Figure 12.16. Effect of vertical advection on borehole inclination.

Figure 12.16. Effect of vertical advection on borehole inclination.

as one would expect and as implied by our opening discussion, du/dz is one of the most important velocity derivatives.

Sensitivity studies suggest that the solutions obtained in these two Barnes Ice Cap experiments do not depend strongly on the assumptions. The most important term is d£i/dt. In instances where the casing bends abruptly, as at joints, w dli/d z also becomes important. In experiments on other glaciers, the results might be more sensitive to some of the other velocity derivatives, and hence to any assumptions made in obtaining them.

Further insight into Equation (12.48) can be achieved by considering the situation in plane strain. Assuming incompressible flow and a uniform longitudinal strain rate, r, we then have du/dx = -dw/dz = r, dw/dx = 0, 4 = sin 0, 4 = cos 0, and 4 = 0, where 0 is the inclination of the borehole from the vertical. Equation (12.48) then reduces to d u d d

The first term on the right is the obvious one, involving a change in inclination of the borehole with time. The second is the one illustrated in Figure 12.12 and discussed earlier. The third is an advection effect. In an area of non-zero vertical velocity, a section of a borehole at depth z2, measured with respect to some constant datum, and with inclination ¿i(z2) will, at the end of a time interval At, be at, say, depth zi (Figure 12.16). If the initial inclination of the borehole at depth z1 was different from li(z2), our measurements would show that the inclination at depth z1 had changed, and this would be true even if du/dz were 0. This is why w dli/dz becomes important near some joints, as just mentioned.

The results of the borehole deformation experiment reported by Hooke and Hanson (1986) will be used to illustrate an application of this analysis. Four boreholes, located approximately along a flowline on Barnes Ice Cap (Figure 12.17), were drilled and cased and inclinom-etry data were obtained from them over a period of up to four years. Figure 12.18a shows the deformation profiles, and Figure 12.18b shows values of 9u/dz calculated from Equations (12.48).

The deformation profiles in most of the holes end at the top of a zone of white ice (Figure 12.18a). Oxygen isotope data demonstrate that this ice is of Pleistocene age (Hooke and Clausen, 1982). The ice is white


Distance from divide, km


Distance from divide, km

| | Holocene deformed superimposed ice | | Holocene glacier ice | | Pleistocene bubbly white glacier ice

Vertical exaggeration, 5x

Figure 12.17. Longitudinal section along a flowline on Barnes Ice Cap showing types of ice encountered in boreholes. The deformed superimposed ice near the margin was overridden during an advance of the glacier (see Figure 5.17). (After Hooke and Hanson, 1986, Figure 2. Reproduced with the kind permission of Elsevier Science.)

because it contains a lot of air bubbles. As a result of these bubbles, the density of this ice is only 870 kg m-3, compared with a density of 920 kg m-3 in the overlying blue ice. We presume that the high concentration of air bubbles is a result of two processes.

1. When the climate warmed at the end of the Pleistocene, meltwater percolation increased, and ice lenses formed. These lenses trapped air in the underlying porous firn.

2. As basal meltwater escaped into the underlying permeable bedrock, air may have been left behind in a sort of physical fractionation process.

As noted in Chapter 11 (p. 311), it is commonly found that such Pleistocene ice is softer than Holocene ice, apparently because impurities lead to smaller grain sizes that then develop strong single-maximum fabrics (Paterson, 1991). The high strain rates implied by the dotted extrapolations of the deformation profiles for holes T061 and T081 in Figure 12.18a are indicative of this weakening. The value of B obtained for this ice from the deformation profile in hole T0975 is 0.1 MPa a1/3 (at -10.1 °C), which is much lower than those ranging from 0.23 to 0.30 MPa a1/3 in the overlying blue Holocene ice in holes T061 and T020 (Table 12.3) and also much lower than other experimental values at this temperature (Figure 12.5).

Also of interest are the values of the parameter A, defined by (see Equation (9.29))

Figure 12.18. (a) Velocity profiles in boreholes; and (b) du/dz as a function of depth. (After Hooke and Hanson, 1986, Figure 3. Reproduced with the kind permission of Elsevier Science.)

Velocity, m a

Figure 12.18. (a) Velocity profiles in boreholes; and (b) du/dz as a function of depth. (After Hooke and Hanson, 1986, Figure 3. Reproduced with the kind permission of Elsevier Science.)

Velocity, m a


a ei


ezx is obtained from the velocity derivatives (Table 12.2) using Equation (9.21), while azx is estimated with the use of z d r B

which is derived from Equation (12.26) in much the same way that we derived Equation (12.29) except that we now retain the T term and also assume that changes in the transverse direction are negligible in an

Table 12.3. Values ofB in MPa a 1/3 for different ice types'






Ice type

Weakly oriented Broad single maximum4 Two maxima Three or four maxima White ice (clean)

White ice (dirty)

0.46 (-9.0)c 0.44 (-8.6) 0.50 (-8.3) 0.18d(-7.8) 0.18e(-7.8)

0.23 (-10.2) 0.26 (-8.4) 0.24 (-7.5) 0.30 (-6.8) 0.10d(-6.4)

a Values given are for zones in which fabric is well developed, and thus exclude transition regions. b Equivalent fabric in T020 is small circle. c Numbers in brackets are mean temperatures in °C. d Velocity profile calculated by assuming no slip on the bed.

e Measured over two weeks, starting three weeks after completion of hole in 1977. No smoothing used in calculation.

ice cap. Equation (12.27) was used to evaluate the T term. If B and n are constant, as might be expected, A should vary inversely with e. The awkward fact is that near the surface where ezx is low, this does not appear to be the case. Figure 12.19a shows that A is effectively independent of e. Even the direction of change of A with depth is not consistent from one hole to the next, as indicated by the arrows on the curves in Figure 12.19a. This problem is not unique to Barnes Ice Cap; Raymond (1973) also found that A was independent of e near the surface of Athabasca Glacier.

Somewhat deeper in the glacier the situation improves, and A decreases steadily with increasing e (Figure 12.19b). Here, the slope and intercept of the log A - log e line can be used to determine B and n (Equation (12.50)). In the present case, however, Hooke and Hanson (1986) chose, instead, to assume that n = 3; they then calculated B as a function of depth, and related changes in B to changes in crystallo-graphic fabric. The results are shown in Table 12.3. Although there was quite a lot of noise in the record, it appears that B is slightly lower in fabrics containing only two maxima, and increases in fabrics with three or four maxima. This stiffening can be seen at the bottoms of the deformation profiles in Figure 12.18a. It is consistent with expectation, as it is the third and fourth maxima in these multiple-maximum fabrics that are inclined to the direction of shear (Figure 4.14f). In other words, the basal planes of crystals with these orientations dip either up- or downglacier,

cc CL

Octahedral strain rate, eo, a-1

Figure 12.19. Octahedral shear strain rate, so, plotted against A. (a) The upper 50 m of holes T061 and T081, and the upper 150 m of hole T020. Arrows show direction of increasing depth. (b) The lower parts of the holes. Depth increases from upper left, following lines of points. Reversals in trends reflect hardening of ice in zones where fabric is changing. (After Hooke and Hanson, 1986, Figure 4. Reproduced with the kind permission of Elsevier Science.)

whereas the basal planes in the first two maxima that form dip in the transverse direction (Figure 4.14e).

Values of B in hole T081 are nearly double those in the other holes. Hooke and Hanson assumed that this was because stresses at this location on the glacier were overestimated. However, they were unable to isolate the apparently erroneous assumption that led to this error, even though they undertook calculations with a finite-element model.

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