducibility of measurements since the material behaves more like a continuous solid. Little work has been performed at very low confining pressures where the contact forces are much weaker and their variability from contact to contact is greater. Most studies on the effect of water involved a few discrete values of water content or covered a limited saturation range.¹ Usually the samples of material were small (a few tens of centimeters). In cases where a pulse propagation technique was used, the dominant frequencies were in the hundreds of kilohertz range. In the experiment described here the sand was not subjected to an external stress. Rather, the material simply resided in a cylindrical container loaded only by its own weight. Compressional and shear waves were generated at a frequency of 4 kHz and their velocities were measured by using a tone-burst transmission technique.

It is an established fact that the presence of moisture modifies the acoustic behavior of both consolidated and unconsolidated granular materials (see for example, Bourbié et al., 1987). For instance, in fully water-saturated sands, the compressional wave velocity (commonly referred to as the speed of sound) is typically around 1700 to 1800 m s^-1. As soon as small amounts of gas are added, the stiffness of the medium lowers and the speed of sound reduces significantly to 150 to 250 m s^-1 (Anderson and Hampton, 1980). At the very low saturation end (<2%) it is found that trace amounts of moisture in a porous matrix of consolidated material also affect the wave propagation dramatically (Tittmann et al., 1980). In the intermediate saturation range (2–98%) the acoustical properties depend on the distribution of liquid–gas mixture, pore geometry and size, solid wettability, and the mechanism by which fluids have been mixed and introduced into the porous matrix.

In this study water was allowed to enter the material under slightly negative tension. The saturation in the granular material ranged from a very low value corresponding with the air-dry state up to 98%. To estimate the compressional and shear wave velocities, simplifying assumptions regarding the state of sand were made and the Digby (1981) model of contact mechanics was applied. The model considers the interactions in random packings of equal spheres under hydrostatic pressure.

	Materials and methods

TOP ABSTRACT INTRODUCTION NOTES Materials and methods Results and discussion Summary and conclusions REFERENCES

 
Ottawa sand is a naturally occurring, 99.9% pure quartz sand originating from Ottawa, IL. The sand used in our experiments, Flint Silica #13, had relatively round grains of nominal radius of 250 µm and density of 2650 kg m^-3. A high-density polyethylene cylindrical tank, 91.5 cm in diameter at the top and 86.5 cm diameter at the bottom and 91.5 cm deep, was filled with air-dry sand during a 10-d period. The granular material was dumped in successive batches of 15 kg each, compacted for 5 min, and leveled. This was followed by tapping of the walls with a light plastic hammer for 5 min. Each batch of sand raised the sand level by 12 mm. It was believed that this procedure provided a uniform porosity and a consistent way of packing. The final porosity was 31.9%, and the bulk density of the material was 1805.4 kg m^-3.

Four compressional wave sources (C1–C4) and four shear wave sources (S1–S4) were arranged at different depths ranging from 694 to 132 mm from the top surface of the sand (Fig. 1) . The transducers were supported by brass rods fixed in the holes of a horizontal aluminum rod placed at the top of the tank along the diameter. The compressional wave sources consisted of two piezo-bimorphs mounted back to back using a brass ring as a spacer. When driven in phase, the arrangement acted as an effective acoustic monopole with a resonance frequency around 4 kHz (Fig. 2a) . Each shear wave source consisted of a driven cylindrical geophone vibrating horizontally along the cylinder's axis and generating shear waves in the direction perpendicular to its axis (Fig. 2b). In the present work, geophones model L-410 manufactured by Mark Products (Houston, TX) were used. They were able to generate shear waves in the frequency range from 0.5 to 4 kHz.

View larger version (49K): [in this window] [in a new window]   Fig. 1 Experimental setup. The wave sources lie at different depths in a vertical plane. S1 to S4 denote the shear wave sources and C1 to C4 the compressional wave sources. A pair of accelerometers was placed colinearly with each source as depicted in Fig. 2

View larger version (15K): [in this window] [in a new window]   Fig. 2 Source-receivers arrangement: (a) for compressional waves and (b) for shear waves

The distance source-accelerometer on one side, d₁, was different from the distance d₂ on the opposite side and d₂ - d₁ = 10 cm. The accelerometer orientation was chosen so as to have the greatest sensitivity to the type of waves generated by the corresponding source (Fig. 2a and b).

The water content in sand was measured by the time domain reflectometry method (TDR; Topp and Reynolds, 1998). The TDR probes were placed horizontally at each transducer level and close to the wall of the tank so as not to interfere with the wave propagation. The probes were attached to a Trase 6050XI system manufactured by SoilMoisture Equipment Corp. (Santa Barbara, CA).

The water was introduced into the sand by capillary flow. Preliminary experiments (Velea, 1998) had shown that this method permanently trapped a few percent of air in the sediment. Furthermore, at water saturations >80% the spatial distribution of gas bubbles appeared to be fairly uniform. In contrast, a liquid injection technique would have disturbed the granular material greatly and the distribution of bubbles would have been less uniform (Brandt, 1960; Domenico, 1976).

Figure 3 illustrates how a vertical water content gradient was gravitationally established in the sand. Tank T contained the sand except for a thin layer of 3-mm glass beads at the bottom. A 2-mm wire screen covered with cotton cloth separated the sand and glass beads. The purpose of the glass bead layer was to ensure that the initial waterfront at the sand–glass beads interface was a well-defined horizontal plane. The water was dispensed from Tank D into the bottom of Tank T. The level of water in Tank D marked the zero tension. Assuming that the waterfront in sand is relatively horizontal and rises uniformly, the degree of saturation with water varies according to a typical tension curve shown in the same figure. It must be noted that below the zero tension level a maximum saturation zone was formed in which there still existed trapped gas pockets (2–3% of the void volume).

View larger version (66K): [in this window] [in a new window]   Fig. 3 Establishing a water content gradient based on a typical tension curve in sand shown on the left side

	Results and discussion

TOP ABSTRACT INTRODUCTION NOTES Materials and methods Results and discussion Summary and conclusions REFERENCES

The initial measurements revealed that the wave velocities were dependent on the room temperature. The variation of the compressional wave velocity is shown in Fig. 4 . A similar behavior was observed for shear waves. This temperature dependence was attributed to the expansion and contraction of the tank with the temperature fluctuations in the laboratory. When the room temperature was rising, the tank would expand, releasing some of the horizontal stress on the sand inside, therefore causing the wave velocities to decrease. Upon cooling, the tank contracted, therefore increasing the horizontal stress and velocities. This illustrates one of the difficulties of working with granular materials at zero or very low confining pressures: small changes in the stress in the porous matrix cause significant changes in the wave velocities. In order to test that the observed velocity change was due to expansion and contraction of the tank, a belt was wrapped around the circumference of the tank and tightened. A velocity increase was observed. When loosening the belt, the velocities decreased back to the previous values proving that the procedure did not alter the actual arrangement of the sand grains.

View larger version (24K): [in this window] [in a new window]   Fig. 4 Temperature dependence of the compressional velocity

View larger version (23K): [in this window] [in a new window]   Fig. 5 The horizontal velocity profile in air-dry sand in the tank

View larger version (24K): [in this window] [in a new window]   Fig. 6 Depth profile of (a) compressional and (b) shear wave velocities in air-dry sand (circles), after the introduction of water at bottom (triangles) and at maximum saturation (squares). The maximum saturations are specified at each level. For the values labeled "water at bottom" (triangles) the amount of water in the body of sand was undetectable by the time domain reflectometry probes

Figure 7 shows the typical evolution of compressional and shear wave velocities at the C2 and S2 levels situated at depths of 452 and 534 mm, respectively. The water content is plotted in the bottom graph for the period of the experiment (2300 h). The air-dry values are shown at time zero. It is very interesting to note that the velocity decreased continuously even after the sand reached maximum saturation (97–98%) at the corresponding depth. Thus, it seemed that the changing saturation at superior levels influenced the velocities at inferior levels that had already reached maximum saturation. The decreasing trend appeared to have ceased after the top C4 level reached a saturation of 86.6%. It may be noted that unexplained fluctuations in the velocity measurements remained even after they were corrected for changes in temperature.

View larger version (23K): [in this window] [in a new window]   Fig. 7 Evolution of compressional and shear wave velocities at S2 and C2 levels

Qualitative Justification of Results in Air-Dry Sand
Despite their seeming simplicity, determining the internal stress distribution of unconsolidated granular materials subjected to low confining stress by the walls of the container is an unsolved problem. Qualitatively, the anomalous depth dependence of the horizontal compressional and shear wave velocities that was found in the air-dry sand indicated that the sand was more tightly packed at the C2, S2 depths than at the deeper levels C1, S1 (refer to Fig. 1), that is, a stress anisotropy developed in the sand. The existence of anisotropic regions that support more stress than others in cylindrical containers is confirmed by studies in the silo design (Blight, 1992) or powder pelleting (Train, 1957). Blight's (1992) measurements showed evidence of horizontal (lateral) stress inversion in silos (i.e., horizontal stress decreasing with increasing depth). In other recent studies (Liu et al., 1995; Radjai et al., 1996) the anisotropic regions are attributed to the formation of the so-called force chains (Fig. 1 in Liu et al., 1995, is a photograph of force chains developed in Pyrex beads). The force chains are formed along paths carrying disproportionately large stresses through the material. They can form arches that transfer vertical axial stress to the cylinder's walls, therefore influencing the value of the coefficient of earth pressure at rest.

It is concluded that in the air-dry case the stress anisotropy precludes a prediction of the velocity based on grain-to-grain contact analysis since it is impossible to estimate the forces that appeared at grain-to-grain contacts. However, in the wet sand the situation changed and these forces could be estimated from the gravitational overburden. This is described in the next section.

Working Model of Initial and Final States of Wet Sand
The introduction of water removed the anomalous depth dependence and a continuous decreasing trend of velocities was observed in time. This trend did not stop when a particular depth had reached a maximum water saturation of 97–98%. This fact suggested that the velocity measurements at one level were influenced by the variation of water content and ongoing flow at other levels. The velocity decrease reached a minimum plateau when the flow ceased. This observed interdependence precludes a quantitative analysis of the effect of water on the granular material except at the two extreme states: the initial damp state (i.e., the minimum saturation state for which the zero tension level was at least 6 cm below the first transducer level, S1) and the final state (i.e., maximum saturation state, just before draining). In the following, a working model is proposed to explain the data in the wet sand. It consists of three successive calculations:

1. Approximate the sand grains as spheres with a nominal radius of 250 µm and apply the Digby (1981) model to explain the values measured in the damp state.
   
2. Assume that the final saturated state is that of the dry state altered by the buoyant force of the water and the increased bulk density of the medium. The walls of the tank are assumed rigid.
   
3. Slightly relax the condition that the walls are rigid and account for the increased pressure on the walls due to the added water.
   

Digby's (1981) model considers a dry material consisting of a uniform random packing of identical elastic and smooth spheres of radius R, density _g, shear modulus µ, and Poisson's ratio . If the stress P due to gravitational loading is purely hydrostatic (i.e., the coefficient of earth pressure at rest is equal to 1), then the normal contact force, N, can be computed by

(1)

where is the porosity and C is the average number of contacts per grain (coordination number). The stress is P = _g (1 - )gz, where z is the depth. The contact area is circular and has a radius:

(2)

Here R' is the local radius of curvature of particle surface at the point of contact. For smooth spheres R = R', but this equality does not hold for sand grains approximated as spheres and having a certain roughness. Note that since N is function of R, a in Eq. [2] is a function of both R and R'.

The standard Hertz-Mindlin approach (Hertz, 1895; Mindlin 1949) gives the normal and tangential contact stiffnesses:

(3)

(4)

The normal stiffness for a contact between two solid spheres is defined as the first derivative of the normal contact force with respect to the distance of (normal) approach between the centers of the spheres. The tangential stiffness is defined similarly for the direction perpendicular to the normal (i.e., tangential to the plane of contact).

The compressional and shear velocities are calculated in terms of elastic frame moduli (Winkler, 1983):

(5)

where

(6)

is the compressional elastic modulus and is the bulk density of the dry material. Similarly,

(7)

where

(8)

Damp State
The mechanism that caused the speed decrease at the beginning of the wetting process required only minute amounts of water, implying that the bulk density of the medium changed negligibly. Therefore, the bulk density of the air-dry sand is equal to the bulk density of the damp state: is the grain density and the porosity is . The density of the air in the voids is neglected. In the following analysis it is useful to consider the ratio of compressional to shear velocity, V_p/V_s. In his "rari-constant" theory, Poisson (1829), predicted the ratio for an isotropic solid characterized by "material points" interacting solely by central forces. In Digby's treatment of an isotropic random packing of identical spheres, this ratio is also if the tangential stiffness, D_t, is zero (Eq. [5]–[8]). Therefore, the value is regarded as an indication that in the granular material there are no tangential forces between particles and the stress distribution is hydrostatic.

The ratio V_p /V_s is illustrated in Fig. 8 for both the damp and maximum saturation states. Since the experimental data provided V_p and V_s at staggered depths, the ratio was calculated by simple linear interpolation. For the damp state V_p/V_s changes little with depth and is nearly equal to . (It is worth noting that is a value found not only in sand; Wang (1997) measured a similar ratio for V_p/V_s in glass beads). Consequently, it is assumed that in presence of water (in the damp state) the contact forces between grains are central forces. The moisture removed the force chains in the sand and consequently removed the anomalous depth dependence of the velocities. Based on this, the stress distribution in the sand is assumed to be hydrostatic and Eq. [1] is used to calculate the normal contact force between grains.

View larger version (21K): [in this window] [in a new window]   Fig. 8 The inferred ratio of compressional to shear velocity in the damp state and maximum saturation state

The compressional and shear wave velocities are now given by Eq. [5] and [7] with the modified compressional and shear moduli of the granular aggregate:

(9)

(10)

D_n is the normal contact stiffness given in Eq. [3].

Even with the freedom of choosing , the Digby model predicts an improper depth dependence. Assuming that the hydrostatic stress increases linearly with depth, Eq. [1] to [8] predict velocities that increase as the depth to the 1/6th power. However the measurements show little depth dependence, and to account for this, Janssen's analysis (in Nedderman, 1992) is applied. Assuming a shear stress at the wall, the vertical stress in the sand varies according to:

(11)

Here z_c = D/4µ_wK is a characteristic depth, D is the diameter of the tank, K is the coefficient of earth pressure, and µ_w is the coefficient of friction between the wall and sand related to the angle of friction at the wall, _w, by µ_w = tan _w. In this case, K = 1 since the state of stress in the sand is considered hydrostatic away from the wall. Nedderman (1992) cites work by Jenike (1961) in which the angle of the friction at the wall, _w, and the angle of internal friction, , are related by µ_w = tan _w = sin . Sherif et al. (1984) published the variation of the angle of internal friction of Ottawa sand, , vs. bulk density in the range from 1540 to 1680 kg m^-3. The Ottawa sand in our case was differently packed and had a bulk density of 1805 kg m^-3. A linear extrapolation of the published data yields an angle of internal friction of 54 degrees. With this value µ_w = 0.81 and the characteristic depth is z_c = 0.27 m.

The calculation of compressional and shear wave velocities based on the modified Digby model is shown in Fig. 9 . The chosen value for the equivalent coordination number is .

View larger version (33K): [in this window] [in a new window]   Fig. 9 The results of three successive calculations of (a) compressional and (b) shear wave velocities in wet Ottawa sand as described in the text. The scattered points are the measured speeds. The air-dry experimental values are provided for comparison

(12)

where s is the saturation and _w is the density of water. The buoyant effect of the water reduces the hydrostatic stress on the sand particles from the value in Eq. [11] to:

(13)

This translates into a lower value for the intergranular normal force in Eq. [1], normal contact stiffness in Eq. [3], and frame moduli in Eq. [9] and [10]. The new moduli are denoted M^wet_frame and N^wet_frame. The velocities at maximum saturation are then calculated by expressions similar to Eq. [5] and [7], with the density replaced by its wet value:

(14)

(15)

The calculated velocities are represented by the curves labeled "+ saturation effects" (Fig. 9a and b).

Wall Effect
The gradual increase in water content in the material slowly added more hydrostatic stress that tended to push the tank walls slightly apart in the radial direction. Just as in the case of a temperature increase, the horizontal stress and the horizontal compressional and shear speeds were lowered. At maximum saturation the pressure in the liquid at depth z is simply _wgz. It is considered that the expansion of the walls does not cause a significant departure of the state of the sand from a hydrostatic state. The effective stress increase on the walls due to the addition of water is evaluated by considering a horizontal surface of area A at depth z in the porous medium assumed of constant porosity. In the case of air-filled sand, the force on A is the weight of the solid material (again neglecting the weight of the gas):

(16)

where is the volume occupied by the solid and v_t is the total volume occupied by the granular material. The effective (or equivalent) stress on the surface A is (1 - )_sgz. Similarly, if water is added to fill the pores completely (100% saturation), the force on the surface A increases by the weight of the liquid:

(17)

where is the volume occupied by water. Therefore the additional effective stress on the surface A is _wgz.

An average value corresponding with the middle position in the tank was used as the horizontal stress increment exerted on the walls by the water. The change in horizontal stress P_h can be related to an equivalent temperature increase T via a radial deformation of the wall. Once the corresponding T is calculated the change in velocity, V, is evaluated from the measured slope of the velocity variation with temperature:

(18)

For the compressional wave velocity and for the shear wave velocity .

The relative deformation in the radial direction of the cylindrical tank when the inside horizontal stress increases by P_h is (Landau and Lifshitz, 1959):

(19)

where R₁ is the inner radius, t is the wall thickness, Pa and is Young's elastic modulus, and is Poisson's ratio of the wall material (polyethylene).

The same deformation of the walls can be achieved by thermal expansion:

(20)

Here _t is the coefficient of thermal expansion for polyethylene and T is the temperature increase. By combining Eq. [18], [19], and [20] the relationship between velocity and horizontal pressure variations is:

(21)

Here, , and has the values mentioned above for the compressional and shear velocities.

Based on the above formula and Eq. [14] and [15], the final velocities are:

(22)

(23)

The results are plotted in Fig. 9 (the curves denote "wall effect"). The agreement with the final measured compressional speeds is good, while the shear speed prediction is poorer because of the scatter in the measured values.

What Happened During Draining?
As mentioned before, at the start of each draining step the velocities at all depths increased notably. The most significant increase (5–9%) occurred after the first draining, which also had the highest draining rate. It would appear that the effect could be explained by the following:

Accounting for the Reduction in the Bulk Density and Buoyant Effect
However, during the first draining, the saturation changed only at the top two levels, C4 and S4, while remaining constant below S4. Thus below S4 the velocities should have stayed constant.

An Increased (Capillary-Type) Cohesion between Particles as the Water Drained
However, by this argument an increase in velocities should have been seen during filling as well, but this was not the case.

Perhaps an explanation can be offered by the following analogy. Reimbert and Reimbert (1976) and Blight (1992) reported that during the emptying of silos (especially in the beginning) an increase of lateral pressures on the walls is commonly observed. This phenomenon is not well understood, but it could account for a greater lateral (horizontal) stress in the sand and therefore greater velocities at the start of draining of the tank. The similarity is, of course, not perfect since what flowed out was not granular material but the water in the pores. However, provided the phenomenon is associated with the change in the support of the load by the bottom of the tank, the parallel with the emptying of a silo can be made.

	Summary and conclusions

TOP ABSTRACT INTRODUCTION NOTES Materials and methods Results and discussion Summary and conclusions REFERENCES

 
The model used to interpret the results assumed that the contacts between particles govern the propagation of elastic waves in this experiment across the whole range of water saturation from air-dry to 98%. In the air-dry Ottawa sand the horizontal compressional and shear wave velocities did not increase uniformly with depth. Instead they reached a maximum value about two-thirds of the way to the bottom of the tank and then decreased. In addition, the velocities were found to be very sensitive to horizontal stress changes in the sand produced by temperature changes in the tank. It is concluded from these results that in the air-dry sand some of the gravitational loading is transferred to the walls of the tank and regions of stress anisotropy may have developed during packing or thermal expansion and contraction of the tank. Thus, in the air-dry case it is not possible to predict the velocity from contact mechanics because the contact forces cannot be calculated from the gravitational loading.

When water was introduced into the bottom layer of sand, the velocities were changed all the way to the top of the tank and the ratio V_p /V_s became approximately 3. The latter was taken to indicate a zero tangential stiffness in the grain contacts, central forces between the grains and a hydrostatic state of stress in the sand. In this case it was possible to calculate the normal contact force from the gravitational loading using Digby's theory. The measured velocities were adequately predicted by the theoretical model when the buoyant force of the water and the restraining force of the wall were included. The water in the contact points had little effect on the normal contact stiffness.

Some authors (Domenico, 1976) suggested that the transition to the high compressional velocity values found in fully saturated sand (1700 m s^-1) would begin at saturations as low as 85%. However, an increase in velocity was not observed in this work even at a saturation of 98% (or 0.6% air by volume), thus emphasizing the huge effect that air presence can have in an almost fully water-saturated sand.Reimbert Reimbert 1975

	ACKNOWLEDGMENTS

 
The authors would like to thank the Office of Naval Research and the USDA for their entire support during this project. The authors would also like to thank Dr. M.J.M. Römkens, Dr. S.N. Prasad, and Dr. Craig Hickey for many helpful discussions about this work.

	NOTES

TOP ABSTRACT INTRODUCTION NOTES Materials and methods Results and discussion Summary and conclusions REFERENCES

 
¹ In this paper, saturation and water content are used interchangeably: saturation is defined as s = volume of voids filled with water/total void volume, and the water content is s where is the porosity.

Received for publication April 22, 1999.

	REFERENCES

TOP ABSTRACT INTRODUCTION NOTES Materials and methods Results and discussion Summary and conclusions REFERENCES

 

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