2.1. Field site

Haut Glacier d’Arolla is a small valley glacier at the head of Val d’Hérens in southwestern Switzerland (Fig. 1a). It has a surface area of ~6km² and ranges from ~2600 to ~3600m in elevation (Fig. 1b). The glacier has been subject to extensive multidisciplinary scientific investigations since 1989. Our study area is located in the eastern part of the glacier’s ablation zone (Fig. 1b), and spans a major melt-season subglacial channel that reopens ephemerally in July and is particularly active in late summer (August) (e.g. Reference SharpSharp and others, 1993; Reference Hubbard, Sharp, Willis, Nielsen and SmartHubbard and others, 1995; Reference Gordon, Sharp, Hubbard, Smart, Ketterling and WillisGordon and others, 1998; Reference Kulessa, Hubbard and BrownKulessa and others, 2003a). The area is known to be at least partially underlain by unlithified sediments whose thickness ranges from ~0.05 m near the channel to ~0.26m several tens of metres away from it (Reference Harbor, Sharp, Copland, Hubbard, Nienow and MairHarbor and others, 1997). The bedrock below the glacier consists mainly of schistose granites and gneisses of the Arolla series (e.g. Reference Mitchell, Brown and FugeMitchell and others, 2001).

Fig. 1. (a) The location of Haut Glacier ďДroiia in Switzerland (open circle). (b) The location of the study area on the glacier (open circle). (c) The surface (open circles) and base (closed circles) locations of the five boreholes used in this study. Boreholes 1(N, N = north) through 4(E, E = east) were used for electrical resistivity measurements, and borehole 5(C, C = centre) for water-pressure (P) and electrical conductivity (EC) logging. All coordinates in (b) and (c) correspond to the Swiss National Grid, and are abbreviated in (c) for clarity.

2.2. Borehole array and associated data

An array of five boreholes (Fig. 1c) was drilled to the glacier bed in early August (day 215) 1996. All boreholes had frozen shut near the glacier surface at the time the measurements were conducted (days 222–223 and 232– 233). A diamond-shaped array was established to allow measurements both parallel and perpendicular to ice flow, although array shape was distorted with depth due to nonvertical drilling (Fig. 1c). The location of all borehole bases was determined by inclinometry, with an accuracy of better than 0.5% of ice thickness (Copland and others, 1997). Borehole bases were located at each corner (boreholes 1(N), 2(W), 3(S) and 4(E)) and in the centre of the diamond (borehole 5(C)). The boreholes connected well with the subglacial water system after drilling, as indicated by fluctuating water levels (Reference SmartSmart, 1996). Borehole 5(C) was used for automated water quality measurements. Solid lead electrodes, soldered to single-conductor cable, were lowered to within ~0.5m of the bases of the remaining four boreholes. A further solid lead electrode was buried in an open glass jar ~1 m deep in the nearby lateral moraine to avoid exposure to surface temperature variations.

Boreholes 1(N) to 4(E) were mainly used for automated long-term (August 1996–June 1997) electrical self-potential measurements (Reference KulessaKulessa, 2000; Reference Kulessa, Hubbard and BrownKulessa and others, 2003a, Reference Kulessa, Hubbard, Brown and Beckerb), which were twice briefly interrupted to allow use of the electrodes for the electrical resistivity measurements reported here. On the basis of self-potential data collected between the two measurement periods considered here, Reference Kulessa, Hubbard and BrownKulessa and others (2003a) inferred that channelized water flow occurred along an axis spanned by boreholes 5(C) and 1(N) (Fig. 1c). Although the subglacial sediments in our study area contained noticeable amounts of clay minerals during the measurement period (Reference SeagrenSeagren, 1999), they were largely depleted immediately adjacent to the area of preferential flow as a result of subglacial erosion by hydraulic interaction between this flow and the surrounding sediments (Reference Kulessa, Hubbard and BrownKulessa and others, 2003a).

2.3. Electrical resistivity: concepts

Electrical resistivity measurements allow characterization of the subglacial environment in terms of its ability to conduct electrical current. The electrical resistivity technique commonly involves a series of four-electrode measurements, where two electrodes are used for current injection and two for voltage measurements. Apparent resistivity (p_a) is calculated using

where G is a factor related to the geometrical arrangement of the electrodes, I is injected current, and ΔV is measured potential difference (e.g. Reference Telford, Geldart and SheriffTelford and others, 1990, p. 542). We assume a three-layer system of basal ice, water-saturated subglacial sediments and underlying bedrock. The bulk resistivity distribution in this system is determined from ρ _a by geometrically constrained data inversion, and depends strongly on the resistivity of subglacial water because the latter contains the vast majority of mobile ions. This implies that the current will be strongly concentrated in pore water within subglacial sediments, where they are present (Reference BlakeBlake, 1992; Reference Blake and ClarkeBlake and Clarke, 1999). We expect this to be the case for our study area at Haut Glacier d’Arolla, where ice resistivity is known to be very large (~10⁸-10⁹ Ωm; Reference Hubbard, Binley, Slater, Middleton and KulessaHubbard and others, 1998) and the resistivity of the crystalline bedrock is also high (several thousand Ωm; e.g. Reference Telford, Geldart and SheriffTelford and others, 1990, p. 290).

Archie’s first formula (Reference ArchieArchie, 1942; Reference Telford, Geldart and SheriffTelford and others, 1990, p.649) has been used widely to relate bulk resistivity (ρ _bulk) to water resistivity (p_water) and the electrical formation factor (F) of the unlithified sediment matrix (Equation (2a)). Archie’s second formula (Equation (2b)) relates F to the porosity (ф_s) and cementation factor (m_s) of clay-free sediments:

The improved variant of the Kozeny–Carman equation derived by Reference Revil and CathlesRevil and Cathles (1999) allows the hydraulic conductivity (K _s) of clay-free sediments to be estimated from median grain radius (R; equal to half the median grain-size), the formation factor (F) and the cementation factor for clay-free sediments (m _s):

where δ _water and μ _water are water density and viscosity respectively, and g is acceleration due to gravity.

The hydraulic conductivity of clayey sediments (K _cs) is related to Ks by:

where m _cs is the cementation factor for clayey sediments and the porosity of clayey sediments (ϕ_cs) is calculated from

where ψ _c and ϕ_c are respectively clay volume fraction and clay porosity (Reference Marion, Nur, Yin and HanMarion and others, 1992).

2.4. Electrical resistivity: field measurements

The electrical resistivity data reported here were collected as repeat measurement frames. A frame is defined as the compilation of all possible direct and reciprocal four-electrode measurements that resulted from switching between the five available electrodes. Respectively four repeat readings were taken for any particular direct or reciprocal electrode combination. This allowed identification and elimination of unacceptably noisy data prior to inversion (e.g. Reference Binley, Ramirez, Daily, Beck, Hoyle, Morris and WaterfallBinley and others, 1995). Such data were clearly distinguishable from acceptable data points, which commonly had associated errors of ~10% or less. Measurement frames were repeated every hour between 1300 h on day 222 and 1800 h on day 223 (first series; 28 frames in total), and 1300 h on day 232 and 1300 h on day 233 (second series;20 frames in total). Collection of a frame took ~15min. During the second series, severe thunderstorms forced interruption of the resistivity surveys on several occasions (1700 h on day 232, and 0200 h and 0400–0600 h on day 233). All resistivity data were collected by manual switching with a Geopulse earth resistance meter (Campus Geophysical Instruments, UK), an instrument described in detail in Reference Hubbard, Binley, Slater, Middleton and KulessaHubbard and others (1998).

Water pressure and electrical conductivity data were collected with an automated data logger at 10 min intervals in borehole 5(C) (the instrumentation is described in detail in Reference Kulessa, Hubbard and BrownKulessa and others (2003a)). For consistency with common notation (Equation (2a)), all values of water conductivity are given in terms of water resistivity in the present study, being a measure of total concentration of dissolved solutes in subglacial water (e.g. Reference Stone, Clarke and BlakeStone and others, 1993).

3.1. Inversion concepts

We have used the finite-difference code DCIP3D (Geophysical Inversion Facility, University of British Columbia, Canada; Reference Li and OldenburgLi and Oldenburg, 2000) to invert our electrical resistivity data. Suffice to say here that the inversion process involves least-squares minimization of the misfit between a current resistivity model and a reference model. The current model is calculated from the field data together with the measurement errors. The reference model (Fig. 2) corresponds to a time-averaged, basal resistivity structure, using estimates of the resistivities of glacial ice (10⁸Ωm), subglacial sediments (10³ Ωm) and crystalline bedrock (10⁵ Ωm). The size of each cell in the finite-difference mesh used in the inversion process is illustrated in Figure 2 (1.25 × 1 × 0.75 m³), which places a lower limit on the spatial resolution in both the reference and current models. On completion of the iterative inversion process, the final current model is assumed to correspond as closely to the actual 3-D resistivity structure of the basal environment as can feasibly be reconstructed from the field data.

Fig. 2. The reference model and core finite-difference mesh used for 3-D inversion of the subglacial electrical resistivity data, corresponding to a volume section of the basal environment. Thick glacier ice with a resistivity of 10⁸Ωm (black cells) is underlain by a thin layer of unlithified sediments (10³ Ωm, shaded light grey) and thick crystalline bedrock (10⁵ Ωm, shaded in darker grey). The dimensions of an isolated cell are shown for scale, and borehole base locations are illustrated for reference where possible. See Figure 1c for geometrical reference.

3.2. Results

Each measurement frame in either of the two series was inverted individually, resulting in a total of 48 3-D models of basal resistivity structure. Comparison of these models reveals that temporal changes in resistivity occur only within the blocks corresponding to the subglacial sediment layer, whilst all other blocks remain largely unaffected. This is consistent with the expectation that daily hydrological changes should largely be confined to this layer. A representative inverted model, together with the borehole base locations, is illustrated in Figure 3. The bulk resistivities are lower in the western part (< 1000 Ωm) than in the eastern part (~1000 Ωm) of the model. This model represents the data frame collected at 1000 h on day 223, which was the time of lowest water resistivity in the hydrological cycle between days 222 and 223 (Fig. 4b). Both water pressure (Fig. 4a) and resistivity (Fig. 4b) were typically high during the day and low during the night.

Fig. 3. The inverted reference resistivity frame recorded at 1000 h on day 223, 1996. Only the cells corresponding to the layer of unlithified subglacial sediments is shown, which slopes from the base of borehole 4(E) towards the bases of boreholes 3(S) and 5(C), and then towards the bases of boreholes 2(W) and 1(N). The north arrow points down-glacier, and water flow is predominantly towards the observer. Note that the view is in the opposite direction compared to Figure 2 as dictated by basal topography. See Figure 1c for geometrical reference.

Fig. 4. Water-pressure (a) and water resistivity (b) data recorded in borehole 5(C) (see Fig. 1c for borehole locations) during both survey periods.

To investigate time-lapse changes in bulk subglacial resistivities, the measurement frame illustrated in Figure 3 was subtracted from those collected at all other times. The difference data were then projected onto the horizontal plane, and ‘trimmed’ to focus on the area between the four borehole bases where measurement coverage was particularly high. Comparison of representative 2-D images (Fig. 5) with water resistivity (Fig. 4b) reveals a close correspondence. Resistivities increase between 1300 and 1500 h on day 222, then drop to a minimum around 1000 h on day 223, and finally increase again through the afternoon of day 223.

Fig. 5. Representative time-lapse resistivity difference images, projected onto the horizontal plane for convenient viewing. The dimensions of an isolated cell are shown for scale, and borehole base locations are shown for reference (see also Fig. 1c). The reference resistivity frame collected at 1000 h on day 223 (see Fig. 3) was subtracted from those collected at 1300 (a), 1500 (b) and 2100 h (c) on day 222, and at 1500 h on day 223 (d). Note that the images were trimmed from a rectangular shape to focus on the area of larger measurement coverage between the boreholes. Resistivity scale is different from Figures 2 and 3 because resistivity differences are shown, rather than absolute values.

4.1. Daily resistivity amplitudes and electrical formation factor

Daily amplitudes of bulk resistivity were calculated for each cell in the inverted models using bulk resistivity maxima and minima measured between 1300 and 1200 h on successive days for both the first and second series (Fig. 6). During the first series (Fig. 6a), amplitudes are high to the west of borehole 5(C) and lower to the east of it. During the second series (Fig. 6b), amplitudes are high near boreholes 3(S), 5(C) and 1(N), and decrease progressively away from this area. Assuming that the channel is marked by maximum daily amplitudes of bulk resistivity, these observations are consistent with channel establishment between the two measurement periods, which agrees with previous inferences by Reference Kulessa, Hubbard and BrownKulessa and others (2003a). Indeed, the locations of the highest daily amplitudes of bulk resistivity calculated for the second series agree approximately with the strike of the subglacial channel as inferred by Reference Kulessa, Hubbard and BrownKulessa and others (2003a) (Fig. 6b), allowing for limited spatial resolution as determined by the size of model cells and some statistical variability in amplitudes along the channel (as explained below). Daily amplitudes of bulk resistivity were subsequently averaged using the cells located along the channel axis (Fig. 6), and likewise the averages of all cells located along ‘virtual’ axes parallel to the channel were calculated. The resulting, mean daily amplitudes of bulk resistivity are illustrated in Figure 7 together with standard errors. Standard errors decrease away from the channel up to a distance of ~5 m, then increase again. Comparison of Figures 6 and 7 reveals that the increase in standard error at distances of >5 m from the channel is probably caused by a decrease in measurement coverage, and thus sensitivity in the inversion process. The same is true for the easternmost data point. Standard errors at the channel are probably high relative to adjacent data points because acquisition of any particular measurement frame took ~15min, a period over which noticeable changes in water resistivity occurred in the channel, increasing uncertainty in the inversion process.

Fig. 6. Daily amplitudes of bulk resistivity for (a) the first (days 222223) and (b) the second (days 232–233) series of measurements, projected onto the horizontal plane for convenient viewing. The dimensions of an isolated cell are shown for scale, and borehole base locations are shown for reference (see also Fig. 1c). Note that the images were trimmed from a rectangular shape to focus on the area of larger measurement coverage between the boreholes. The black arrow in (b) indicates the direction of channelized water flow as inferred by Reference Kulessa, Hubbard and BrownKulessa and others (2003a), which agrees approximately with the location of the highest daily amplitudes of bulk resistivity.

Fig. 7. Measured daily amplitudes of bulk resistivity (circles with error bars) and calculated exponential decay (solid line). Closed and open circles respectively represent data points of high and low confidence based on measurement coverage. See text for further explanation.

At the channel the electrical formation factor (F) is readily calculated using Equation (2a) since both bulk and water resistivity are known. Here, F has a value of 5.1 ± 0.2. The decrease in daily amplitudes of water resistivity with distance from the channel can be approximated by an exponential decay (Reference Kulessa, Hubbard and BrownKulessa and others, 2003a). Equation (2a) suggests that daily amplitudes of bulk resistivity should therefore also decrease exponentially in the same direction, unless F changes. Daily amplitudes of bulk resistivity close to the channel are fit well, in a least-squares sense, by an exponential decay (Fig. 7). This implies that F is approximately constant up to a distance of ~2.5 m from the channel. At larger distances daily amplitudes of bulk resistivity lie consistently above the calculated exponential decay, implying that F is higher at these distances than at the channel. We find that F has values of 11.4 ± 0.8 at a distance of 5 m from the channel, and 37.0 ± 16.0 at the farthest distance of 8.75 m (Fig. 8a). We emphasize that values calculated for distances of >5m are subject to considerable uncertainty since measurement coverage is relatively poor (see above).

Fig. 8. Calculated values of electrical formation factor (a), porosity of clayey sediments (</>_cs) (b) and hydraulic conductivity of clayey sediments (K_cs) (c) with distance from the channel. Closed and open circles respectively represent data points of high and low confidence based on measurement coverage. See text for further explanation.

4.2. Determination of sediment porosity and hydraulic conductivity

Estimation of porosity is complicated by the fact that Equation (5) contains the cementation factor of clayey sediments (m_cs) and clay volume fraction (‘_c), both of which are uncertain and potentially change with distance from the channel. Similarly, estimation of hydraulic conductivity using Equations (3) and (4) requires knowledge of median grain radius (R) and the cementation factor of clay-free sediments (m _s). Both parameters are uncertain, and additionally R potentially changes with distance from the channel. Here, we select a best estimate together with a range of uncertainty for each parameter, based on parameter values and relevant associated information previously published in the literature. Best estimates and uncertainty ranges are summarized in Table 1 and were chosen using the following criteria:

In either of these cases, worst scenarios are considered. Actual uncertainty was probably much less in most cases. Standard errors were calculated for porosity and hydraulic conductivity as each parameter is varied individually within the given range of uncertainty, all other parameters being kept constant at the respective best estimate. An overall standard error for porosity and hydraulic conductivity was subsequently calculated. Sediment porosity (Fig. 8b) was found to decrease from 0.34 ± 0.01 at the channel to 0. 26 ± 0.01 at a distance of 5 m from it, and hydraulic conductivity (Fig. 8c) was calculated to decrease from (6.4 ± 2.1) × 10^–2ms^–1 to (3.3 ± 2.2) × 10^–2ms^–1 over the same distance. At distances of >5 m from the channel, porosity was found to decrease further and hydraulic conductivity to increase again. However, confidence in these findings must be low as discussed above, and illustrated further by high standard errors in the case of hydraulic conductivity (Fig. 8c), in particular.

4.3. Discussion

The hydraulic conductivity of the unlithified sediments near the subglacial channel in our study area was previously estimated to have values of several 10^–2 ms^–1 depending on exact location, based on slug-test data collected in the 1995 summer melt season (Reference Kulessa, Hubbard and BrownKulessa and others, 2005). An earlier hydraulic conductivity estimate by Reference Hubbard, Sharp, Willis, Nielsen and SmartHubbard and others (1995), based on reconstruction of diurnal water-pressure waves measured in the 1993 summer melt season, was smaller (~10^–4ms^–1). This discrepancy is explained by Reference Kulessa, Hubbard and BrownKulessa and others (2005) in terms of progressive removal of fine sediments between 1993 and 1995 by channel-induced erosional processes, the existence of which was originally inferred by Reference Hubbard, Sharp, Willis, Nielsen and SmartHubbard and others (1995). Since the present study was conducted 1 year after the 1995 melt season, we assume that the slug-test derived hydraulic conductivity estimate of several 10^–2ms^–1 may be more representative for the present (1996) subglacial hydraulic conditions than that of Reference Hubbard, Sharp, Willis, Nielsen and SmartHubbard and others (1995). Indeed, we observe that the present estimates of hydraulic conductivity ((6.4 ± 2.1) × 10^–2ms^–1 decreasing to (3.3 ± 2.2) × 10^–2ms^–1) agree well with those inferred by Reference Kulessa, Hubbard and BrownKulessa and others (2005).

In contrast to hydraulic conductivity, the porosity of the subglacial sediments in the present study area has so far not been known well. It is commonly assumed that sediment porosities of 0.3–0.4 may be typical for unlithified subglacial sediments (e.g. Reference Fountain and WalderFountain and Walder, 1998), although values as low as 0.2 have also been considered potentially realistic (e.g. Reference ClarkeClarke, 1987). The present porosity estimates (0.34±0.01 decreasing to 0.26±0.01) thus lie well within the expected ranges.

The individual dependence of hydraulic conductivity and porosity on the four uncertain parameters involved in the electrical–hydraulic relationships is illustrated in Figure 9. The overall ranges of cementation factors (m_s, m_cs), clay volume fraction (ψ_cs) and median grain radius (R) in Figure 9 are defined approximately by the maximum and minimum values for each parameter in Table 1, irrespective of distance from the channel. We emphasize that these overall ranges must therefore not be confused with individual ranges of uncertainty used for particular distances from the channel in the calculation of hydraulic parameters (Table 1). Hydraulic conductivity increases with cementation factor of clayey sediments (mcs) and median grain radius (R), and decreases with clay volume fraction (ψ_cs) and cementation factor of clay-free sediments (m _s) (Fig. 9). Porosity increases with cementation factor of clayey sediments (m_cs) and decreases with clay volume fraction (ψ_cs) (Fig. 9a and b). The dependence of both hydraulic conductivity and porosity on m_cs is particularly strong, and relatively minor in the case of ψ_cs. The dependence of hydraulic conductivity on R is also considerable, and strong for m _s<1.5. Despite the marked dependence of hydraulic conductivity and porosity on several parameters and the fact that we have chosen worst-case scenarios in all cases when selecting individual ranges of parameter uncertainty in section 4.2 (Table 1), the resulting overall standard errors were found to be low (Fig. 8). In practice, it is important that at least constraints on median grain radius (R) and clay volume fraction (ψ_cs) are available. Knowledge of ψ_cs can also guide the choice of a best estimate and a range of uncertainty for the cementation factor of clayey sediments (m_cs) together with values previously published in the literature (section 4.2).

Fig. 9. Sensitivity of hydraulic conductivity (K _cs; solid lines and right ordinates) and porosity (ϕ _cs; dashed lines and left ordinates) of clayey sediments to changes in: (a) cementation factor of clayey sediments (m _cs); (b) clay volume fraction (‘_c); (c) median grain radius (R); and (d) cementation factor of clay-free sediments (m_s). Ordinate scale is the same in all plots for convenient comparison, although in (a) hydraulic conductivity reaches much higher values for m_cs>2.2 than those shown on the ordinate.

In summary, our inferences suggest that hydraulic conductivity and porosity of unlithified subglacial sediments beneath Alpine glaciers can decrease noticeably over distances as small as a few metres away from subglacial channels. This implies that the speed of subglacial water flow towards or away from such channels probably changes on similar distance scales, which is likely also true for the ability of subglacial sediments to deform as their composition and hydraulic properties change.