Appendix A Range of Wave Speeds
As a first approximation, it is sometimes assumed that porosity is an indirect measure of the till dilation that accompanies deformation, and further that wave-speed measurements are a good indicator of porosity (e.g. Reference Alley, Blankenship, Bentley and RooneyBlankenship and others, 1987; Reference Atre and BentleyAtre and Bentley, 1993; Reference SmithSmith, 1997). However, these assumptions require caution. For example, Reference Eberhart-Phillips, Han and ZobackEberhart-Phillips and others (1989) have made a study of seismic wave speeds in fully saturated, unconsolidated ocean sediments as a function of porosity, clay content (C) and effective pressure. They round that clay content is as important as porosity in determining wave speeds. This means that spatial gradients in clay content may create effects as large as those in porosity. Several other spatially varying properties also affect wave speeds at this level, and they must be considered before making morphological interpretations based on wave speeds alone.
The importance of grain-size distribution on seismic properties has long been noted (e.g. Reference MorganMorgan, 1969; Reference HamiltonHamilton, 1970, 1976; Hamdi and Reference Hamdi and Taylor-SmithTaylor-Smith, 1981; Reference Marion, Nur, Yin and HanMarion and others, 1992). Reference Prasad and MeissnerPrasad and Meissner (1992) found that V
P of unconsolidated samples varied by as much as 250 m s-1 between different grain-sizes and angularity. At the low effective pressures common in subglacial conditions, V
P ranged from 1750 to 2000 m s-1 and front 550 to 700 m s-1 for fully saturated and dry samples, respectively. V
S was found to vary little with grain-size, and for low P
eff it was found to be about 300–350 m s-1 for both fully saturated and dry samples.
Clay content is especially important in determining wave speeds. Reference Marion, Nur, Yin and HanMarion and others (1992) found a peak in V
P as clay content was experimentally increased in an unconsolidated sand/clay mixture. At P
eff = 10 MPa, they found that V
P was 2149 m s-1 for pure sand and 1950 m s-1 for clay, but when C was 40% by weight, the speed reached a maximum of 2500 m s-1. This effect was amplified at higher effective pressures.
These grain-size effects make comparisons between different tills difficult, if not impossible. For example, the wave-speed measurements of till beneath Ice Stream B: (Reference Alley, Blankenship, Bentley and RooneyBlankenship and others, 1987) may not apply to till beneath Black Rapids Glacier. This is because the Antarctic till has an unusually high clay content of > 30% (Reference Tulaczyk, Kamb, Scherer and EngelhardtTulaczyk and others, 1998), whereas Black Rapids Glacier till has < 7% (personal communication from M. Truffer, 1996). We also might expect till wave speeds to vary along ice streams and long glaciers if erosional clay content increases in the downstream direction, or if the type of bedrock from which the till is derived varies spatially.
The microstructure (the shape of the grains and their arrangement) also has a large influence on the acoustic properties of rocks and tills. Bourbié and Reference Bourbié and ZinsznerZinszner (1985) compared sandstones with nearly the same grain-size and porosity. They found that wave speeds and attenuation in those sandstones with thinner, flatter pores were more sensitive to changes in effective pressure than those in sandstones with round pores. The change in V
P due to a change in effective pressure from about 0.1 to 5 MPa was < 50 m s-1 in a sample with round pores, while it was ten times larger in one with flatter pores. At 100 kPa the P-wave speed was 5270 and 4250 m s-1 for these two samples, respectively. These authors found a stronger correlation between microstructure and V
P than between porosity and wave speed.
This effect of microstructure is clue to “squirt flow” (Reference Mavko and NurMavko and Nur, 1975; Reference Palmer and TravioliaPalmer and Traviolia, 1980; Reference Dvorkin and NurDvorkin and Nur, 1993; Dvorkin and others, 1991, 1995). These studies show that, at both seismic and ultrasonic frequencies, pore fluid is temporarily squeezed out of thin, compliant pores and into stiller, rounder pores by a passing wave. This fluid motion is the dominant mechanism of attenuation and velocity dispersion, and controls the relationship between effective pressure and wave speeds. Materials with many thin pore spaces will have a higher attenuation and lower wave speeds — particularly V
P, as the bulk modulus is influenced much more than the shear modulus.
For a fixed grain-size distribution and microstructure, there is a good correlation between porosity and wave speed. The dotted curves in Figure 4 (labeled “Regression Curves”) show several empirical relationships between porosity and V
P, most of which were measured for unconsolidated ocean-floor sediments (Reference MorganMorgan, 1969; Reference HamiltonHamilton, 1976; Hamdi and Reference Hamdi and Taylor-SmithTaylor-Smith, 1981; Reference Eberhart-Phillips, Han and ZobackEberhart-Phillips and others, 1989). They show the trend in relations between V
P and φ at a range of effective pressures, and they loosely constrain a range of physically acceptable values for fully saturated sand/silt mixtures. Note that decreasing porosity leads to higher wave speeds.
Various measurements on fully saturated, unconsolidated sediments and tills (regression curves in Fig. 4) indicate that for φ = 0.25 ± 0.05, V
P should be in the range 1700–2070 m s-1. It should be noted that many of these studies were done at effective pressures well in excess of those expected in our study. At P
eff = 100 kPa, the laboratory measurements of Reference Marion, Nur, Yin and HanMarion and others (1992) on saturated mixtures of sand and clay showed a range of 1550–1750 m s-1 as porosity ranged from 0.4 to 0.2, respectively. The seismic observations of Reference Alley, Blankenship, Bentley and RooneyBlankenship and others (1987) on in situ Ice Stream B till at low effective pressure (~40 kPa; Reference KambKamb, 1991) and anomalously high porosity (> 0.32) and clay content give an estimated V
P of 1550 ± 300 m s-1 (our error bar).
Data on the shear wave velocity are more sparse than those for P-waves. Reference HamiltonHamilton (1976) reports average V
S for coarse sands, sand-silts and sand-silt-clays in ocean sediment samples to be 240, 430 and 390 m s-1, respectively (as corrected to 0°C following Reference ShumwayShumway, 1958); the actual range within the sample set was 100–550 m s-1. V
S for Ice Stream B till (Reference Alley, Blankenship, Bentley and RooneyBlankenship and others, 1987) was 145 ± 30 m s-1 (our error bar); Reference HamiltonHamilton’s (1976) summary suggests that this low value of V
S is indicative of a high porosity and clay content, as is the case there. Reference Prasad and MeissnerPrasad and Meissner (1992) report a range for V
S of 300–600 m s-1 (depending on P
eff) for a range of single grain-size samples.
Considering the uncertainty due to grain-size, micro-structure and porosity, we estimate the permissible range of V
P for a fully saturated till beneath Black Rapids Glacier to be 1500–2100 m s-1, with the central value of 1800 m s-1 being the most likely. A permissible range for V
S is 150–550 m s-1, with the higher values at effective pressures near 800 kPa, and the lower at 100 kPa. It is likely that P
eff at reflector PPN is within 10% of overburden (i.e. 450 kPa), so we consider the central value of 350 m s-1 to be the most likely. For reasons described in the text, the permissible range for V
S for partially saturated sediments is the same.
As described in Appendix B, partially saturated sediments can have P-wave speeds below 500 m s-1. From Figure 6, these low values of V
P do not correspond with any permissible values of V
S that would yield R
PP (50°) = 0. Therefore, we consider the lower bound of partially saturated V
P to be 1000 m s-1, corresponding to the lower bound of V
S in Figure 5. As an upper bound for partially saturated V
P, we use 1500 m s-1 — the lower bound of the fully saturated range — although the actual permissible range for partially saturated V
P includes most of the fully saturated range.
Appendix B Seismic Changes due to Saturation
As shown in Figure 9a, as saturation (S) decreases from 100% to 97%, there is a significant drop in V
P but little change in V
S. Changes in saturation below about 97% have only minor effects on V
P or V
S; this is the cause of a well-known interpretative difficulty with “bright spots” in hydrocarbon exploration: the bright spots signify the presence of gas within the pores, but not its volumetric percentage (below about 97%). An explanation for how saturation affects wave speeds (Reference MurphyMurphy, 1982) can be found by examining the formulae for V
P and V
where K and μ are the bulk and shear moduli, respectively. In a fully saturated sample, K (the inverse of compressibility) is dominated by the compressibility of water (~51 × 10-11 m2 N-1). As the saturation decreases to below 97%, K decreases to that of the more compressible rock matrix (< 3 × 10-11 m2 N-1) and V
P decreases. As S decreases further, the density continues to decrease; this slowly increases V
P to that of the fully dry sample. The amount of gas-filled pore space does not affect μ, so there is no decrease in V
S. However, V
S has the same 5% increase due to the decreased density as V
This change in wave speed can be demonstrated by a simple experiment. Place a sealed beer bottle on a table and tap it with a wooden spoon. Then shake it up, place it on the table and tap it again. The decrease in pitch and lone duration alter shaking is due to the change in compressibility arising from the presence of bubbles in the liquid — in effect, a drop in saturation. The frequency (pitch) of the sound is equal to V
P/2L, where L is approximately the length of the bottle. The attenuation (which controls the duration) is proportional to V
-3. Therefore, as the wave speed decreases, both the frequency and tone duration decrease. This works with wine and hard liquors as well; it is not a function of carbonation.
Reference HamiltonHamilton (1976) reports data on coarse Ottawa sands with porosity of about 36%. At 100 kPa, V
P dropped from 1900 to 500 m s-1 from 100% to 0% saturation, respectively. At 400 kPa, V
P dropped from 2000 to 700 m s-1. Thus, in our range of effective pressure, V
P dropped by 64–74% from the fully saturated value.
Reference Elliott and WileyElliott and Wiley (1975) studied unconsolidated sands (φ = 30%) at 700 kHz. At 7 MPa, V
P dropped from 2200 to 1280 m s-1, a 42% decrease.
Reference Prasad and MeissnerPrasad and Meissner (1992) studied unconsolidated silts and sands with fixed grain-sizes at 100 kHz. At about 200 kPa, V
P of a coarse silt/fine sand decreased from 1875 to 500 m s-1 going from 100% to 0% saturation. At the same effective pressure, V
P of a medium-coarse sand decreased from 1950 to 650 m s-1. These two samples bracket their range of variation: a decrease of 67–73% from fully saturated to dry conditions.
Consolidated rocks show similar but smaller changes in V
P with saturation: Reference Winkler and NurWinkler and Nur (1982) studied sandstone (φ = 23%) at frequencies of 500–1700 Hz. At about 100 kPa, V
P dropped from 4000 to 1600 m s-1 when saturation dropped below 97%, a 60% decrease.
Reference DomenicoDomenico (1976) studied Ottawa sandstone at about 10 MPa and 200 kHz. V
P dropped from 2073 to 1280 m s-1, a 38% decrease.
Reference GregoryGregory (1976) studied sedimentary rocks (φ = 4–41%) and gabbros at 1 MHz. He found that saturation effects were more pronounced at porosities less than 25%, but his curves of V
P vs S were significantly different than those of most researchers. For example, at 100 kPa, V
P of a sandstone with φ = 4% decreased from 4665 to 4268 m s-1 as S changed from 100% to 80%, but the decrease continued to 3506 m s-1 as saturation went to zero.
There is some uncertainty in the saturation at which the abrupt transition in V
P occurs. For example, experiments by Reference MurphyMurphy (1982) on sandstones indicate that most of the change in V
P has occurred by S = 97%, while those of Reference DomenicoDomenico (1976) indicate that the transition occurs at about S = 90%. The discrepancy in experimental results is likely due to the difference in frequency used in the studies (Reference Castagna, Castagna and BackusCastagna, 1993). These results show that a decrease of 25% in V
P and zero in V
S, as illustrated in Figure 8, is a conservative estimate if S goes from 100% to 97% or lower.
Appendix C Seismic Changes due to Effective Pressure
Changes in effective pressure affect V
P and V
S in the same qualitative way in unconsolidated silt, sands and clays, as well as sedimentary and igneous rocks (Reference GregoryGregory, 1976; Reference HamiltonHamilton, 1976; Hamdi and Reference Hamdi and Taylor-SmithTaylor-Smith, 1981; Reference Winkler and NurWinkler and Nur, 1982; Reference Eberhart-Phillips, Han and ZobackEberhart-Phillips and others, 1989; Reference Prasad and MeissnerPrasad and Meissner, 1992). When the pore space is fully saturated with water, V
P is fairly insensitive to changes in P
eff. When partially saturated, V
P decreases with decreasing P
eff. The effects on V
S are comparatively insensitive to changes in saturation. As described in the text, the single valued concept of effective pressure (i.e. P
eff = P
o – P
pw) breaks down for partially saturated media because gas pressure becomes important. Most researchers nevertheless continue to report wave speeds as functions of P
eff = P
o – P
pw and, for belter or worse, we continue their practice here.
These effects are qualitatively shown in Figure 7b. This figure was adapted from data of Reference DomenicoDomenico (1976) and Reference HamiltonHamilton (1976) on unconsolidated sands, and Reference Winkler and NurWinkler and Nur (1982) on limestones. The actual relationship is strongly affected by the grain-size distribution and microstructure, and for the Black Rapids till we can only estimate the change in fully and partially saturated wave speeds due to a change in effective pressure from 100 to 800 kPa.
The data of Hamdi and Taylor-Smith (1982) are particularly useful in this light (reproduced in Reference NolanNolan, 1997). They show that as P
eff is increased from 96 to 772 kPa, V
P is increased by 50–100 m s-1. An increase in effective pressure from 100 to 800 kPa leads to an increase of about 200 m s-1 in V
S. These samples were saturated ocean sediments (clay to fine sands) with relatively high porosities (φ = 34–52%), the porosity decreasing with increasing effective pressure.
Results on other unconsolidated samples are similar. Reference HamiltonHamilton (1976) reports V
P varied less than 100 m s-1 in dry sands and over 200 m s-1 in brine-saturated sands, and that V
S varied about 180 m s-1 at either saturation over this range of P
eff. Reference Prasad and MeissnerPrasad and Meissner (1992) studied samples of single grain-sizes and found that fully saturated V
P increased about 150 m s-1 as P
eff increased by 1 MPa, while fully dry V
P increased about 30 m s-1 MPa-1. V
S increased about 70 m s-1 MPa-1 for both dry and saturated samples.
Combining these results, we estimate that an increase in P
eff from 100 to 800 kPa will increase V
P by about 100 or 200 m s-1 under fully or partially saturated conditions, respectively. V
S should increase about 200 m s-1 at any saturation.
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