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The deforming bed beneath a surge-type glacier : measurement of mechanical and electrical properties Blake, Eric Weston 1992

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THE DEFORMING BED BENEATH A SURGE-TYPE GLACIER:MEASUREMENT OF MECHANICAL AND ELECTRICAL PROPERTIESbyERIK WESTON BLAKEB.A.Sc., The University of Toronto, 1986A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of Geophysics and AstronomyWe accept this thesis as conformingTHE UNIVERSITY OF BRITISH COLUMBIAMarch 1992to the required standard® Erik Weston Blake, 1992Signature(s) removed to protect privacyIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)_______________________Department of ‘ kST,eO (‘toM (The University of British ColumbiaVancouver, CanadaDate MCH Z7 ,19’2DE-6 (2/88)Signature(s) removed to protect privacyABSTRACTGlacier surging is a flow instability characterized by short periods of rapid glacierflow separating longer periods of normal flow. It is accepted that sustained high subglacial water pressure causes glacier surging by decoupling the glacier from its bed, buthow this high subglacial water pressure is developed and sustained is the subject ofdebate. The current focus of glaciological research is on the interaction of subglacialprocesses with the subglacial drainage system.We have developed new investigative techniques for exploring two subglacial processes: basal deformation and electrical phenomena. These techniques have been applied in research undertaken on Trapridge Glacier, a small surge-type glacier in the St.Elias Mountains, Yukon, Canada; these are the first in situ measurements of deformation, electrical resistivity, and streaming potentials beneath a surge-type glacier.The development of a reliable rheological description of subglacial material required field observations of its stress—strain response; this was the motivation for oursubglacial deformation experiments. We have demonstrated that no clear relationshipexists between values of shear stress and effective pressure calculated using acceptedmethods and deformation rate; the absence of an expected relationship suggests thatalternate methods for quantifying subglacial shear stress and effective pressure need tobe found.Data from our subglacial electrical resistivity measurements and from the deformation measurements provide strong evidence that the distribution of normal and shearloading at the glacier bed is not even and that subglacial deformation rates can beaffected by distant changes in subglacial pressure conditions. We have also observedtemporal changes in electrical flow paths within subglacial sediments.We have shown that temporal variations in natural potential observed beneathTrapridge Glacier are caused by streaming potentials; streaming potentials result fromcross-coupling between fluid flow and electric currents. Our data suggest that inexpensive subglacial electrode arrays may be used to supplement pressure sensors.11TABLE OF CONTENTSAbstractList of FiguresList of TablesList of SymbolsAcknowledgementsChapter 1. INTRODUCTION1.1 What is glacier surging71.2 How does one identify a surging glacier?1.3 What happens during a glacier surge?1.3.1 Hard bed versus soft bed1.3.2 Seasonal timing .1.4 The study site1.5 Thesis scope1.5.1 Basal deformation1.5.2 Electrical phenomena1.5.3 Peripheral activities1.5.4 How does it fit together?Chapter 2. BED DEFORMATION2.1 Introduction2.2 Technique for sensor insertion2.3 Qualitative measurements .2.3.1 Bed casting2.3.2 Rubber rod2.4 Quantitative measurements2.4.1 Electrolytic tilt sensors11xxl”xivxvi1245789• • . . 13151516• . . . 1718181821• 21• 22• 2526AND METHODTHEORYIII2.4.2 Leaf spring tilt sensors2.5 Potential sources of error2.5.1 Sensor scale effects2.5.2 The ice—bed interface2.5.3 Sensor attitude2.5.4 Sediment intrusion into the borehole2.5.5 Connecting wiresChapter 3. BED DEFORMATION: DATA ANALYSIS3.1 Introduction3.2 Experiment design3.2.1 Ancillary information3.2.1.1 Internal deformation3.2.1.2 Basal sliding3.3 The 1988 experiment3.3.1 Correlation with effective pressure3.3.2 Effective viscosity3.4 The 1989 experiment3.4.1 1989 tilt results3.4.2 1989 strain rates3.4.3 Subglacial pressure3.4.4 Effective viscosity3.4.5 Net strain and mean strain rate3.5 Negative strain rates3.5.1 Fluid models3.5.1.1 Sheet flow3.5.1.2 Extrusion3.5.2 Roller bearing models343838394142444646474849535357616670707374757677777882iv3.5.3 The shadow box computer .3.6 Discussion3.6.1 Boulton and Hindmarsh flow models3.6.2 Shear stress and normal stress3.6.3 Effective viscosityChapter 4. ELECTRICAL PHENOMENA4.1 Introduction4.2 The rock—electrolyte interface4.2.1 Gouy—Chapman model4.2.2 Stern model4.3 Conduction mechanisms4.4 Electrical resistivity4.4.2 Measuring electrical resistivity . Potential fields4.4.2.2 Interpretation4.4.2.3 Current switching4.4.2.4 Transient effects4.5 Natural Potentials4.5.1 Irreversible thermodynamics4.6 Electrokinetic phenomena4.6.1 Observing streaming potentials .4.6.2 A theoretical development4.6.3 Hydrodynamics and boundary layers4.6.4 Zeta potentials4.6.5 The reverse current4.6.5.1 The charge accumulation model4.6.5.2 The charge conservation model8485868789THEORY 919191929496979999100103104107108112113113115115118119119V4.6.6 Calculating the streaming potential 1204.7 Electrical phenomena relevant to this study 121Chapter 5. ELECTRICAL PHENOMENA — METHOD 1235.1 Introduction 1235.2 The 1987 apparatus 1235.2.1 Electrode configurations 1245.2.2 Electrode design 1245.2.3 Voltage measurement 1255.2.4 Current source and current measuring 1265.2.5 Additional control 1285.2.6 Technical specifications 1295.3 The 1988 apparatus 1295.3.1 EPROM programs 1295.3.2 The current multiplexer 1305.3.3 The potential multiplexer 1315.3.4 Electrode design 1335.4 The 1989 apparatus 135Chapter 6. ELECTRICAL PHENOMENA DATA ANALYSIS 1366.1 Introduction 1366.2 Predicted forcing/response relationships 1376.2.1 Diurnal phenomena 1386.2.2 Episodic phenomena 1396.3 Assumptions 1396.4 The 1987 experiments 1416.4.1 Forefield operational test 1416.4.2 Experimental design 1426.4.3 Diurnal cycling of d.c. resistivity 144vi6.4.4 Geometrical corrections 1456.4.5 Polarity reversals 1456.4.6 Streaming potentials 1476.4.7 Recapping the 1987 field season 1486.5 Dedicated electrode arrays 1496.6 The 1988 Experiments 1496.6.1 Experimental design 1506.6.2 Telluric noise 1546.6.3 Potential error 1556.6.4 Potential gradients 1576.6.5 Hole connections 1586.6.6 Overwintering events 1626.7 1989 Experimental design 1646.7.1 Fall shutdown 1666.8 Manipulation experiments 1686.9 Conclusions 1736.9.1 Apparent resistivity 1736.9.2 Streaming potentials 174Chapter 7. CONCLUSIONS 1767.1 General comments 1767.1.1 Shear stress and normal stress 1767.2 Electrical phenomena 1787.2.1 Streaming potentials 1787.2.2 Electrical resistivity . . . 1797.3 Basal deformation 1797.3.1 Rheology 1797.3.2 Effective viscosity 180vii198198200200200201202203205• . . . 207• • . . 208• . . . 217220223227229229230234REFERENCES.. 182Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING 197A.1 Introduction 197A.2 Historical overviewA.2.1 Basic dip and azimuth measurementsA.3 Electronic incinometryA.3.1 Measuring tiltA.3.2 Measuring azimuthA .3.2.1 External azimuth controlA.3.2.2 Internal azimuth controlA.4 The UBC inclinometerA.5 Coordinate systemsA.6 Data analysisA.6.1 CalibrationA.6.2 TransformationsA.6.2.1 NormalizationA.6.2.2 Eulerian Angles . .A.6.2.3 Eulerian TransformationA.6.2.4 Universal application of transformationsA.6.3 Inverse problemA.7 Interpolation SchemeA.8 SensitivityA.9 Discussion211212213215217Appendix B. SUBGLACIAL WATER AND SEDIMENT SAMPLERSB.1 Niskin samplerB.2 Subglacial vacuum samplerB.3 ConsiderationsyinAppendix C. SUBGLACIAL DRAG SPOOL 235C.1 Introduction 235C.2 The “Slide-O-Meter” or “drag spool” 235ixLIST OF FIGURES1.1. Location of Trapridge Glacier .1.2. Topographic map of Trapridge Glacier2.1. The borehole percussion hammer .2.2. Bed casts2.3. Results from rubber rod experiment2.4. Electrolytic tilt sensors2.5. Data from electrolytic tilt cells2.6. Leaf spring tilt sensor2.7. Data from leaf spring tilt sensors .2.8. Drag spool data3.1. Location of Trapridge Glacier3.2. Velocity proffle through Trapridge Glacier3.3. Internal deformation proffle3.4. Location map for 1988 experiment .3.5. Data from 1988 experiment3.6. Comparison with Boulton and Hindmarsh3.7. Location map of 1989 experiment .3.8. Data from 1989 experiment3.9. Strain rate data from the 1989 experiment3.10. Pressure data from the 1989 experiment3.11. The vertical displacement record .3.12. Roller bearing model3.13. The shadow box4.1. The rock—electrolyte interface4.2. Detail of the Stern model interface .1012• 202324• 27323537434849525456data 65676971748183859395x4.3. Pseudo-depth calculation.4.4. The d.c. resistivity current waveform4.5. Induced polarization4.6. Streaming potential equivalent circuit5.1. The 1987 d.c. resistivity apparatus configurations5.2. High voltage current limiter5.3. A Cu—CuSO4porous pot electrode6.1. Forefield pseudo-section6.2. Resistivity record P1, 19876.3. Resistivity record P2, 19876.4. Polarity reversing electrode pattern6.5. Natural potential record from 1987 experiment6.6. Relative locations of electrode arrays6.7. Electrode array template6.8. The 88DC01 electrode array6.9. The 88DC02 electrode array6.10. Effect of telluric noise on measurements .6.11. Electrode noise6.12. Natural potential fluctuation6.13. Borehole forcing response6.14. Diurnal cycling6.15. Overwintering apparent resistivity record .6.16. Overwintering natural potential record . .6.17. The 89DC01 electrode array6.18. Fall shutdown electrical phenomena6.19. Map of 88DCO1B electrode array6.20. Subglacial phenomena during manipulation104105106120125127135142143146147148150151152153156157159160162163164165167170172xiA.1. Block diagram of UBC inclinometer 204A.2. The inclinometer coordinates systems 206A.3. The inclinometer orientation vectors 208A.4. Eulerian angles transformations 214A.5. The circular arc interpolation method . . . 220A.6. Detail of circular arc interpolation method . 221A.7. Interpolation of coplanar vectors 222A.8. Monte Carlo analysis of inclinometer sensitivity 226B.1. Niskin sampler 231B.2. Subglacial Hoover 233C.1. The drag spool 237XIILIST OF TABLES1.1. Geometrical characteristics of Trapricige Glacier 131.2. Characteristics of Trapridge Glacier study site 134.1. Established phenomenological relations . . 1115.1. Control line function, 1987 apparatus 1285.2. Control line function, 1988 current multiplexer . 1315.3. EPROM programs, 1988 current multiplexer 1325.4. EPROM programs, 1988 potential multiplexer 1335.5. Control line function, 1988 potential multiplexer 133xliiLIST OF SYMBOLSSymbol Meaninga glacier surface slope/3 glacier basal slopee charge on an electronE electric fielde dielectric constantstrain ratemean strain ratedynamic viscosityJ electrical current densityJ generalized flow densityg gravitational accelerationC d.c. resistivity geometrical factor7 sediment yield stressh layer or glacier thicknessH pore geometrical factorI electrical currentk reflection coefficientkB Boltzmann’s constantK1 thermal conductivityK2 hydraulic conductivityphenomenological coefficients1u ion chemical potentialn number density of ionsfl surface normal vectorP water pressure potential4 interface potentialo interface potential at surfacetotal electric potentialq fluid flow densityxivSymbol Meaningp scalar electrical resistivityPap apparent resitivitypi density of icepv volumetric charge densitypv. density of watero electrical conductivityT temperaturer shear stressV electric field potentialX generalized flow forcing(-potentialz valence of an ionz1, pseudo-depthxvACKNOWLEDGEMENTSTo my parents, Ir&grid and Wes, for giving me the motivationand skills to attempt a task such as this....and to Sabine, for her support from afar.I wish to express my sincere gratitude to my graduate supervisor, Garry K. C. Clarke.Throughout my sojourn at U.B.C., Garry’s door has never been closed to me (althoughit has been locked). Because of his generosity, I have enjoyed five wonder-filled fieldseasons in the glorious Kluane mountains, a winter sabbatical at the Scott Polar Research Institute, and numerous conferences. Perhaps Garry’s greatest attribute is hisunobtrusive and gentle manner; I have enjoyed what must be an uncommon freedomto explore subjects that have not always served Garry’s or my best interests.I wish to thank my friends and peers in the Department, in particular Marc G&in,Francis Jones, Tavi Murray, B. Barry Narod, Jeffrey Schmok, and Dan Stone, forentertainment and thoughtful discussion. I am grateful for the accommodations mycommittee members made for a rather hasty completion schedule: Drs. Robert M.Ellis, Rosemary Knight, Douglas Oldenburg, and Tadeus Ulrych.I owe special thanks to K. Dieter Schreiber, William Siep, and Harry Verwoerdfor their comments and help on instrumentation design and for their fine machiningskills. I also want to thank the Department of Geophysics and Astronomy as a wholefor providing a comfortable and friendly atmosphere eminently suitable for researchand education; this is no small thing.Funding for the research presented in this thesis, and for food to feed its author, has been provided by the Natural Sciences and Engineering Research Council ofCanada, the Canadian Northern Studies Trust of the Association of Canadian liniversities for Northern Studies, the University of British Columbia, and the NorthernScience Training Program of Indian and Northern Affairs Canada.xviChapter 1INTRODUCTION“I say, ‘Why don’t you write an anti-glacier book instead?’- Kurt Vonnegut, Jr., 1969, Slaughterhouse Five“Make as many corrections as you consider needed.”- Chinese fortune cookie, 17 February 1992Glaciers have long held the interest of humans. Anyone who has travelled througha glaciated landscape has experienced the sense of awe and wonder that these rivers ofice inspire. But glaciers do not always play a benevolent role in the landscape; peoplewho dwell near glacier fringes would be well advised to cultivate a sense of respect forglaciers and an appreciation for the dynamics of glacier flow.Encroachment of glaciers on structures is a clear threat, but it is likely that indirect consequences of glacier advance, such as ice-dammed lakes, ice avalanches, andoutburst floods wreak greater destruction. In the Swiss Alps, alpine glaciers havesent lethal torrents of ice and water into the inhabited valleys below and sometimes theglaciers themselves have crept down into the trees (Hoinkes, 1969; Röthlisberger, 1969a,1981). Similar events have been recorded in the Karakoram (Desio, 1954; Hewitt, 1969).Wonderful woodcuts, etchings, and paintings record the variations and consequencesof glacier motion in the Alps since the late 16th century (e.g. Hoinkes, 1969; Zumbühland others, 1981). On the west coast of North America, the flooding of land and theoutburst floods associated with ice-dammed lakes (formed at the margins of glaciersand by the damming of rivers by advancing glaciers) have been recorded in Indian oralhistory (e.g. Cruikshank, 1981) and in geological records (Clarke and Mathews, 1981;Clarke, 1982, 1986; Clarke and others, 1984a; Waitt, 1984; Schmok, 1986). Catastrophic1Chapter 1. INTRODUCTION 2sea-level rise resulting from melting of polar ice caps is perhaps the most stealthy ofglaciological disasters because the fateful event would transpire in a remote and generally unfamiliar area.The subject of this thesis is glacier surging, an uncommon mode of glacier flowthat has been associated with each of the glaciological mishaps listed above.1.1 What is glacier surging?Not all glaciers move with slow, steady persistence; some glaciers exhibit a periodicflow instability known as surging that is characterized by short periods of rapid glacierflow separating longer periods of normal flow. The quiescent phase of the cycle lastsfrom 10 to 100 years and the surge phase lasts for 1 to 6 years; for a given glacier,the periodicity of the surge cycle is thought to be regular (Post, 1960; Meier andPost, 1969). During a surge, the average flow speed of the glacier increases by one tothree orders of magnitude and results in the transport of ice from a reservoir area to areceiving area (Meier and Post, 1969; Clarke and others, 1986; Raymond, 1987). Thisdefinition of glacier surging is practical rather than exhaustive because, as with mostnatural phenomena, variety abounds and exceptions exist.Surging glaciers are a subset of a larger group, broadly described as fast flowing. Jacobshavns Isbr, a tidewater glacier in western Greenland, flows at more than8000 myr’ and is considered the world’s fastest glacier (Lingle and others, 1981), butexcept for a tidal influence near the terminus, its flow is relatively constant (Bindschadler, 1984; Echelmeyer and Harrison, 1990). It is tempting to speculate that Jacobshavns Isbr is surging since this sustained velocity is comparable to maximumobserved flow rates of surging glaciers elsewhere — for instance, the upper parts ofVariegated Glacier, a surging glacier in the Alaska Panhandle, flowed at a similarspeed during the 1982—1983 surge (Kamb and others, 1985) — but Jacobshavns IsbrChapter 1. INTRODUCTION 3is not delivering more ice to the ocean than is accumulated in its drainage basin (Bindschadler, 1984) so we could reasonably expect the glacier to flow at this speed indefinitely. Ice Stream B in Antarctica flows at a speed of about 800 m yr’ (Whillansand others, 1987) and has shown no signs of slackening its pace, yet this ice streamis operating at a deficit — it delivers more ice to the sea than is accumulated in itscachment area (Clarke, 1987a). Perhaps Ice Stream B is destined to slow down and“shut off” as evidence indicates its neighbour, Ice Stream C, has done (Shabtale andothers, 1987). This suggests that the ice streams controlling the mass balance (andperhaps the stability) of the Antarctic ice sheet are large analogues to surging glaciers,but operating at correspondingly large temporal scales.Surge-type glaciers tend to be clustered geographically, although why they arefound in particular regions is not well understood. In Canada, concentrations of surgingglaciers are found in the St. Elias mountains of the Yukon Territory, and on EllesmereIsland, N.W.T. Elsewhere, surging glaciers are found in the Pamirs, the Caucasus,the Tien-Shan, the Karakoram, the Himalayas, the Andes, Svalbard, Iceland, EastGreenland, the Alaska Range, and Switzerland (Dolgoushin and Osipova, 1975; Paterson, 1981, p. 283; Clarke and others, 1986). Not all glaciers within these geographicalconcentrations surge, and glaciers in close association with surge-type glaciers (thosethat share accumulation zones, for instance) do not necessarily surge. The clusteringof surge-type glaciers is probably related to subglacial conditions such as geology andhydrology rather than climatic ones (Post, 1969). For example, glaciers on the “wet”and “dry” sides of the St. Elias Range, Yukon, receive disparate amounts of annualsnowfall, yet surging glaciers are found in both areas (Clarke and others, 1986).Chapter 1. INTRODUCTION 41.2 How does one identify a surging glacier?Surging glaciers are best identified by direct observation of one or more surges. Particularly in uninhabited regions, such information is not available and we must rely onother clues (Paterson, 1981; Clarke and others, 1986).The most striking features often found on large surging glaciers are wavey, looped,contorted, or sheared medial moraines. These features are formed when tributaryglaciers surge or when the main trunk of the glacier surges and ice piles up againststagnant ice and debris near the terminus. Some caution must be used in attributingdeformed medial moraines to surging since these features can also be formed by changesin mass balance, movement of ice divides, and other prosaic changes in glacier systems.During the quiescent phase, a surging glacier may develop a characteristic longitudinal proffle that shows progressive thickening of the glacier in the reservoir zone andthinning in the receiving zone. This evolution of the longitudinal profile reflects thebuildup of ice mass in preparation for the next surge phase. The proffles of MedvezhiyGlacier (Pamir Mountains) and Variegated Glacier (St. Elias Mountains), two well-studied surge-type glaciers, have been shown to evolve in this manner (Dolgoushin andOsipova, 1975, 1978; Raymond, 1987; Raymond and Harrison, 1988) and Post (1960)has observed similar changes in Muldrow Glacier (Alaska Range). The longitudinalprofile of Trapridge Glacier (St. Elias Range) has evolved in a different manner as theglacier approaches its next surge phase: a prominent bulge has developed where theglacier flows into the stagnant ice left over from the previous surge; the formation ofthe bulge is probably controlled by thermal conditions at the base of the ice (Clarkeand others, 1984b; Clarke and Blake, 1991).Following a surge, the ice in the lower reaches of the glacier typically stagnates.Progressive melting during the quiescent phase can leave portions of this ice strandedbelow the main body of the glacier. The stranded ice, often covered with debris,may persist until the next surge cycle. It may also be possible to identify surge-typeChapter 1. INTRODUCTION 5glaciers by comparing the balance mass flux with the actual down-glacier transport ofice (Clarke, 1987a; Clarke, 1991). In accordance with the buildup of ice in the reservoirzone, the latter should consistently exceed the former for a quiescent surge-type glacier.1.3 What happens during a glacier surge?Glacier surging presents some fascinating questions: What causes the transition froma quiescent state to a surge state? How is a surge state maintained for a period of oneor more years? How does a surge terminate? Most of the work on surging glaciers hasfocused on the first of these questions.Several hypotheses for a surge trigger have been proposed. Tarr and Martin, (1914,chapter 10) observed rapid advances of several glaciers in Alaska following the 1899earthquake at Yakutat Bay and suggested that earthquakes may trigger surges. Subsequent work by Post (1960, 1967) discredited this hypothesis, but Gardner and Hewitt (1990) argued that rockslides onto a glacier surface (which can be triggered byearthquakes) may cause a surge in a glacier that is predisposed to surge behaviour.Robin (1955) proposed thermal instability in a surge-type glacier could permit therapid basal sliding rates which seem necessary to allow high surge speeds. A similarhypothesis developed by Clarke and others (Jarvis and Clarke, 1975; Clarke, 1976) isbased on a model containing a transition from a warm bed at the pressure meltingpoint (which allows sliding) to a cold frozen bed (which inhibits sliding), but this hypothesis is suspect as a generalized surge trigger because temperate surge-type glaciersexist (Bindschadler and others, 1976) (the bulk of a temperate glacier is at the pressure melting point). Later work (Clarke and others, 1977; Paterson and others, 1978;Cary and others, 1979; Yuen and Schubert, 1979) explored the effects of strain heatingon the stability of glaciers and ice sheets (earlier suggested by Weertman, 1957), butthe possibility that thermal instability could represent a surge trigger for glaciers wasdiscounted in part because the predicted surge cycle periods were too long.Chapter 1. INTRODUCTION 6The only hypotheses that retain wide acceptance are those which focus on theinteraction of subglacial water drainage, subglacial water pressure, and sliding processesas a means for explaining aspects of the surge cycle. The reason for this is that thehigh flow velocities encountered during a surge must be accompanied by rapid slidingat the glacier bed since internal deformation of the glacier cannot possibly contributesignificantly to motion. It is now widely accepted that glacier surging is caused by highsubglacial water pressure decoupling the glacier from its bed, and thus allowing theglacier to flow rapidly downhill. This high water pressure results from the destructionor disabling of the normal subglacial drainage system. The disabling of the drainagesystem also causes a rise in the quantity of water stored at the bed. What is notunderstood is how these alterations are initiated and sustained during a surge.High subglacial water pressure and water storage lead to fast flow rates by reducing the normal force exerted by the glacier on its bed (thereby reducing the maximum available friction force available to retard ice flow), by lubricating the ice-bedinterface, and perhaps by softening subglacial sediments. The positive correlation between water pressure and flow velocity has been observed experimentally (Iken, 1972;lodge, 1979; Iken and Bindschadler, 1986; Engelhardt and others, 1987; Kamb and Engelhardt, 1987, Meier, 1989) and explored theoretically (Bindschadler, 1983; Weertmanand Birchfield, 1983).The forerunners of current sliding models were Weertman’s “tombstone” model(1957) and Lliboutry’s “washboard” model (1959) — the names give an idea of the representation chosen for basal roughness. Lliboutry (and later Weertman (1962, 1969))introduced the idea that increases in the basal water pressure could raise the glacierclear of small obstacles in its path and thereby allow it to slide more easily. There followed a period of heated exchange between Lliboutry and Weertman regarding whosemodel more accurately portrayed the processes of sliding over a hard bed and thedisposition of subglacial water (Weertman, 1962, 1964a, 1966, 1967a, 1967b, 1969;Chapter 1. INTRODUCTION 7Lliboutry, 1964, 1966, 1967, [discussion following Weertman, 1967a}, 1968). Duringthis period of debate, other researchers made interjections of a theoretical and observational nature (e.g. Nyc, 1958; Kamb and LaChapelle, 1964 [commented on byWeertman, 1964b]; Kamb, 1970). Reviews of this debate can be found in Lliboutry(1968, 1979) and Weertman (1979). The salient difference between the two modelslies in how they deal with subglacial water: Lliboutry’s model proposes a system ofliziked cavities in the lee of basal obstacles; the pressure in and the movement of waterbetween cavities are controlled by narrow, self-adjusting channels, Walder (1986) andKamb (1987) incorporated these ideas into their proposed glacier surge mechanisms.Weertman’s model proposes a largely continuous film of water under the glacier withobstacles protruding through the film serving to retard glacier flow. It is easy to seehow Weertman argues that any thickening of the water film will increase glacier flowrate by drowning obstacles, but the principal flaw in this argument is that there is nofeedback mechanism to prevent the excess water draining into subglacial channels —the surge would halt straightaway.1.3.1 Hard bed versus soft bedUntil quite recently, most of the theory concerning basal sliding and surge mechanismshas been based on a hard bed model where the glacier is assumed to move over a hard,non-deforming substrate (e.g. Fowler, 1987a, 1987b; Kamb, 1987). Some work began onsoft (i.e. deformable) glacier bed models in the late 1970’s (Boulton, 1979a, Boulton andJones, 1979), but it was not until the late 1980’s that the community at large shiftedits thinking towards soft-bedded glaciers (Clarke and others, 1984b; Blankenship andothers, 1986; Shoemaker, 1986; Alley and others, 1987a, 1987b; Boulton and Hindmarsh, 1987; Clarke, 1987; Alley, 1989a, 1989b; Alley and others, 1989). A soft bedhas interesting ramifications for glacier motion and subglacial hydrology. Movement ofthe ice can occui through deformation of the sediments as well as sliding. SubglacialChapter 1. INTRODUCTION 8drainage can be through channels excised in the ice (Röthlisberger- or R-channels) orsediments (similar to Nye- or N-channels), through a linked cavity system, or throughthe sediments themselves, but the most important aspect of soft-bed glacier dynamicsis that sediment deformation can cause rapid changes in the mode of drainage.Some glaciers and ice sheets are underlain by a deforming layer of water-saturatedsediment (Boulton and Jones, 1979; Clarke and others, 1984b; Alley and others, 1986,1987a; Blankenship and others, 1986; Blankenship and others, 1987; Brown and others, 1987; Clarke, 1987b; Kamb and Engelhardt, 1989; Engelhardt and others, 1990;Alley, 1991; Blake and others, 1991; Clarke and Blake, 1991). High subglacial waterpressure associated with glacier surges has been attributed to morphological change inthe subglacial drainage system (Röthlisberger, 1969b; Clarke and others, 1984b; Kamband others, 1985; Alley, 1989a), so we are investigating the possibility that subglacialdeformation causes the rise in water pressure by disabling the normal drainage system. Competition between processes that establish the drainage system and those thatdestroy it could easily degrade the efficiency of subglacial drainage. If the drainagesystem of the bed is adversely affected to the point where water input exceeds drainagecapacity, then high water pressure and surge conditions might arise.1.3.2 Seasonal timingIn several instances, the initiation and termination of surges has been observed tooccur in certain seasons. These observations may provide clues as to the nature ofthe surge mechanism. During the 1982—1983 surge of Variegated Glacier, both surgepulses began in winter and terminated in early summer (Kamb and others, 1985).Recent surges of Medvezhiy Glacier in 1963 and 1973 also conformed to this seasonalschedule (Dolgoushin and Osipova, 1975), but a recent surge of Peters Glacier initiatedin summer and terminated in mid-winter (Echelmeyer and others, 1987). Accordingto Raymond (1987), no contradictory evidence has been found for seasonal initiationChapter 1. INTRODUCTION 9timing, but he notes that generalized statements concerning the timing of surge onsetand termination should be avoided since observation of glacier surge events is haphazardand spotty.1.4 The study siteThe St. Elias Range, Yukon Territory, has one of the world’s highest concentrations ofsurge-type glaciers. Many of the larger ones, such as the Lowell, Tweedsmuir, Steele,and Donjek Glaciers, cause flooding and outburst floods when they surge across therivers they abut. As recently as 1852, the land now occupied by the town of HainesJunction and a good portion of the Alaska Highway were flooded by an episode ofNeoglacial Lake Alsek. This lake is created when the Lowell Glacier surges acrossthe Alsek River (Schmok and Clarke, 1989) and pinches off the flow against GoatherdMountain.Trapridge Glacier is a small subpolar surge-type glacier in the later stages of itsquiescent phase (the temperature profile of a subpolar glacier begins with below-freezingtemperatures in its upper layers and warms to the pressure melting point at the base).Trapridge Glacier is located in the St. Elias Range near the northern extent of KluaneNational Park (Figure 1.1). The glacier flows eastward off a flank of Mt. Wood into avalley that contains two other glaciers that have been identified as surge-type: Rustyand Backe Glaciers (Clarke and Classen, 1970) (the names of the glaciers have changedsince this paper was published; Trapridge, Rusty, and Backe Glaciers were earlier knownas Hyena, Fox, and Jackal Glaciers respectively). Trapridge Glacier is the study sitefor the experiments discussed in this thesis.Chapter 1. INTRODUCTION 10Fig. 1.1: (a) The location of Trapridge Glacier in the St. Elias mountains,southwestern Yukon. (b) Trapridge Glacier is located near two large surge-type glaciers, the Steele and Hodgson Glaciers. (c) Trparidge Glacier flowsroughly north by east off the flanks of Mt. Wood (ice is white, bare groundis shaded).Evidence for the surge behaviour of Trapridge Glacier is threefold: (1) Photographic evidence from two expeditions into the Steele Glacier/Mt. Wood area in thelate 1930s through to 1941 indicate that Trapridge Glacier was then in a pre-surge state(Wood, 1940, cover plate; Sharp, 1947, Fig. 5; Sharp, 1951, Plate 5B). Sharp (1947) describes “Glacier 13” as advancing rapidly, but photographs of Trapridge Glacier takenin recent years look almost identical to those taken during this 1941 expedition, andno surge is currently in progress. The events of World War II disrupted exploration ofthe area and it was not until 1951, when the first vertical air photographs of TrapridgeGlacier were taken (Clarke and others, 1984b, Fig. 3), that the observational recordresumed (in 1943, a vertical air photography pass up the Steele Glacier valley stoppedtantalizingly short of Trapridge Glacier). In the 1951 air photo, Trapridge Glacier hasadvanced far down the valley; sometime between 1941 and 1951, Trapridge Glaciersurged. Subsequent vertical air photography through the 1970s shows the stagnationChapter 1. INTRODUCTION 11and melting of the lower reaches of the glacier (Clarke and others, 1984b, Fig. 3).(2) Stagnant ice in the valley below the glacier is visible in both the 1941 photographsand those taken in recent years, although the amount of stranded ice present in the1941 photographs is significantly greater. The vertical air photo sequence shows clearlythat this ice is the remains of the ice delivered to the receiving zone during the previous surge. (3) Since 1969, surface measurements on Trapridge Glacier have followedthe evolution of a wave-like bulge in the upper, active parts of the glacier (Clarkeand others, 1984b; Clarke and Blake, 1991). The formation of this bulge reflects theaccumulation of ice in preparation for the next surge.Trapridge Glacier is an ideal study site precisely because the glacier is small andthin. Its small size (roughly 1 km by 4km) allows us to roam the entire surface withease (Figure 1.2). Because the glacier is only about 75m thick, we can also drill manyholes to the glacier sole; our record for holes drilled was 83 in 1989. Researchers onlarger (and thicker) glaciers and ice streams are limited to a fraction of this number ofboreholes. Holes to the bed of the glacier are vital for studying basal phenomena andour ability to drill so many holes presents unique opportunities for the UBC glaciologygroup.Figures 1.1, 1.2 and 3.1 show the general location of the study area used for thebed deformation studies. Extensive hot-water drilling in this area indicates that theglacier has a very uniform thickness of about 72 m; during a given drilling season, we canoften predict the depth of a new hole to within 10—20 cm. For several hundred metresto the north, west, and east, the glacier topography is gently undulating. To the south,there is a slight rise associated with heavy crevassing. The surface and basal slopes, asdetermined by surveying and drilling, are both 7° in the direction of glacier flow. Theflow direction, as determined by stake surveys, is 11° north of east (bearing 79°). Datafrom vertical strain sensors indicates that the area is under moderate compression.Chapter 1. INTRODUCTION 12Fig. 1.2: Topographic map of Trapridge Glacier showing the entire catche—ment area. The contours and boundaries are based on 1981 air photography.The glacier has advanced about 300 m since that time in a north by eastdirection.Chapter 1. INTRODUCTION 13Tables 1.1 and 1.2 summarize the geometrical and thermal characteristics of TrapridgeGlacier and the study site.Elevation range 2250—2900mElevation of equilibrium line —2400 mGlacier length 3.5 kmTable 1.1: The geometrical characteristics of Trapridge Glacier (comparewith Figures 1.1, 1.2 and 3.1).Elevation —‘2370mSurface flow direction 110 north of eastWidth 0.95 kmThickness ‘-72 mSurface slope 7°Basal slope 7°Temperature at lOm depth —6°C to —3°CBasal temperature 0°CNet annual ablation (estimated from 0.5—1.Omyr’annual_survey_of flow marker_poles)Mean annual flow rate 33 + 1 m yr1(1984—1991)Table 1.2: Thermal and geometrical characteristics of the TrapridgeGlacier study area (sources: Clarke and Blake, 1991; unpublished data)1.5 Thesis scopeIt is our belief that a greater understanding of surging phenomena can be gained bystudying the basal processes of surge-type glaciers. To date, theoretical developments inthe study of surging have outpaced complementary and elucidating experimental work,principally because of the difficulties in making useful measurements in a subglacialenvironment.Experimental geophysicists are largely in the business of making remote sensingmeasurements of natural phenomena. When the object of investigation is the deforming bed beneath a glacier, measurements are necessarily of a remote nature and can beChapter 1. INTRODUCTION 14divided into two classes: (1) measurements made from the glacier surface and (2) measurements made near the ice-bed interface. Surface measurements are remote in theclassical sense; some energy field, either a natural one or one arising from the application of an artificial primary field, is observed. Basal measurements may be similar tosurface measurements or they may be direct measurements of basal properties in thevicinity of the sensor.Measurement techniques that have been used to investigate basal processes arelisted together with representative references: Surface measurements include radarsounding (Jones, 1989), seismic refraction (Blankenship and others, 1987), electricalresistivity (Röthlisberger, 1967), and gravity (Bull and Hardy, 1956). Basal measurements include temperature (Clarke and others, 1984b; Clarke and Blake, 1991), seismictomography (Clarke and Blake, 1990), electrical resistivity (Haeberli and Fisch, 1984;Hooke and others,. 1988), water pressure (Iken and Bindschadler, 1986), and deformation (Boulton and Hindmarsh, 1987; Kohier and Proksch, 1991).Prior to the development of efficient hot-water drilling techniques, researchers werelimited to making surface measurements and basal measurements requiring a smallnumber of holes. In recent years, basal measurements, some involving many holes,have become in vogue because they provide an opportunity for detailed examination ofthe subglacial system. Nevertheless, surface measurements are still yielding astonishingresults (for example, seismic refraction techniques were used to discover the deformingsediment layer beneath Ice Stream B, Antarctica).The glaciology group at UBC is currently using a suite of specialized instrumentsfor making basal measurements. Many of the instruments developed in the course ofthis thesis work continue to be used; more instruments are being developed every year.In preparation for my first data collection season in 1987, we decided to pursue twolittle-used basal measurement techniques that we felt had great potential for clarifyingbasal processes associated with a surge-type glacier: basal deformation and electricalChapter 1. INTRODUCTION 15resistivity. A serendipitous failure of the electrical resistivity equipment that summerled us to expand our investigations to include naturally occurring electrical potentials.1.5.1 Basal deformationAn understanding of subglacial deformation processes is crucial for developing a rheology for subglacial sediments and for understanding the role that these sediments mightplay in surging. In 1987, we were aware of only one other research group who had madein situ measurements of subglacial deformation (Boulton, 1979a; Boulton and Hind-marsh, 1987). As will be discussed in Chapter 3, their work has various limitations thatthrow into question the value of the rheologies they derive; our experiments constitutean attempt to correct these faults and to make definitive measurements of subglacialdeformation. Our measurements are also the first to be made beneath a surge-typeglacier.1.5.2 Electrical phenomenaSubglacial electrical resistivity and streaming potential measurements have the abilityto monitor the movement of water in the subglacial environment. Electrical resistivityprovides a measure of the bulk water content of the sediments and streaming potentialsprovide a measure of the water pressure gradient through the sediments; measured overtime, these electrical phenomena should be sensitive to the morphological changes inbasal sediments that lead to elevated subglacial water pressure and surge initiation.In 1987, we were also only aware of one other researcher who had made basal measurements of subglacial electrical resistivity (Haeberli and Fisch, 1984) although measurements were not made over a period of time. Several researchers have made surfacemeasurements of resistivity (e.g. Hochstein, 1967; Röthlisberger, 1967; Röthlisbergerand Vögtli, 1967; Vögtli, 1967; Fisch and others, 1977), but there are obvious difficulties in observing a conductive subglacial system through a thick and effective insulatorChapter 1. INTRODUCTION 16— the glacier. Since the initiation of our work, Brand and others (1987) have publishedmeasurements of subglacial resistivity beneath Storgiaciliren, Sweden, but their experiment was on a much smaller scale, both temporally and spatially, and did not involvemeasuring the evolution of subglacial streaming potentials over time. Our observationsof subglacial streaming potentials are a first for glaciology and our measurements ofelectrical phenomena in general are the only ones spanning a considerable period oftime. Additionally, no other researchers have made electrical measurements of any sorton surge-type glaciers.1.5.3 Peripheral activitiesThe electrical phenomena experiments require not only the drilling of many holes, butalso the calculation of borehole trajectories so that the position of subglacial sensorscan be determined. This requirement became evident when the difficulty in interpreting the data from 1987 was encountered. In 1988, we borrowed an inclinometer fromthe U. S. Geological Survey, and in 1988, we had our own prototype inclinometer built.The processing of inclinometer data is by no means a trivial procedure and considerable effort was expended in developing a robust processing algorithm. The analyticalprocedures were derived specifically for the UBC instrument but are applicable to inclinometers in general. This work is detailed in Appendix A. In the future, we hopethat accurate measurements of internal deformation within Trapridge Glacier will bemade with the UBC inclinometer.Interpretation of subglacial deformation and speculation concerning the surgemechanism are facilitated if samples of subglacial material are available. Samples fromthe forefield of the glacier provide a good hint as to what material is found underthe ice, but it is desirable to have samples taken directly from the actively deformingbed. Two original designs for subglacial samplers have been developed at UBO: theChapter 1. INTRODUCTION 17“Hoover” and the modified Niskin sampler, although I can only claim to have designedthe former. These instruments are described in Appendix B.1.54 How does it fit together?The relationship between subglacial electrical phenomena and subglacial sediment deformation might seem obscure, but both these investigative avenues provide information concerning the disposition of water in the subglacial system and may supply cluesrelating to the glacier surge mechanism operating at Trapridge Glacier.Chapter 2BED DEFORMATION: THEORY AND METHOD2.1 IntroductionDeformation beneath soft-bedded glaciers may be a physical mechanism that contributes to flow instabilities such as surging. A rheological description is requiredif the role of bed deformation is to be understood, but the development of a rheologyis hampered by a paucity of in situ stress and strain measurements. In this chapter,we describe four new techniques for measuring subglacial strain. Three of these givecontinuous strain measurements, a capability that permits calculation of instantaneousstrain rates and allows comparison of strain data with other time series. We alsodescribe a technique for inserting sensors into subglacial sediments.We limit the scope of this chapter to a description of experimental techniques andinclude data only to illustrate strengths and weaknesses of each approach, and to drawobvious qualitative conclusions concerning sediment rheology. A more complete dataset and analysis are presented in Chapter 3.The substance of this chapter has been published previously in the Journal ofGlaciology (Blake and others, 1991).2.2 Technique for sensor insertionA hollow percussion hammer allows a flexible instrument to be inserted into the stiffglacier bed (Figure 2.1). The hammer consists of a 2 m long tubular stainless steel bodyhaving an inside diameter of 10.87mm and an outside diameter of 17.3mm (0.375 in,schedule 80 seamless pipe). Brass blocks of 3.81 cm (1.500 in) diameter are fixed toboth ends of the body. The lower block serves as an anvil for a tubular striker that18Chapter 2. BED DEFORMATION: THEORY AND METHOD 19slides over the body. This striker, which is 60 cm long and weighs 2.1 kg in water, issuspended by a 1.59mm (0.0625 in) stainless steel wire yoke that threads through theupper stop block. The lower body thread extends below the anvil and allows variousaccessories to be attached. When installing a bed strain instrument, we attach aninsertion sheath of sufficient internal diameter to allow a loose fit for the instrument.Each instrument is terminated with a conical brass tip that fits snugly into the sheathwhen the instrument and its cable are threaded through the percussion hammer. Thetip has a larger diameter than the sheath, so that the lip of the cone forms a bluntannular barb.While the hammer, with the bed strain instrument loaded inside, is lowered downa hole, its weight is carried by the wire rope. The instrument cable is held just tightenough to keep the instrument from sliding out of the hammer. To avoid jostling theassembly, the rope and cable are laid out on the ice surface and the operator walkstoward the hole from a distance greater than the thickness of the glacier, holding bothlines. When bottom is reached, the taut instrument cable is marked level with theice surface. As the wire rope is used to drop the striker repeatedly against the anvil,percussive forces are transferred to the brass tip; the instrument remains protectedwithin the sheath. The cable is left slack during hammering so that impulsive advancesof the hammer and instrument will not yank on the cable. Insertion depth is measuredby observing the displacement of the mark with respect to the ice surface, although thisdisplacement does not reflect the position of the instrument with respect to the ice—bedinterface. Hydraulic excavation by our hot water drill creates a soft, decimetres-thickdisturbed layer in the subglacial material through which the hammer penetrates byvirtue of its static weight. The hammering procedure results in penetration of theunderlying undisturbed basal material. Thus, even if the length of the instrument isless than the insertion depth into undisturbed material, the entire instrument may bebelow the ice—bed interface.Chapter L BED DEFORMATION: THEORY AND METHOD 20WIRE ROPESTOPSTRIKERBODYINSERTIONSHEATHINSTRUMENT TIPFig. 2.1: Schematic diagram (not to scale) of the borehole percussionhammer used to insert flexible strain instruments into the soft bed of TrapridgeGlacier. The tool is sufficiently narrow that a figure drawn to scale would hidedetails.The hammer weight is light (2.1 kg in water), but repeated light blows have a cummulative effect similar to a few heavy blows and are certainly gentler on the instrument.As many as one hundred blows may be required before the hammer ceases to advance.When the hammer is withdrawn, the lip of the instrument tip catches in the sedimentand draws the instrument from the sheath. Once the harmner has been lifted to theANVILChapter 2. BED DEFORMATION: THEORY AND METHOD 21surface, the instrument cable is tied off at the surface with 1—1.5m of slack cable letdown the hole.2.3 Qualitative measurementsBed strain measurements were first attempted during the 1987 field season. Our primary goal was to determine if measurements of bed deformation were possible using thepercussion hammer insertion technique. With this in mind, we developed two simplemethods for making qualitative strain measurements: the bed cast and the rubber rod.2.3.1 Bed castingThe bed cast is a simple method for measuring total strain. A length of vinyl tubing,containing freshly mixed casting resin and a heating wire, is hammered into the bed. Atfreezing temperatures the catalyzing reaction is effectively halted, so the tube remainsflexible. After several days of deformation, the resin is heated and hardened by passinga current through the wire. The stiff tube, now a cast of the deformation to which ithas been subjected, is then pulled out. In most cases, the cast retains enough elasticityto survive passage through the borehole undamaged; if the cast should crack, its shapeis easily reconstructed.In our experiments, we use vinyl tubing having an outside diameter of 6.35mm(0.250 in) and an inside diameter of 4.76 mm (0.1875 in). The polyester casting resin isdoped with the recommended amount of catalyst for hardening at room temperature.The nichrome heating wire has a resistance of 35 Il m1. Hardening is accomplishedby circulating 0.9—1.3 A of current through the wire for —‘3 h. This represents a powerdissipation of 28—59Wm’ into the resin. The nichrome wire is sheathed in teflon“spaghetti” tubing to prevent the vinyl tube from melting. Because the resin is attackedby water, care must be taken to ensure that the instrument is watertight.Chapter 2. BED DEFORMATION: THEORY AND METHOD 22Figure 2.2a shows a cast which was hammered 12 cm into the bed and then catalyzed immediately. As expected, no deformation is evident. The slight curling of thecast was caused by shrinkage of the resin as it set. Figure 2.2b shows a cast whichwas inserted 7 cm into the undisturbed bed, and left 92 h before heating began. Thedashed line in the figure represents the surface of the undisturbed bed material, andthe dotted line indicates the estimated location of the ice—bed interface (20 cm abovethe undisturbed material). The distinct double bend offset near the bottom of thecast suggests that the zone containing the bend was deforming, but because the castmay have extended up into the ice, the bending could be attributed to glacier sliding.During the experiment, the glacier surface above the instrument moved approximately40cm.Two other bed casts were successfully installed in 1987 (a fifth cast was not properlyanchored in the bed and detached when the hammer was withdrawn). One, installedto a depth of 12 cm, had a break in its heating circuit and could not be catalyzed.The other, installed to a depth of 35 cm, was held so tightly by the glacier bed that itsnapped in two and only the upper 10cm of the cast was recovered.2.3.2 Rubber rodTo ascertain whether strain rate varies with time, we built a rudimentary strain gaugecapable of making continuous qualitative strain measurements. The rubber rod consistsof a 20 cm length of 6.35 mm (0.250 in) square-section rubber onto which two straingauge networks are bonded. The gauge networks are bonded onto the faces of the rodnear its midpoint and can record fiexure along two axes. The rod is sheathed in aglycerin-filled protective vinyl tube.Results from two days of observation using this instrument confirmed that temporal changes in the strain rate occur (Figure 2.3). The experiment ended when excessiveChapter 2. BED DEFORMATION: THEORY AND METHOD 23a bñ-4----_••e•..... .... ..... .... ....... — — — — — I I I I I— •41OcmFig. 2.2: Tracings, from photographs, of Trapridge Glacier bed casts ona 10 cm by 10 cm grid. (a) A bed cast catalyzed and withdrawn immediatelyafter insertion. (b) A bed cast left in the deforrning layer for 92 h before removal (the tip was lost upon withdrawal). The dashed line indicates the upperboundary of the undisturbed bed into which the bed casts were hammered; thedotted line indicates the estimated location of the ice—bed interface. Glaciermotion is to the left.strain caused breaks in the strain bridge wiring. The glacier surface velocity was approximately 12 cm day1 during the course of the experiment. The axes of strain areChapter 2. BED DEFORMATION: THEORY AND METHOD 2410 11AUG 1987Fig. 2.3: Results from the 1987 rubber rod experiment. The vertical scaleis in arbitrary linear strain units, as the device was uncalibrated. The solidline represents strain in the direction of ice flow. The dotted line representscross-flow strain.labelled down-flow and cross-flow assuming that the upward trending record (solid line)represents strain in the direction of ice flow, and the second record (dotted line) represents strain perpendicular to this direction. We believe these assumptions are correctbecause the physical characteristics of the rod cause it to twist within the vinyl tube sothat the bending strain is shifted to one pair of faces. A striking feature of the down-flow strain record is that it is not monotonically increasing. The slope of the curve,which is proportional to strain rate, is occasionally zero or negative. Because deformation counter to the direction of ice flow is unlikely, the negative strain rates suggestsporadic extrusive flow within the deforming layer. Extrusive flow within the layer isnot excluded because, unlike the glacier itself, the layer is confined from above and below by rigid boundaries. The glacier forms a rigid upper boundary, and non-deformingsediment or bedrock a lower boundary.Cl)IZz—Dfri —I‘dl109 12 13Chapter 2. BED DEFORMATION: THEORY AND METHOD 252.4 Quantitative measurementsOur success with qualitative techniques led us to design two types of tilt sensors capableof making continuous quantitative measurements of strain rate at several levels withinthe bed. Data from subglacial tilt sensors can be used to compute instantaneous strainrates (averaged over the length of the cell) by numerical differentiation of the tilt timeseries. For the purposes of this derivation, we assume that the tilt cells are experiencingsimple laminar shear in a fluid. A fluid model is reasonable for a fine-grained materialwhere the tilt cell is much larger than the largest clast, but as the subglacial materialbeneath Trapridge Glacier is inhomogeneous and contains clasts of a size comparableto that of the tilt cells, this model is not particularly suitable. It is likely that at thescale of observation of the tilt cell, the deformation cannot be treated as simple shearand that the tilt cells are influenced by the rotation or rocking of neighbouring clasts.It is also likely that the strain experienced by the tilt cell is not uniform over its length.Nevertheless, the fluid model does provide a framework for the interpretation of basaldeformation data. The question of scale of observation is addressed in a subsequentsection.Consider a coordinate system having its x axis positive in the direction of iceflow, and the z axis normal to the bed and positive upward. In a macroscopicallyhomogeneous material, the strain rates and are defined aslfOu t9w’\=+(2.1)andlfOv thv’\=+(2.2)Chapter 2. BED DEFORMATION: THEORY AND METHOD 26where u, v, and w, are the down-flow, cross-flow, and upward components of velocityrespectively. If Ow/Ox and Ow/Oy are negligible, these equations become1 Ou 1 OtanOdUt (2.3)and1 Ov 1 Otan8 (2.4)where the velocity gradients are expressed as the rate of change of down-flow tilt angle9, and cross-flow tilt angle 9. Tilt angle is measured with respect to the z axis; positivetilt corresponds to tilt in the positive x and y directions.24.1 Electrolytic tilt sensorsElectrolytic tilt cells operate on the principle that the conductance between two electrodes immersed in an electrolyte is proportional to the total wetted surface area ofthe electrodes. A single aids electrolytic tilt cell consists of three electrodes partiallyimmersed in an electrolyte (Figure 2.4). Electrodes may descend from the top of thecell or ascend from the bottom. As the cell is rotated about the plane of the electrodes,the central electrode remains immersed to the same level and the lateral electrodesare dipped in and out of the electrolyte. To make a tilt measurement, the tilt cell isconnected to an alternating cycle (a.c) bridge containing two reference resistors (a.c.excitation prevents electrode polarization). The magnitude of the output voltage ofthe cell is roughly proportional to the tangent of the tilt angle, and the sign determinesthe tilt direction. A dual axis tilt cell has five electrodes arranged in a “+“ patternwith the central electrode common to both circuits.Chapter 2. BED DEFORMATION: THEORY AND METHOD 27Fig. 2.4: Schematic diagram of the electrolytic tilt cell and electronicsused in the 1988 experiment. The cell is 57mm long and 16 mm in diameter.An a.c. excited bridge is formed between the cell and two reference resistors.As the cell is tilted, the lateral electrodes dip in and out of the electrolyteand their resistance with respect to the central electrode changes. The bridgeimbalance is measured to give the degree and polarity of tilt. For dual axistilt sensitivity, a second pair of lateral electrodes is mounted at right anglesto the first pair and the central electrode is shared by both circuits.Our tilt cells have copper electrodes suspended from the top of an acrylic cell.The electrolyte is a weak solution of Alconox, a common laboratory detergent. Thespace above the electrolyte is filled with naphtha to prevent water inifitration underpressure. When microfractures are etched in the surface of the acrylic by rinsing it for5 s in methanol, no visible meniscus forms between these two fluids; the fluid interfaceBRIDGE CIRCUITELECTRODESELECTROLYTEremains horizontal as the cell is tilted. The presence of a meniscus would interfere withChapter 2. BED DEFORMATION: THEORY AND METHOD 28accurate measurements of tilt because the electrodes are in close proximity to the wallof the cell.Our choice of immiscible fluids represents the best formula picked during a seriesof desperate in-field tests. During the 1988 field season, we discovered that the originalelectrolyte choice — saturated copper sulphate solution — did not behave properly inthe presence of naphtha or any other insulating fluid available to us. We are certainthat better choices exist (such as those in commercial devices) and do not recommendthat this electrolyte formulation be used. Since the tilt cells are not recovered after theexperiment is completed, the use of commercial tilt cells for these experiments was tooexpensive.Three cells, each 57 mm long and 16 mm in diameter, are assembled into a stringat a centre-to-centre spacing of 10 cm. The string is cased in a protective sheath of heatshrink tubing, and the lowermost cell is fitted into the insertion tip. Prior to insertion,the cells are calibrated on a special jig. The jig allows a cell to be rotated about itslong axis while the tilt of the axis is fixed at various angles. Our cells are calibrated attilt angles ranging between 00 and 60° from vertical. For each calibration tilt angle Oj,the output voltages from each tilt circuit are fitted, using least squares, to functions ofthe formV(8, q5) = A sin(4i + B) + C (2.5)where the subscript i enumerates the discrete tilt angles, A are the fitted signal amplitudes at those tilt angles, B are the azimuth offsets, C are voltage offsets, q’ representsthe azimuth of the tilt, and V is the predicted voltage output. It is important to notethat the azimuth q is measured in a local coordinate system where the z-axis alignedwith the long axis of the tilt cell; the azimuth is in no way equivalent to orientation ina geographical coordinate system.Chapter 2. BED DEFORMATION: THEORY AND METHODFor each tilt angle 8, the least squares fit is found using the equations29wherea/3a— a sinj=1)(2.6a)(2.6b)(2.6c)(2.6d)(2.6e)and where Vj are the measured output voltages at azimuths cEj. The symmetry evident(msin qf sin ci) Os7% n n-7% 7% nA = i/(a2 +132)B2=tan—( 7% /7%=— ( >::1=1 \j1/7%sin=(nsinc-.InsiIt.cos = injcosqj—\ j=1(2.6f)(2 .6g)(2.6h)sinq5,in the expressions for a and /3 suggests, as is indeed the case, that the least squares fitChapter 2. BED DEFORMATION: THEORY AND METHOD 30is derived for the equationV(9, 4) = a sin(q5) + /3 cos(4) + C (2.7)Equations (2.6c) through (2.6e) are used to determine A, B, and C.A tilt cell consists of two tilt circuits, so each cell has two sets of functions of thisform associated with it. For a given tilt angle, the circuits in a perfect cell would havevalues of B differing by exactly 90°. In practice, the electrode groups are imperfect andthe values of B differ by roughly 90°. When the field data are analyzed, a natural cubicspline (Press and others, 1986, p. 86) interpolates values of A, B, and C at tilt anglesintermediate to those for which calibrations were performed. This transforms the twosets of discrete calibration functions for a specific cell into the continuous system ofequationsV(,4) =A(8)sin[4+B(6)] +C(O)(2.8)V,(8, q.’) = A(O) sin{ + B(9)] + C(O)where the subscripts x and y distinguish the orthogonal tilt circuits. The NewtonRaphson method for nonlinear systems of equations (Press and others, 1986, p. 269) isused to calculate the values of 9 and q that would give the observed output voltagesV and Vi,. The error on tilt and azimuth for these sensors was estimated by using thecalibration data as input to the inversion scheme. The tilt angle error is +1° at 0° oftilt, increasing to +2° at 30° of tilt, and to ±3° at 60° of tilt. The azimuth error is ±5°.Though these errors are large, they are systematic. We believe that the relative errorbetween successive measurements is negligible because the tilt and azimuth recordsappear smooth. Based on zero relative error, the error in strain rate is 1% at 0° of tilt,increasing to 4% at 30° of tilt, and to 20% at 60° of tilt.On 10 August 1988, an electrolytic tilt sensor string was hammered 8 cm into theundisturbed bed of Trapridge Glacier. Although this insertion depth is shallow, this isinsertion depth into undisturbed sediments; because the insertion hammer penetratesChapter 2. BED DEFORMATION: THEORYAND METHOD 31through disturbed sediments before coming to rest at the bottom of the borehole, thetilt cells are actually inserted more than 8 cm below the ice—bed interface (section 2.5.2).Figure 2.5 shows the results for 23 days of observation. The data are presented in theraw form generated by applying the calibration functions: the solid lines indicate tiltfrom vertical, and the dotted lines indicate the azimuth of the tilt with respect to theinternal coordinates of the cell. During the course of the experiment, the glacier surfacein the study area moved about 10 cm day’.The data for the lowest two tilt cells indicate an initial net tilt angle of about 45°;this is a surprisingly large value. The deformation instrument is inserted co-axiallywith the bottom of the borehole, but this borehole had a 3° up-glacier tilt. Some otherdisturbance caused the tilt cells to “fall over” shortly after the percussion hanuner wasremoved. We believe that the heat shrink sheath stiffened the instrument sufficientlyso that when slack cable was lowered down the borehole, the instrument was pushedover. Even the bottom cell, which was inserted into undisturbed material, was affected.Evidently, the disturbed layer of the bed was not firm enough to prevent this motion.The rapid onset (within one day) of independent cell motion suggests that the disturbedlayer recovered quickly, allowing further shear deformation to be recorded.If we assume that the principal direction of tilt is in the down-flow direction, wecan further decompose the net tilt and azimuth values into down-flow and cross-flowcomponents of tilt. Variations in the azimuth of a tilt cell result from three motions:(1) rotation of the tilt cell about its long axis, (2) shear-like movement of the tilt cellin a direction perpendicular to the vertical plane passing through its axis (this willalso change the net tilt angle), and (3) pivoting of the tilt cell through a near-verticalorientation. This last movement results in a distinctive 180° shift in azimuth and asharp dip in the net tilt angle. In the absence of information on the orientation of thetilt cell in a geographical coordinate system, the relative contribution of each of thefirst two motions to changes in azimuth are unknown. In order to interpret the data,Chapter 2. BED DEFORMATION: THEORY AND METHOD 32(I)wwwI-1800_Cl)—180ww80c.w=180 1—180Fig. 2.5: The 23 day record from the 1988 electrolytic tilt sensor string.The upper, middle, and lower cells are labelled (a), (b), and (c), respectively.Solid lines indicate tilt from vertical (left scale); dotted lines indicate theazimuth of the tilt with respect to the internal coordinates of the cell (rightscale). The records from the two lower cells indicate that they were within adeformation zone that sometimes experienced zero and negative strain rates.The arrows indicate these times. The cartoon below the graphs shows theposition of the individual cells within the deforming layer at specific times.Ice movement is to the right.we make the assumption that azimuth changes due to rotation are generally long-term• a111111111 I 11111111111 I900.90090010 20 30AUG 198880glacier flow—..(over several days) and that changes due to shearing are generally short-term (less thanChapter 2. BED DEFORMATION: THEORY AND METHOD 33one day). The basis for this assumption lies in the observation that the azimuth timeseries tends to be a slowly-varying function upon which fluctuations are superimposed.By attributing the azimuth drift to cell rotation (which has no effect on strain rates),we can isolate the azimuth fluctuations, which in turn are attributed to both rotationand shear. This assumption is of little consequence because altering the partitioningof azimuth change between these two motions has a surprisingly small effect on thecomputed strain rates. In our analysis, the azimuth drift is removed by subtractinga linear drift function. The decomposition process must also account for the initialorientation of the tilt cell relative to the glacier flow direction; the initial cross-flow tiltmay be non-zero.The azimuth records shown in Figure 2.5 show little variation from piecewisestraight lines and, for this sensor string, the dip direction of the hole is aligned withthe glacier flow direction. We conclude that almost all the tilt for this particular dataset is in the down-flow direction so the data separate naturally into down-flow andcross-flow tilts. The component decomposition for these data are not shown becausewe wish to show the smooth character of the raw data, but the decomposition wouldshow a near-zero cross-flow tilt and a down-flow tilt mimicking the net tilt. Early in theexperiment, each of the lower two cells experienced a 180G shift in azimuth associatedwith a sharp decrease in tilt angle. This suggests that these two cells originally tiltedup-glacier; the azimuth shift indicates when each cell pivoted to tilt down-glacier.It is evident that all the cells quickly reached an orientation outside their calibratedrange. Indeed, the uppermost cell seems to have passed the whole experiment restingon its side. Late in the experiment, this cell was tipped over, probably by the glacierpulling on the cable. As with the strain record from the rubber rod, brief intervals ofzero and negative strain rate are observed. These are indicated by arrows on the figure.Chapter 2. BED DEFORMATION: THEORY AND METHOD 3424.2 Leaf spring tilt sensorsThe disadvantages of our electrolytic tilt cell are that it has a limited range over whichtilt measurements can be made and that the measurement error is large. Specialattention must be paid to sealing the cell and care must be taken in choosing a suitablestable electrolyte. These problems are overcome with the leaf spring sensor illustratedin Figure 2.6.Two leaf spring pendula measure the tilt along perpendicular axes. Each leafspring consists of a small clamped strip of 50.8 um (0.002 in) spring brass having amass attached to the free end. A pair of small strain gauges is bonded to the faces ofthe spring. As the cell tilts, the weight of the mass bends the spring, and strains thegauges. The output voltage is roughly proportional to the sine of the tilt angle, so thatthe tilt angle itself is ambiguous when the cell is tilted more than 90° from vertical.These cells are 47mm in length and 16 mm in diameter. Each cell is made of acrylic,and is filled with Dow Corning 200 fluid. This inert silicone fluid is non-conductive andnon-corrosive, serves to dampen vibration of the masses, and thwarts water inifitration(all these characteristics prove necessary). The damping fluid has a kinematic viscosityu of 50 cSt (1 cSt = 1 mm2s’; the centi-Stoke is the cgs unit often used to describefluids in chemical supply house catalogues). Since the density p of the silicone fluid isclose to that of water, the dynamic viscosity = p is about 0.05 Pa s. The pendulumweights are made of split lead shot. The cells are assembled into strings of three havinga centre-to-centre spacing of 7.5 cm. The cells are connected using 36 AWG solid copperwires. The strain bridge circuit is such that eight wires lead up to the surface from theupper cell, six wires connect the upper and middle cells, and four wires connect themiddle and lower cells. Because the protective heat shrink sheath created problemsin 1988, no sheath was placed over these sensor strings. Some permanent straining ofthe brass springs was observed to occur during the hammering process, but this shiftin the calibration is easily corrected by using tilt recordings made before, during, andChapter 2. BED DEFORMATION: THEORY AND METHOD 35LEAF SPRINGDAMPING FLUIDBULKHEADSTRAIN GAUGESPENDULUM MASSFig. 2.6: Schematic diagram of the leaf spring tilt cell used in the 1989experiment. The cell is 47 mm long and 16 mm in diameter. As the cell istilted, the pendulum masses bend the two leaf springs mounted at right anglesto each other. The leaf springs are sensitive only to bending in one direction, sothe two pendula respond to tilt along different axes. The bending is measuredby strain gauges bonded to the surfaces of the leaf springs.after the insertion. In future, this problem could be avoided by using a higher viscositydamping fluid and plastic leaf springs. Certain plastics have a much higher elastic limitthan does brass and would accommodate greater strain.The calibration process for leaf spring tilt cells is identical to that for electrolytictilt cells with the exception that the calibration range is extended to 900 of tilt. The pro-cessing of in situ deformation data is somewhat different because the Newton-RaphsonChapter . BED DEFORMATION: THEORY AND METHOD 36method does not yield a stable inversion. When the field data are analyzed, values 9and ç for the cell are computed from Equation (2.8) using a two-dimensional simplexalgorithm (Press and others, 1986, p. 289) that minimizes the difference between predicted and observed output voltages. As with the electrolytic tilt cells, the error ontilt and azimuth for these sensors was estimated by using the calibration data as inputto the inversion scheme. The sine of the tilt has a maximum error of +0.005, whichtranslates into a +0.3° error at 00 of tilt, increasing to ±0.6° error at 60° of tilt, andto ±5° error at 90° of tilt. The azimuth error is ±4° at 0° of tilt, decreasing to +1° at9Q0 of tilt. The error in strain rate is 0.003% at 0° of tilt, increasing to 2% at 600 of tilt.Two sensor strings were hammered into the glacier bed during the summer of1989. Unlike our experience in 1988, the initial tilt of these sensors was close to thatof their respective insertion angles. Omitting the stiff protective sheath appears tohave resulted in instruments that are flexible enough so as not to be forced from theirinsertion position. Figure 2.7 shows the down-flow strain rate data sensed by one ofthese leaf spring tilt cell strings. This string was inserted 20 cm into the undisturbedbasal sediments and had an overall length of 27 cm. After the insertion, the instrumentcable was tied off with 1.5 m of slack. The signals from all three tilt cells indicate thatthey lay within actively deforming basal material. Because the insertion depth was notsufficient to place the uppermost cell within solid basal material, and because the celldid not have support from the cable, it must have been supported by the disturbedsediment layer. This sets the minimum thickness of the disturbed layer for this holeat about 10 cm. We estimate that, in general, the thickness of the disturbed layer is15—25 cm.wcrzICl)I I I I I I6 8 10 12AUG 1989Fig. 2.7: Sample results from the 1989 leaf spring tilt sensor experiment.Tilt records for each of the three tilt cells have been differentiated to givestrain rate parallel to ice flow. Strain rates normal to ice flow have beenomitted for clarity, but are of comparable magnitude.These records were produced by performing a decomposition of the raw tilt andazimuth data, as was discussed above. Because the tilt measurements are made withrespect to vertical, we modify Equation (2.3) to include the basal slope of the glacier/3 so thatChapter 2. BED DEFORMATION: THEORY AND METHOD 3725O0—2501%1v’1sJv250w—250V18Ez = j— tan(Od — /3) (2.9)Chapter 2. BED DEFORMATION: THEORY AND METHOD 38The basal slope in the region surrounding the study area is70, as determined by drillingdepths.Cross-flow strain rates are not shown (see Figure 3.10 for complete data set), butare of comparable magnitude. This is a surprising result, but one we accept becausethe alternative seems to be unacceptable; if the down-flow/cross-flow decomposition isconstrained to prohibit cross-flow strain, then the implied motion of the cells requiresback-and-forth rotation about their long axes by as much as 1000. There may bepartitioning between cell rotation and cross-flow tilt, but the effect on down-flow tilt isslight. Any fraction of cross-flow tilt that is assigned to cell rotation reduces cross-flowtilt by a similar fraction and down-flow tilt increases slightly to maintain the correctnet tilt.Strain rates were computed by applying a five point running first derivative ifiterto the strain record (Abramowitz and Stegun, 1965, p. 914), followed by a Gaussiansmoothing filter having a standard deviation of 50 mm.2.5 Potential sources of errorHere we discuss sources of error that may arise from the insertion procedure and fromsensor characteristics.2.5.1 Sensor scale effectsWhen making detailed measurements of deformation within an inhomogeneous material, a fundamental concern is choosing an appropriate scale of observation. If the scaleof observation is too small (on the scale of individual clasts in the material), the concept of bulk viscosity becomes meaningless because there is no macroscopic, smoothvelocity gradient along which shear stress may be transferred and viscosity defined.Undersized sensors would measure the random wanderings of individual clasts ratherthan the macroscopic properties of the deforming material. Conversely, if we observeChapter 2. BED DEFORMATION: THEORY AND METHOD 39the bed on too large a scale, the deforming layer appears as a single unit with nointernal structure.We expect the deforming layer beneath Trapridge Glacier to be decimetres thick(Boulton and Hindmarsh, 1987; Clarke, 1987b), so we want to observe deformation on ascale of centimetres. Were the basal material beneath Trapridge Glacier homogeneous,our 5 cm long tilt sensors would make clear measurements of subglacial strain, but thisis not the case. A dry volume fraction analysis of Trapridge Glacier basal materialshows that 50% of the clasts are less than 0.25 mm in diameter and that 80% of theclasts are less than 2 cm in diameter (Clarke, 1987b); the size of our tilt sensors is suchthat they behave as large clasts. There is no alternative but to accept this limitationbecause any technique for making local deformation measurements will be subject tothe same conditions. Tilt cells placed near a clast of similar or larger size will beinfluenced by the movement of the clast, and Equations (2.3) and (2.4), which rely onthe homogenous nature of the basal material, may not be directly applicable.2.5.2 The ice—bed interfaceThe base of Trapridge Glacier in the study area is at the pressure melting point (Clarkeand others, 1984b, Fig. 4, above site 11). For the bed to remain at this temperature,the net heat input to the basal ice must be positive and the basal ice must be melting.Basal melting produces a clean contact between the ice and the bed unless ice creepinto subglacial sediments is faster. Nevertheless, we believe that the ice—bed interfacebeneath our study site is sharp; the basal sliding and deformation discussed elsewherein this thesis will inhibit ice intrusion.When using the percussion hammer to insert instruments into the glacier bed,the final position of the instrument relative to the ice—bed interface is uncertain. Wecannot observe the insertion procedure directly, and must rely on physical constraintsand tactile information to infer what is happening at the bottom of the borehole.Chapter 2. BED DEFORMATION: THEORY AND METHOD 40Our hot water drill uses a 1.14mm (0.045in) diameter jet with a pressure drop of7—14 MPa (1000—2000 psi) across the nozzle. We believe that hydraulic excavation bythe drill loosens material to a depth of several decimetres below the ice—bed interface.Evidence for this excavation includes the sediment laden basal water samples that weobtain shortly after completing a borehole. When the percussion hammer is loweredto the bottom of a borehole, its dead weight exerts a pressure of about 1 MPa on thetip (for comparison, the ice overburden pressure beneath our study site is 0.64 MPa).This pressure pushes the hammer through the loosened material; the insertion processbegins, and is measured from, the upper surface of the undisturbed bed material.Thus, the upper parts of the instrument may be below the ice—bed interface, evenif the insertion depth into the undisturbed material is less than the length of theinstrument. Typically, an insertion proceeds quickly for the first 5 cm, and then slowsdown, suggesting that there is a gradation between the disturbed and undisturbedmaterial. Evidence for the existence of the disturbed layer is found in the data andwas discussed above.Unfortunately, we have not yet developed a reliable technique for measuring thedistance between the top of undisturbed sediment and the ice—bed interface. In 1990,a crude device for measuring this distance was constructed. The device consists ofa 15 cm long steel rod with two holes drilled through it at positions 5 cm and 10 cmalong its length. By suspending this device from two thin wire ropes, the bar can beoriented in a vertical or horizontal position. The bottom of the borehole is measuredwith the bar in the vertical position and the level of the ice—bed interface with the barin the horizontal position (when in the horizontal position, the effective diameter ofthe bar is 15 cm and the device is too large to fit back up the borehole). The elevationdifference between these two positions is a measure of the separation we seek, but theresults must be interpreted with two limitations in mind: (1) We cannot guarantee thatthe rod is binding at the ice—bed interface. (2) As the weight of the rod is much lessChapter 2. BED DEFORMATION: THEORY AND METHOD 41than that of the percussion hammer, the penetration depth of the rod into disturbedsediment is apt to be less than that of the hammer. Therefore, the measurementsprobably underestimate the separation between the ice—bed interface and the top of theundisturbed sediment. In the two holes where we managed to make this measurement(in the third hole, the wire ropes became irretrievably tangled), separations of 13±5cmand 63 ± 5 cm were measured.Additional information about the ice—bed interface is revealed during the drillingprocess. When completing the drilling of a “connected” hole, the meitwater in thehole drains rapidly into the bed; the equilibrium level of the water column representsthe subglacial water pressure. On a number of occasions we have encountered artesianoutflow conditions where silty subglacial water flowed up through the borehole at arate of 2—3Lmin’; in these holes, the subglacial water pressure was well above iceflotation pressure and an equilibrium was not possible (Clarke and Germ, 1989). Conversely, the water in an “unconnected” hole does not drain into the bed, or does sovery slowly. For some holes, the characterization of a given borehole as connected orunconnected is subjective, but for most holes, it is quite clear. The holes that drainquickly are connected to a well developed drainage system, but we have evidence thatthe spatial extent of connected regions of the glacier bed changes with time (Smart andClarke, 1988; Clarke and Blake, 1991). lodge (1979) has observed similar behaviourat South Cascade Glacier.2.5.3 Sensor attitudeThe diameter of the percussion hammer is 3.81 cm and, based on our experience withlarger subglacial instruments, the diameter of the bottom of a borehole is just over5 cm. These radial dimensions, in combination with the 2 m length of the percussionhammer, ensure that the hammer fits snugly in the borehole and that the instrumentis inserted into the subglacial sediment co-axially with the bottom of the hole. EachChapter 2. BED DEFORMATION: THEORY AND METHOD 42of the holes used for bed deformation experiments was profiled using an inclinometerequipped with dual-axis tilt sensors and a compass. By determining the azimuth of theborehole as it intersects the ice—bed interface, we determine the initial azimuth of thebed deformation instrument relative to the glacier flow direction. The inclinometer iscapable of determining the location of the bottom of a 70 m borehole to within 60 cm.As they reach the bed, our boreholes commonly deviate from vertical by as much as502.54 Sediment intrusion into the boreholeAfter the borehole is completed, several hours typically pass before the insertion procedure begins. This raises the concern that we may not insert our sensors into subglacialmaterial, but rather into sediment that has squeezed into the borehole. If we accedethat the sensors are located within the borehole, then the deformation observed by thesensors would be controlled by ice deformation; since the observed shear deformationrates exceed reasonable ice deformation rates by several orders of magnitude, the sensors cannot be located above the ice—bed interface. Moreover, using similar insertiontechniques, we install other types of sensors into subglacial sediment and have unequivocal evidence that these sensors are implanted in bed material rather than extrudedsediment. We discuss these two observations in turn.The mean basal shear stress beneath the study site is 77 kPa (based on geometricalcalculations) and the basal temperature of Trapridge Glacier in the study area is close tothe pressure melting point (Clarke and others, 1984b, Fig. 4, above site 11). EvaluatingGlen’s flow law for simple shear(2.10)Chapter . BED DEFORMATION: THEORY AND METHOD 4370E 60C.)•— 50C0 402010at 000 gives a strain rate of 0.076yr when the shear stress r = 77kPa, A =5.3 x 1O’ skPa3,and n = 3 (Paterson, 1981, p. 39). Assuming Newtonian simpleviscous shear deformation of the formT (2.11)this strain rate represents a dynamic viscosity of 2 x 1013 Pa s. This ice viscosity valueis two orders of magnitude greater than published values of measured or predicted tillviscosity (Boulton and Hindmarsh, 1987, fig. 7; Clarke, 1987b). Given the low strainrate of the ice, we could not expect to measure strain rates exceeding 100 yr1 withina borehole (see Figure 2.7) if the straining of extruded material within a borehole werecontrolled by ice deformation.In 1990, we ran a series of experiments using “drag spooi” instruments. Our intentwas to obtain some measure of the sliding rate beneath Trapridge Glacier. A drag spoolconsists of a multi-turn potentiometer connected to a spooled string (Fig. 0.1). Thedrag spool is suspended within the ice close to the bed, and measures continuously thelength of string payed out to an anchor in the bed. Figure 2.8 shows an example ofdata obtained from a drag spool.208 212 216 220Time (days)Fig. 2.8: Relative displacement between a drag spool and its anchorrecorded in the summer of 1990. The data is from 901133, located withinthe study area.Chapter 2. BED DEFORMATION: THEORY AND METHOD 44The percussion hammer was used to insert three drag spooi anchors in different boreholes to a depth within the undisturbed basal material similar to that of the1989 deformation instruments. The data indicate that, on average, the anchor pointsmoved away from their respective boreholes at a rate of about 4 cm day’. This largerelative velocity could not persist for many days if the anchor were placed in intrudedmaterial. Although these measurements were made at different subsurface locationsand during a different field season than the 1987—1989 bed deformation experiments,surface velocity in each season was about 10 cm day’ and we have not made anychanges to our drilling and insertion techniques during this time; it seems unlikely thatbasal conditions have undergone dramatic changes between the 1987—1990 field seasons.Thus we conclude that the anchors for our bed deformation instruments are also belowthe ice—bed interface. If the upper parts of a deformation instrument were to extendabove the ice—bed interface, the data should show this as a rapid increase in tilt to90° as the tilt cells are drawn out of the borehole and in under the ice; we do not seethis behaviour in any of our instruments.2.5.5 Connecting wiresThe two quantitative instrument designs require that thin wires connect the individualtilt sensors together and that a multi-conductor cable transmit the strain informationto the surface.The main instrument cable freezes to the side of the borehole, and could conceivably pull the entire sensor string out of the bed as the glacier slides forward. By lettingslack cable down the hole after the instrument insertion is completed, we alleviate thisproblem. There is evidence from the 1988 results that this slack cable technique issufficient to prevent interference from the cable for several weeks.The wires connecting the individual sensors are thin enough (36 AWG) so thatthey cannot prevent the sensor string from bending. Additional cladding, such as theChapter 2. BED DEFORMATION: THEORY AND METHOD 45heat shrink sheath used in 1988, may stiffen the sensor string so that problems ariseduring the insertion procedure, but if the cladding is omitted, no problems arise. Theconnecting wires may also act under tension to align the sensors with each other. Withthe sensor string transfixing a shearing layer, any tension effect will become more acuteas time passes. There is not much that can be done about this problem, other thanusing telemetry techniques that do not require wires. We have considered coiling thewires between the sensors in order to provide stretch, but in order to maintain thespacing between sensors during insertion, a strong non-extensible connection betweenthe cells is required. Despite precautions, the insertion process places considerabletensional force on the instrument.Chapter 3BED DEFORMATION: DATA ANALYSIS3.1 IntroductionIn order to develop models for subglacial deformation and drainage, we must havea rheological model for the subglacial material. Quantifying the sediment rheologyrequires measurements of shear stress, subglacial water pressure (an indirect measureof normal stress), and strain rate.Boulton (1979b) has, to our knowledge, published the only measurements of subglacial shear stress. The fixing of shear stress sensor plates onto a glacier substratecalls for some particular circumstances: (1) physical access to the bed and (2) a solidsubstrate into which the sensors can be set. Boulton installed his sensors on bedrockunderlying short-lived cavities beneath the Glacier d’Argentière, France. He succeededin measuring time-varying normal and shear stresses, although these variations wereattributed to clasts scraping over the sensor plates rather than gross changes in basalshear stress. The demanding logistical requirements for this type of experiment makeit highly improbably that measurements of shear stress beneath a soft-bedded glacierwill ever be made.Those in situ measurements of strain rate that have been made (Boulton andHindmarsh, 1987; Fahnestock and Humphrey, 1988) are of total strain over a numberof days. An average strain rate can be computed from this “before and after” look atthe bed, but any time-varying behaviour is obscured. Temporal changes in strain rateare expected because subglacial water pressure, and hence effective pressure on the bed,can fluctuate dramatically (Mathews, 1964; Iken, 1972, 1978; lodge, 1976, 1979; Ikenand Bindschadler, 1986; Kamb and Engelhardt, 1987; Engelhardt and others, 1990).46Chapter 3. BED DEFORMATION: DATA ANALYSIS 47Boulton and Hindmarsh have fitted total strain measurements made near the terminus of Breidamerkurjökull to two non-linear viscous fluid models. A problem withtheir analysis is that they combine data from different years and different sites to produce a single rheology (Boulton and Hindmarsh, 1987, Figure 7). The necessary butunstated assumption is that the sediment is both spatially and temporally homogeneous. A proper rheological description may require continuous and contemporaneousstrain measurements at a number of spatially distributed sites. In our experiments, wehave succeeded in making continuous measurements of subglacial deformation, but logistical constraints have limited the spatial coverage to one or two sites per field season(Chapter 2; Blake and others, 1991). Nevertheless, our data reveal some interestingaspects of the basal processes beneath Trapridge Glacier.This chapter presents data collected from in situ strain instruments placed in theactively deforming bed of Trapridge Glacier during the 1988 and 1989 field seasons.Additional deformation measurements were made in the surnnier of 1990, but the dataanalysis has not been completed. Several models are presented that seek to explain theobserved deformation patterns.The deformation data were collected using electrolytic and leaf-spring tilt cellsinserted into the glacier bed; the function, design, calibration, and data processing ofthe tilt cells and the design of the percussion hammer used to insert them are describedin Chapter 2.3.2 Experiment designFigures 1.1 and 3.1 show the general location of the study area used for the bed deformation studies. Extensive hot-water drilling in this area indicates that the glacier hasa very uniform thickness of about 72 m; during a given drilling season, we can oftenpredict the depth of a new hole to within 10—20 cm. For several hundred metres tothe north, west, and east, the glacier topography is gently undulating. To the south,Chapter 3. BED DEFORMATION: DATA ANALYSIS 48there is a slight rise associated with heavy crevassing. The surface and basal slopes, asdetermined by surveying and drilling, are both 70 in the direction of glacier flow. Theflow direction, as determined by stake surveys, is 110 north of east (bearing 79°). Datafrom vertical strain sensors indicates that the area is under moderate compression.Fig. 3.1: Trapridge Glacier is located in the St. Elias range of southwestern Yukon, Canada. The rectangle indicates the area where most holes,including those used in the subglacial deformation experiments, were drilledduring the 1988, 1989, and 1990 field seasons. The dotted line indicates theapproximate location of the fall snowline. The purpose of the magnetometerand telluric array will be clarified in Chapter Ancillary informationHard-bedded glaciers can move only by processes of basal sliding and internal deformation of the ice. Figure 3.2 shows that three processes contribute to the surface velocityof a soft-bedded glacier: internal deformation, basal sliding, and basal deformationChapter 3. BED DEFORMATION: DATA ANALYSIS 49(Alley and others, 1986; Alley, 1989a). The experiments described in this paper attempt to elucidate the process of basal deformation, but estimates of basal sliding andinternal deformation are necessary for data interpretation.zIinternal deformationICE /BED )\\Nsliding veIotybasal deformationFig. 3.2: The surface velocity of Trapridge Glacier (about 10 cm day’during the summer) is made up of contributions from internal deformation(0—1 cmday’), basal sliding (‘—‘ 4cmday’), and subglacial deformation (‘—i5 cmday). Internal deformationInformation on internal deformation of Trapridge Glacier is largely empirical. Much ofthe glacier is cold (Clarke and others, 1984b; Clarke and Blake, 1991), which discouragesplastic deformation. Although we do have problems with instrument cables beingChapterS. BED DEFORMATION: DATA ANALYSIS 50cut, this is an annoyance primarily in crevassed areas. In the region where the beddeformation experiments were conducted, several cables connecting pressure sensorsplaced at the glacier bed to the surface have survived intact for more than a year.Following the development in Chapter 6 of Paterson (1981.), we can compute atheoretical estimate of the internal deformation character of Trapridge Glacier basedon Glen’s flow law for ice (Glen, 1952). Computing internal deformation in this wayis notoriously error-prone because the parameters describing glacier ice rheology areknown to vary between different ice samples and because the behaviour of ice near thepressure melting point is not well described or understood.The derivation of a theoretical deformation rate proceeds as follows: considera coordinate system with the x axis pointing down-flow and parallel to the ice—bedinterface; the z axis is positive upward. We assume deformation in the x-z plane only,with no variation in strain, stress, or temperature with x or y. Under these conditions,Glen’s flow law may be written as= A(T)r’r (3.1)= A(T)r’1r (3.2)withr2=rI+r (3.3)Tz = pgsina(h — z) (3.4)A(T) = 5.2 x 1016 exp{ ( — 263.15) } kPa3 s_i (3.5)and where r2 is the normal deviatoric stress along the x axis and r, is the only nonzero shear stress component. The activation energy Q = 139 kJ mol’, the ideal gasconstant R = 8.314 J mol’ K—’, the density of ice p = 900 kgm3, the gravitationalacceleration g = 9.81 ms2, the glacier thickness h = 72m, the surface slope a = 70,ChapterS. BED DEFORMATION: DATA ANALYSIS 51and the flow law exponent n 3. Temperature T is in degrees Kelvin; the value ofA(T = —10°C) is 5.2 x 10_16 kPa3 (Paterson, 1981).We estimate the value of è by measuring the vertical strain rate c2. Vertical strain rate is measured by freezing a 1.2m long, 0.076mm (0.003 inch) diameterconstantan wire into a shallow 15 m borehole. The wire forms one arm an electricalbridge whose output is monitored over time; if the glacier is under compression, thewire will be stretched and its resistance will increase. This vertical strain instrumentwas designed by W. D. Harrison (personal communication).Equations (3.1) through (3.5) were used to calculate the internal deformation ofthe glacier assuming a linear temperature distribution between 0°C at the bottom and—6°C at the surface (Clarke and Blake, 1991). The dotted line in Figure 3.3 shows thecomputed velocity profile with no longitudinal strain; the solid line shows the effectof adding a longitudinal strain rate of = —0.019 yr’ (equal in magnitude to theobserved vertical strain rate of +0.019yr1); this implies a compressive longitudinalstress. The computed velocity contribution from internal deformation accounts for2.8myr’ (0.76 cmday’) of the average annual surface flow rate of about 3Omyr’.This rate of internal deformation seems consistent with the lifespan of our subglacialsensors.We have analyzed only one long-term field record of internal deformation. Hole891165 was redrilled a year later as 901140 by following along a conductivity sensorcable; over a full year, the total lateral deformation of the hole was less than the erroron the inclinometer which is about 30 cm (Appendix A) (the hole designations— e.g.881156 — indicate the year of drilling and the ordinal position of the hole in the glacierdrilling program for that year).Theory predicts that the largest deformation occurs near the bed where the ice iswarmest. Should there be any debris in this basal ice (note that we do not belive thatthere is debris near the ice—bed interface beneath our study site), evidence suggests thatChapter 3. BED DEFORMATION: DATA ANALYSIS 52basal deformation should be even higher (Hubbard and Sharp, 1989; Echelmeyer andWang, 1990; Huang and Wang, 1990; van der Veen and Whillans, 1990). It is thereforesurprising that the theoretical prediction overestimates the observed deformation. Overa year, the theoretical calculation predicts lateral borehole deformation of at least 1.5 meven if longitudinal compression is neglected — deformation of this magnitude should beclearly evident in the inclinometry data. This incongruity means that Trapridge Glacierice is stiffer than is predicted by Glen’s flow law (using parameters from Paterson, 1981)and that almost the entire surface motion of the glacier is accommodated by slidingand deformation processes at the bed.70ci)-Qci)>0-oo 30ci)20cm 10 -00.0 0.4 0.6 0.8Velocity (cm/day)Fig. 3.3: The computed internal velocity distribution for Trapridge Glacier. The solid line indicates the solution with the observed compressive longitudinal strain included and the dotted line indicates the solution with nolongitudinal strain.II//0.2Chapter 3. BED DEFORMATION: DATA ANALYSIS 533.2.1.2 Basal slidingExperiments with basal drags spools in 1990 (Chapter 2; Appendix C; Blake andClarke, 1991a) placed an upper limit on glacier sliding of about 4 cm day’ (becausethe spool anchors were placed within the deforming sediments, some of the observedrelative motion between the anchors and the ice could have been caused by deformation of the interposed sediment). During the summer, Trapridge Glacier moves at aspeed of about 10cm day’ (this speed was consistent during the 1988, 1989, and 1990field seasons), so the above estimates for glacier sliding and internal deformation leave5—6 cm day1 of motion to be accounted for by basal deformation.3.3 The 1988 experimentOn 10 August 1988, a deformation instrument consisting of three vertically-spacedelectrolytic tilt sensors was hammered 8 cm into the bed of Trapridge Glacier. Thetilt cells were spaced at 10 cm intervals and the overall length of the sensor string was27cm. A signal cable led from the instrument to the surface; following insertion, 1.5 mof slack cable was fed into the borehole.In the weeks prior to inserting the tilt sensor string, two subglacial water pressuresensors and a vertical strain sensor were installed in neighbouring holes. Figure 3.4shows a location map of the experimental site. The indicated positions of subglacialsensors are those determined by borehole inclinometry. The indicated position of thevertical strain sensor, placed 15 m below the surface, is that of the borehole collar (thetop of the borehole). Hole 881104, which contains one of the pressure sensors, wasthe only connected hole; the other holes reaching the bed (881135 and 881156) wereunconnected.Chapter 3. BED DEFORMATION: DATA ANALYSIS 54I88H04.PRESSURE88H37 88H56. 088H35 VERTICAL TILTPRESSURE 0 10Fig. 3.4: The location map for the 1988 experiment. The indicated positions of the subglacial sensors is that of the hole bottom as determinedby borehole inclinometry. Solid dots indicate connected holes and holes notreaching the bed; circles indicate unconnected holes.Data from the 23 day experiment are shown in Figure 3.5. Figure 3.5a shows thetilt angle from vertical decomposed into down-flow (heavy line) and cross-flow tilt (lightline). All of the observed azimuth changes were attributed to cross-flow tilt (see section 2.4.1). Inclinometry results indicate that the hole through which the deformationinstrument was lowered intersected the glacier bed in a near-vertical orientation. Thedesign of the percussion hammer ensures that the instrument is inserted coaxially withthe bottom of the borehole (Chapter 2; Blake and others, 1991), which in this casetilted 30 up-glacier. The large initial tilt angle reported by the tilt cells suggests thatthe disturbed sediment at the bottom of the borehole was not able to support the sensorstring once the hammer was withdrawn, perhaps because the stiff heat shrink sheathon the instrument allowed the slack instrument cable to press down on the instrument.Chapter 3. BED DEFORMATION: DATA ANALYSIS 55Nevertheless, the rapid onset of independent motion in the lower two cells suggests thatthe bed quickly reconsolidated following the drilling disturbance.The uppermost tilt cell appears to have passed most of the experiment resting onits side, perhaps lying at the ice—bed interface. Late in the experiment this cell wastipped over to point down-glacier, probably by the glacier pulling on the cable. Thetwo lower tilt cells quickly reached their calibration limit of 600 tilt from vertical.Figure 3.5b shows strain rate curves generated by applying Equations (2.4) and(2.9) to the tilt data from the lower two sensors (the basal slope /3 is 7°). The time serieswere truncated at the time when the net tilt of each cell exceeded the calibration limitof 60° and then smoothed using a Gaussian smoothing filter having a standard deviationof 50 mm. The two striking characteristics of the strain rate curves are that the strainrates are large and sometimes negative. Assuming a 1 m thick deforming subglaciallayer supporting 5 cm day1 of uniform shear deformation, the expected down-flowstrain rate is +9yr’, but in Figure 3.5b, we observe strain rates ranging between—200 yr1 and +400 yr1. The large positive down-flow strain rates observed on bothtraces for the first day are probably associated with consolidation of the disturbedsubglacial sediments in which the sensors were located, but large excursions are alsofound at later times. For instance, early on 15 August, the middle tilt cell experiencedrapid changes in down-flow tilt angle that translated into large strain rate fluctuations.Negative strain rates are expected in the cross-flow record — we can picture thetilt cell wobbling from side to side as down-flow deformation progresses — but thenegative excursions in the down-flow strain rate record indicate that, at times, thedown-glacier velocity gradient through the deforming layer is negative. Both the strainrate extremes and negative strain rates are probably caused by interaction between thetilt cells and large neighbouring clasts; we shall explore this and other options below.Chapter 3. BED DEFORMATION: DATA ANALYSIS 56>‘UizF-Cl)EUiDU)U)Ui0.-90..MIDDLEBOTFOM-90-J+90C,)UiUiC,Ui.1-90+90 -0-1111111111 II 1111111111abCFig. 3.5: Data records from the 1988 experiment. (a) The tilt records forthe three tilt sensors indicating tilt from vertical. The uppermost record isfor the uppermost tilt sensor. The heavy line indicates down-flow tilt and thelight line cross-flow tilt. (b) The strain rate records corresponding to the lowertwo tilt cells. Strain rate is not calculated for the top cell, so the uppermostrecord is for the central tilt cell. Down-flow strain rates are drawn with heavylines and cross-flow strain rates with light lines. (c) The pressure records fromholes 881104 (light line) and 881135 (heavy line).1111111111111111111111AUG 1988Chapter 3. BED DEFORMATION: DATA ANALYSIS 573.3.1 Correlation with effective pressureEffective pressure is a poorly-defined term used to describe the distribution of iceoverburden pressure on the two load-bearing components of the subglacial system:the water and the sediment. The effective pressure Fe is a measure of the loadingexperienced by the sediment. In a definition analogous to that describing the partialpressures of constituents making up a gas, the conventional expression for Fe isF=Fe+Fw (3.6)where F is the overburden pressure of the ice and F is the pore water pressure. Subglacial measurement of pore pressure poses obvious logistical difficulties, so researchersusually substitute measurements of water pressure made at the bottom of boreholes.Unfortunately, the low hydraulic conductivity usually associated with glacial sediments means that changes in subglacial water pressure take some time to diffuse downinto the sediments. In addition, sediment deformation can drastically alter the porepressure without affecting subglacial water pressure. Pore water pressures at depthwithin the subglacial material may be quite different from the borehole water pressure.We can compute an estimate of the rate of intrusion of water at the ice—bedinterface into subglacial sediments by using Darcy’s Law (Equation (4.24c)). Darcy’sLaw is usually written asq = —KVh (3.7)where h = F/pg is the pressure head, P is the pressure potential (fluid pressure lessthe hydrostatic pressure), pm is the density of water, g is gravitational acceleration,q is fluid flux, and K is the hydraulic conductivity. Typical values of K for glacialtill and clay range between 10—12 and 106 ms (Freeze and Cherry, 1979, table 2.2),although there is evidence from borehole response tests that a thin (5 cm) layer ofChapter 3. BED DEFORMATION: DATA ANALYSIS 58material with K = 10—2 to 10_i ms (Daniel Stone, personal communication) mayexist next to the ice—bed interface. If we choose a large head gradient of 50 (thisrepresents the entire overburden pressure of Trapridge Glacier expressed across a 80 cmthick sediment layer) with a hydraulic conductivity of 1O ms1 (suggested by TaviMurray, personal communication), then the intrusion rate is about 0.05 IIms. Thisis a low flow rate; if this pressure head gradient is sustained for an entire day, water atthe ice—bed interface will penetrate only 4 mm into the bed.Translating fluid flux into temporal changes in pore pressure requires knowledge ofthe hydraulic diffusivity of the bed and an understanding of the effect of deformation onpore pressure. Since these are both unknown quantities, we carl oniy make qualitativestatements about pore pressure: we know that deformation can cause dilation whichlowers the pore pressure (Murray, 1990), and, based on the above calculations, weexpect that diffusion of pressure from the ice-bed interface will be slow (on the orderof hours or days).Equation (3.6) underestimates the loading actually experienced by the contactsurfaces of the sediment grains; it is the grain-to-grain contact pressures that controlthe rheological behaviour of the sediments. A more accurate definition for effectivepressure is obtained by scaling the contributions of effective pressure and water pressureto overburden support. It can be argued that some fraction of the planar bottom surfaceof the glacier (if indeed the bottom is planar) is supported by water and the remainderby sediment. A trivial proof demonstrates that if a planar cut is made through arepresentative volume of material, the ratio of the area of the cut intersecting waterto the area of the cut is equal to the porosity. The weighted effective pressure is thusdefined asP = (1 fl)Pe +flPu, (3.8)Chapter 3. BED DEFORMATION: DATA ANALYSIS 59where n is the porosity of the sediment. Typical values of porosity for glacier tillare 20—30% for consolidated sediment and 40% for dilated sediment (Kamb, 1991),so Equation (3.8) produces larger values of Fe than does Equation (3.6) for the samevalues of P and P,.Nonetheless, as Scott (1963) points out, Equation (3.8) also underestimates thegrain-to-grain contact pressure. In a water-saturated sediment, neglecting surface tension effects (which may not be insignificant), the proper expression for effective pressureisP = aPe + (1 — a)F (3.9)where a is the area ratio of the grain contacts. Since a << 1, Equation (3.9) is oftenwritten asP=aPe+Pw (3.10)Clearly, if a is small, Fe can get very large indeed. Practically, we cannot make insitu measurements of n or a, so we make do with Equation (3.6) using water pressuremeasured with subglacial pressure sensors rather than with pore pressure sensors. Wemust keep in mind that the effective pressure calculated in this way is wrong.Figure 3.5c shows the contemporaneous pressure records collected from the twoneighbouring subglacial pressure sensors. The heavy line represents the pressure inhole 881135 and the light line the pressure in 881104. The similarity of the two pressure records, except for a brief four-day period, suggests that hole 881135 has becomeconnected to the subglacial drainage system since it was drilled. Migration of connected zones has been observed at Trapridge Glacier and is not unusual (Smart andClarke, 1988). The two pressure sensors are separated by 23 m. It is noteworthy thatlate on August 15th, the pressure recorded by these two sensors diverged sharply forChapter 3. BED DEFORMATION: DATA ANALYSIS 60a period of four days after which they resumed strong mutual agreement; the pressure pulses (or dips) produce a strong diurnal cycling of horizontal pressure gradientgreater than 1 m(H2o)m1. It is unlikely that such a large pressure gradient could existwithin a single connected zone; the hydraulic properties of connected zones are suchthat a pressure disturbance in one part is felt elsewhere in the zone almost immediately(Smart and Clarke, 1988).To explain this pressure event, we imagine the following scenario: On August 15,the drainage system in the neighbourhood of 881104 becomes temporarily unconnected;at the same time, some morphological change in the subglacial drainage system orsome change in diurnal meltwater input causes the pressure in 881135 to begin largefluctuations, raising the local water pressure near to the flotation pressure of 65 m(H2o)(flotation pressure is equal to the ice overburden pressure). The separation betweenholes 881104 and 881135 is roughly half the thickness of the ice; over this distanceand on a diurnal time scale, the glacier is stiff enough to support a lateral transfer ofnormal stress on the bed. In response to an upwards force on the glacier at 881135, theoverburden pressure in the neighbourhood of 881104 is reduced; the water pressure in881104 drops accordingly.Unfortunately, we have no pressure record from 881156, the location of the bed deformation instrument, but because 881156 is roughly equidistant from the two pressuresensors, we expect that some changes in water pressure — and hence effective pressure— occurred at 881156 during this four-day period. Given a change in effective pressureand no change in driving stress, we also expect that the basal strain rate should change.This response is not evident in Figure 3.5b. The only discernable correlation betweenthe onset of the diurnal pressure cycles and strain rate is a transition from near-zerovalues of strain rate to positive values. We are not convinced that this correlation isanything but coincidental, particularly since the strain rate of the middle tilt cell doesnot revert to near-zero values once the pressure event terminates.Chapter 3. BED DEFORMATION: DATA ANALYSIS 613.3.2 Effective viscosityThe classical definition for dynamic viscosity of a linear viscous fluid isTb = ?7— (3.11)or, substituting Equation (2.3),Tb = 2ij (3.12)The velocity gradient ôtt/Oz is measured along a line perpendicular to the interfaceacross which the shear stress Tb 15 measured. This relationship can be derived byconsidering the transfer of momentum across the interface by the motion of particlesmaking up the fluid (e.g. Joos, 1988, p. 568). The statistical analysis fails when thenumber of particle interactions considered is too low; this situation arises when thedynamic system is observed at too large a magnification (so that only interactions witha few particles are visible) or when the system is observed for too short a period oftime.In situ measurements of deformation within granular subglacial material sufferfrom a problematic scale of observation. The basal material contains particles having asize comparable to that of the tilt sensors. Since it is these macroscopic particles thatare the vehicles of momentum transfer, determining sediment viscosity by observing themotion of a tilt cell is analogous to determining the viscosity of a gas by watching onemolecule bouncing about. In addition, subglacial particle interactions are not elasticcollisions as envisaged in the momentum transfer model, but are bumping and grinding interactions. These aspects of the deformation process suggest that the conceptof viscosity may be entirely inappropriate for describing the proportionality betweenstress and strain in granular material. Nevertheless, we can attempt to mitigate theseweaknesses by averaging the observed instantaneous strain rates over time, therebyChapter 3. BED DEFORMATION: DATA ANALYSIS 62increasing the number of particle interactions. We must keep in mind this limitationof the experimental design.We can compute an estimate of which we call the effective viscosity. The meandown-flow strain rate ê for the middle tilt cell over the four day period beginning on20 August is 36±16 yr1. This is a period during which both subglacial water pressureand strain rate appear relatively stable. The average basal shear stress Tb0) = 77kPa is computed using Equation (3.4) assuming a glacier thickness h of 72m,a glacier slope a of 70, and no longitudinal stress gradient. Applying Equation (3.12)gives a value for j of 1.7 ± 0.8 x 1010 Pa s. The failure of the viscosity concept becomesclear when one considers individual points on the strain rate curve. We have no reasonto believe that the shear stress r changes, yet the strain rate is sometimes zero (resultingin infinite viscosity) or negative (persisting negative viscosity would suggest that theglacier is being dragged down-slope by fast flow of subglacial sediments — this is apreposterous idea).To estimate the thickness of the deforming layer by assuming that the strain rateis uniform throughout the layer (although the two strain rate records in Figure 3.5bappear to contradict this), we can rewrite Equation (3.11) as=(3.13)If we assume that shear deformation accounts for 5 cm day’ of glacier motion (Lu)then .z, the deforming layer thickness, is 12 ± 5 cm at this time and place. If this isindeed the thickness of the deforming layer, it might explain why we were able to drivethe tilt sensor string only 8 cm into undisturbed subglacial sediments.Boulton and Hindmarsh (1987) surmnarize a series of subglacial deformation experiments undertaken between 1977 and 1983 from within ice tunnels excavated beneaththe terminus of Breidamerkurj6kull. They fit the observed average strain rate overChapter 3. BED DEFORMATION: DATA ANALYSIS 63the course of each experiment to computed values of effective pressure Fe and basalshear stress Tb. The seven data points are fitted to equations of the form= A(m —7)ThP (3.14)and= A-1P (3.15)which represent a Bingham fluid model and a nonlinear viscous model respectively (-y isa yield stress). Since there are a small number of data points and a relatively largenumber of free parameters, it is perhaps not surprising that Boulton and Hindmarshachieved a rather good fit of their data to Equations (3.14) and (3.15). With theparameters m, n, and A substituted, Equations (3.14) and (3.15) become0.625 —1.25= 7.62 (T7) (-) (3.16a)where7 = 0.625Pe + 3750 (3.16b)and—1.80=(rb)1.33 () (3.17)The strain rate is in yr’, and Fe and Tb are in Pa. According to these equations,strain rate is proportional to stress, as we expect, and inversely proportional to effectivepressure.The average effective pressure Fe, as calculated using Equation (3.6), for holes881104 and 881135 over the four day period beginning on August 20 is 292+6 kPa. If wesubstitute this value of Fe into Equations (3.16) and (3.17), together with Tb = 77 kPa,we get a predicted strain rate of 0.41 yr for the viscous fluid model and no deformationfor the Bingham fluid model (the shear stress is below the predicted yield stress 7).Since we have ample evidence that the strain rate was about 36 yr during this time,Chapter 3. BED DEFORMATION: DATA ANALYSIS 64it is clear that Equations (3.16) and (3.17) are not appropriate rheological descriptionsfor subglacial material under Trapridge Glacier.Boulton and Hindmarsh based their values of effective pressure on measurementstaken from pressure sensors buried in the subglacial sediments; they were attemptingto measure pore pressure rather than water pressure at the ice—bed interface, but wesuspect that their pressure measurements are no better at estimating pore pressure thanours are. Boulton and Hindmarsh placed their pressure sensors less than 10 cm intothe sediments. If Breidamerkurjökull has a thin layer of more hydraulically conductivesediment near the ice bed interface, as we believe Trapridge Glacier does, then we wouldexpect shallow pressure sensors to measure the same pressure as at the ice—bed interface.Indeed, Boulton and Hindmarsh observed diurnal pressure cycling with peaks in theafternoon; this pattern of pressure change is consistent with daily meltwater input andshows no time lag resulting from diffusion of pressure into the sediments.Figure 3.6 reproduces the results for the Bingham fluid model (Boulton and Hind-marsh, 1987, Fig. 7) with the our data point for the 1988 experiment superimposed.Assuming that the fluctuations in basal water pressure observed in 881135 were presentin 881156, the effective pressure near the deformation instrument varied over the rangeindicated by the dotted line. As noted above, we observed strain rate variations between—200 and +200 yr’ with no correlated change in effective pressure.For the effective pressure and shear stress at our study site, the rheology functionsproposed by Boulton and Hindmarsh greatly underestimate the observed strain rate.The different experimental locale is surely responsible for part of this discrepancy. Forinstance, a comparison of the particle size distributions for the two glaciers (Boulton andHindmarsh, 1987, Fig. 3; Clarke, 1987b, Fig. 4; Murray, 1990; Tavi Murray, personalcommunication) reveals that Trapridge Glacier has a greater fraction of silt and clayparticles; at high effective pressures, clay-rich sediments are more easily deformed thancoarser sediments because the clay has a lubricating effect (more precisely, the internalChapter 3. BED DEFORMATION: DATA ANALYSIS 65CCCC(Cfriction angle of clay is low (Murray, 1990)). In addition, it is possible that Boulton andHindmarsh have overestimated the shear stress present beneath the Breidamerkurjökullice tunnels. Driving stress calculations involve using longitudinal stress gradients andthe difference between surface and basal slope angles to make corrections to the simplegravitationally-driven basal shear stress. Boulton and Hindmarsh may have improperlyused the slope of the over-steepened ablating glacier snout (Boulton, 1979a, Fig. 6) andneglected the longitudinal compressive stresses usually present at a glacier terminus(Robin, 1968; Collins, 1968; Paterson, 1981, Chapter 6, page 98).The mean basal shear stress for Trapridge Glacier is computed from knowledge ofthe glacier thickness, surface slope, and basal slope. We have recorded a near-surfacelongitudinal compressive strain of 0.019 yr’, but determining the effect of this longitudinal stress on basal shear stress requires a knowledge of how and vary alongthe glacier flow line (x axis); this is information we do not have. Because our experimental site is located under a uniform part of the glacier and only 10—20 m in elevationEffective pressure — kPaFig. 3.6: A facsimile of Boulton and Hindmarsh (1987), Figure 7, withthe data from the 1988 experiment superimposed. Note the discontinuity inthe effective pressure axis. The dotted line indicates the range of effectivepressure encountered in hole 881135 during the 1988 experiment, and the solidbar indicates the effective pressure range for the 1989 experiments.Chapter 3. BED DEFORMATION: DATA ANALYSIS 66below the equilibrium line, we suspect that there are no significant longitudinal stressgradients that we need to consider; we can make due with a gravitationally-driven basalshear stress (Equation (3.4) evaluated at z = 0).The unfortunate corollary to having a uniform glacier is that our study site providesonly one mean basal shear stress value; this makes the development of a basal rheologyfor Trapridge Glacier difficult. Nevertheless, we have evidence from both electricalexperiments (Chapter 6) and deformation experiments (discussed below) that shearstress beneath Trapridge Glacier has a far from uniform distribution. Had we a methodfor measuring in situ local shear stress, we might be able to gather the data necessaryfor deriving a rheology; unfortunately, we have no such method.3.4 The 1989 experimentOn 5 August 1989, a deformation instrument consisting of three vertically-spaced leafspring tilt sensors was hammered 20 cm into the glacier bed at the bottom of hole891178 (Figure 3.7). A subglacial water pressure sensor was placed in the same hole.The next day, a second deformation instrument was hammered 17 cm into the glacierbed at the bottom of hole 891180, located 17.5m northeast of 891178. No pressuresensor was placed in this second hole. For both deformation instruments, 1.5 m of slackcable was fed into the borehole before securing the cable at the surface. Hole 89H79was drilled on 5 August, but the attempt to install a deformation instrument in thishole failed; the instrument did not penetrate into the bed far enough for the subglacialsediments to provide a sufficient anchor.Chapter 3. BED DEFORMATION: DATA ANALYSIS 67mFig. 3.7: The location map for the 1989 experiment. The indicated positions of the sensors are those of the hole bottoms as determined by boreholeinclinometry. Solid dots indicate connected holes; circles indicate unconnectedholes.A second pressure sensor had been installed on 20 July in hole 891150, locatedmidway between the two deformation instruments. As a demonstration of the largelateral variability in the subglacial drainage system, holes 891178 and 891180 were connected holes and 891150 was unconnected; there is no evidence in the 891150 pressurerecord that the drainage system at 891150 had evolved into a connected system bythe time holes 891178 and 891180 were drilled. A third pressure sensor, installed on/N89H14• PRESSURE89H80TILT89H500PRESSURE89H78PRESSURETILTI I0 10Chapter 3. BED DEFORMATION: DATA ANALYSIS 6812 July in connected hole 891114, was located about 40 m southwest of the deformationexperiment.Tilt data from the six-day experiment (terminated upon our departure from theglacier) is shown in Figure 3.8. The sampling interval is 5 mm. Figure 3.8a shows thetilt data from 891178 (due to data logger failure, the first day of data was lost) andFigure 3.8b shows the tilt data from 891180. Note that the vertical scale in Figure 3.8bis twice that in Figure 3.8a. As in Figure 3.5, the heavy line indicates down-flowdeformation and the light line indicates cross-flow deformation.All the tilt cells registered initial tilt angles close to the terminal tilt angles oftheir respective boreholes (7° for 891178; 5.5° for 891180) which suggests that the instrument cable did not interfere with the deformation instruments. The improvementin maintaining initial orientation may have resulted from a design change: the two1989 deformation instruments did not have a stiff heat-shrink protective sheath as didthe 1988 instrument and this may have prevented the slack cable from pushing on theinstruments. The 1989 deformation instruments were also haimnered further into theundisturbed sediments.Hole 891178 intersected the glacier bed at a 7° tilt angle pointing roughly up-glacier,but 12° away from the flow direction. This initial orientation produces an offset in thecross-flow tilt as is evident in Figure 3.8a. Hole 891180 pointed more directly up-flow,so the initial cross-flow tilt angle in Figure 3.8b is almost zero. The tilt records for thetwo upper cells are truncated because as the tilt cells approached a net tilt of 90°, theinversion scheme that recovers tilt and azimuth (Chapter 2; Blake and others, 1991)became unstable.Because of the significant cross-flow tilt excursions present in these records, wechose to attribute only 50% of the observed tilt cell azimuth fluctuations to cross-flowtilt; the remaining azimuth changes were attributed to rotation of the tilt cells on theirChapter 3. BED DEFORMATION: DATA ANALYSIS 69I—C,)wUiC!,UiC,)UiUiUi -8O0I I I I I I6 7 8 9 10 11 12 13AUG 1989Fig. 3.8: Tilt data from the 1989 experiment. (a) The tilt records from891178. The uppermost record is for the uppermost tilt cell, and the lowestrecord is for the lowest tilt cell. The heavy line indicates down-flow tilt andthe light line cross-flow tilt. The labels on the top record show how differentmean strain rates can be computed for different time intervals (section 3.4.5).(b) The tilt records from 891180. The data are arranged as in part(a).axes (Chapter 2; Blake and others, 1991). The consequence of this decision is to reduceCTOP40040n0400MIDDLEaBOflOMI I I I I IMIDDLEthe amplitude of cross-flow tilt fluctuations by a factor of 2.Chapter 3. BED DEFORMATION: DATA ANALYSIS 7034.1 1989 tilt resultsThe tilt records in Figure 3.8 suggest that the three tilt cells within each deformationinstrument experienced the same local deformation events but that basal deformationat the two sites was of quite different character.The three tilt sensors in 891178 were moving in almost perfect concert; even thefiner details are present in all three records. Unlike the tilt data from 1988, there is adiurnal signal evident throughout the 891178 traces, although it is strongest in the lastthree days; in general, there appears to be an increase in tilt angle in the morning andthen a decrease in tilt angle in the afternoon.The three tilt records from 891180 also exhibit some similarity; for instance, all tiltsensors experienced a sharp increase in tilt angle early on 8 August followed by a sharpdecrease in tilt angle at noon on the same day. Unlike the 891178 data, fine details arenot consistent across the three records and there appears to be more high-frequency tiltactivity. In the final three days, the lowest tilt sensor shows the same diurnal behaviouras the records from 891178, but the tilt oscillations are delayed by several hours.34.2 1989 strain ratesThe strain rates computed from the tilt records are shown in Figure 3.9. As in Figure 3.5b, the strain rate curves have been smoothed. Figure 3.9a shows the strain ratesfrom 891178 and Figure 3.9b shows the strain rates from 891180. Note that the verticalscale in Figure 3.9b is ten times that in Figure 3.9a. The synchronized motion of thetilt cells in 891178 is again evident in Figure 3.9a.Chapter 3. BED DEFORMATION: DATA ANALYSIS 71>wzIU)ibAUG 1989I I11 12 13Fig. 3.9: Strain rate data from the 1989 experiment. (a) The strain raterecords from 891178. The uppermost record is for the uppermost tilt cell, andthe lowest record is for the lowest tile cell. The heavy line indicates down-flowstrain rate and the light line cross-flow strain rate. (b) The strain rate recordsfrom 891180. The data are arranged as in part (a).The data for both 891178 and 891180 share several interesting characteristics:I I I I I IabTOPwzU)+15000--150Q+15000--1500+15000--1500MIDDLEBOTTOM-(1) Negative and zero strain rates are observed. Although the mean down-flow strainChapter 3. BED DEFORMATION: DATA ANALYSIS 72rate for each cell is positive, indicating net down-flow strain within the deforming layer,the periods of negative strain rate are frequent. (2) In the second half of the experiment, there is evidence of diurnal synchronization. (3) The cross-flow strain rate iscomparable in magnitude to the down-flow strain rate. For the 891178 sensor string,all three records are similar; this indicates that each sensor was moving in concert withthe others and suggests uniform deformation over the 20 cm length of the sensor string.When contemplating the large strain rates, it is important to keep in mind thatthese peaks result from the differentiation and smoothing of step-like changes in tiltangle— these high strain rates are not sustained over long periods of time. Becausethese step changes in tilt angle are often observed to occur simultaneously for all thetilt sensors in one deformation instrument, we are confident that the events are notartifacts of unstable data inversion. In subsequent sections, we will discuss mechanismsthat might generate these tilt events.Net cross-flow strain over the period of the experiment is small for all the tiltsensors, but the raw data indicate unequivocally that the tilt sensors were experiencinglarge azimuth fluctuations. As mentioned above, we have attributed 50% of theseazimuth changes to rotation of the tilt cells about their long axes and feel that it wouldbe unreasonable to increase this fraction. As the only other source of azimuth changesis cross-flow tilt, large cross-flow tilt changes must have occurred; large cross-flow tiltmay not be an unreasonable consequence of deformation of granular material.In the absence of overwhelming, persistent extrusive flow within the subglacialsediments, we expect that the average down-flow strain rate over some reasonable periodof time will be positive. Although their appearance may not be convincing, all the strainrate curves in Figure 3.9 have positive integrals over the course of the experiment. Thesurprising aspect of these curves is that the deformation must sometimes be integratedover many days to ensure the expected positive net down-flow strain.Chapter 3. BED DEFORMATION: DATA ANALYSIS 7334.3 Subglacial pressureFigure 3.10 shows the data from the three subglacial pressure sensors. The steppedcharacter of the 891114 and 891178 records is caused by the limited resolution of thedigital data logger. The pressure records from 891114 and 891178 are well matched,although the observed pressure fluctuations are very much smaller in amplitude thanthose observed in 1988. Measurements of borehole depth and surveying results indicatethat the pressure sensor in 891114 is roughly 3 m above the sensor in 891178. Fora horizontal piezometric surface, the pressure sensor in 891178 should read 3 m(1120)greater than that in 891114. Since the measured pressure difference is only 1 m(H2o),we infer that a steady pressure gradient exists between these two holes — the polarityof this gradient is appropriate for driving water down-glacier.The pressure record from 891150 indicates a falling pressure that is consistentlywell above the flotation pressure of 65 m(H2o). Recalling that 891150 was an unconnected hole and that 891178 was a connected hole, it is astonishing to see such a largelateral pressure gradient. We are certain that 891150 was drilled to the bed and suspect that the high pressure in the hole was caused by freezing of the water in theupper reaches of the hole. Because both 891178 and 891180 were connected holes, andbecause previous experience with local connected systems at Trapridge Glacier suggests that neighbouring connected holes can be remarkably well interconnected (Smartand Clarke, 1988), we will assume that the pressure record from 891178 reflects thesubglacial water pressure in the vicinity of 891180.Chapter 3. BED DEFORMATION: DATA ANALYSIS 74!68 89H5058-AUG 1989Fig. 3.10: The pressure records from 891114, 891150, and 891178 are shown.Note that the 891150 has a different vertical scale.The correlation between the 891178 pressure record and the 891178 and 891180strain rate curves is not impressive, but they share a clear diurnal cycling towardsthe end of the experiment. Periods of higher subglacial water pressure in the earlymorning hours of 10 August and 11 August correlate well with positive strain rates in891178 and less well with those in 891180. Because the pressure fluctuations are smallin amplitude (about 10 kPa), we are surprised to see such a large change in strain rate.Compared with the effects of the pressure event in 1988, the strain rate variations seemcompletely out of proportion. We suspect that the correlation between local subglacialwater pressure and strain rate is not direct, but that there is some other mechanism atwork; we will discuss this in a later section.34.4 Effective viscosityUsing the same procedure as for the 1988 experiment, we can derive an effective viscosity value for the bed. The mean strain rates for the three sensors in 891178 are+8.5yr’, +10.8yr’, and +4.7yr’ for the top, middle, and bottom sensors respectively. The mean of the down-flow strain rates measurements from all three sensors is+8 + 2yr1 (the error estimate is based on the deviation of the three sensor means,Chapter 3. BED DEFORMATION: DATA ANALYSIS 75not on the standard deviation of all the data points). Using Equation (3.11) we findthe effective viscosity is 8 + 2 x 1010 Pas. This estimate of effective viscosity is almostfive times that found in 1988.If we make the same calculations for the 891180 sensor string, we get a “global”mean strain rate of +30 + 15yr and an effective viscosity of 2 + 1 x 1010 Pas.34.5 Net strain and mean strain rateRather than integrating strain rate curves over an interval of interest, it is simpler tocompute the mean strain rate by recasting Equations (2.4) and (2.9) in the form1 (tan 82 —tanSi) (3.18)2and applying this equation to the tilt values at the endpoints of an interval. Thesubscripts in Equation (3.18) indicate the values of 8 and time t at the endpoints of theinterval [1,2]; is mean strain rate over that interval. Note that for down-flow strain,the tilt angle 8 must be corrected for basal tilt as in Equation (2.9).When we apply Equation (3.18) to the down-flow tilt angle records for the top tiltcell in 891178 (Figure 3.8a), we can arrive at a number of estimates of mean subglacialstrain rate. The mean strain rate using interval [A, B] is 32 yr’, but the mean valueusing interval [A,C] is 4.1yr. This latter value is close to the strain rate predictedby the Boulton and Hindmarsh models for an shear stress of 77kPa and an effectivepressure of 78 kPa (the mean pressure in 891178 over the course of the experiment givesan effective pressure of 78kPa, as calculated using Equation (3.6)). The eight-folddifference in mean strain rate that arises simply by considering net strain over differenttime intervals demonstrates the danger in relying on measurements of net strain for thedevelopment of basal rheologies, but the tale is even worse: it is also clear that intervalscan be chosen so that the mean strain rate is zero (an infinite effective viscosity) ornegative (an irrational effective viscosity).Chapter 3. BED DEFORMATION: DATA ANALYSIS 763.5 Negative strain ratesStrong negative strain rates are striking characteristics of Figures 3.5b and 3.9. Anegative strain rate results when the down-flow velocity of the top of the cell is negativewith respect to the cell bottom; the bottom of the cell is displaced, in a positivedirection, relative to the top of the cell. Note that we have not specified any constraintson the net velocity of the tilt cell with respect to the geographical coordinate system;the tilt cell can be moving down-glacier, up-glacier, or rotating in place. In this section,we explore several mechanisms that could produce the observed data. In the process,we discover that negative strain rates and extreme values of strain are not difficult toproduce.Before describing basal deformation mechanisms, the effects of instrument designon tilt data should be discussed. Problems related to the relative size of the tilt sensorsand basal clasts have been addressed above. We also believe that the stiff protectivesheath used on the 1988 deformation instrument may have caused some problems atthe start of the experiment before the disturbed sediments reconsolidated, but once thishad occurred, the sheath would have had negligible effect on bending of the instrumentbecause the effective viscosity of the sediment was so large. It is clear that tensionin the wires became important late in the 1988 experiment when all the tilt sensorshad high tilt angles (shortly after the tilt records in Figure 3.5a end, the instrumentwas pulled apart). No protective sheath was used in 1989 and the tilt sensors wereconnected only by flexible 36 AWG wires. These deformation instruments were verydelicate; several of the connecting wires were broken during careful handling on theglacier surface. Since none of these wires broke during the course of the experiment, webelieve that almost no tension existed between tilt cells in the 1989 experiment. Forthis reason, we have ignored any possible interference from connecting wires and/orsheaths.Chapter 3. BED DEFORMATION: DATA ANALYSIS 773.5.1 Fluid modelsImagine, for the moment, that the particles making up the bed are all much smaller thanthe tilt sensors; this allows us to model the deformation as a fluid flow. Equations (2.4)and (2.9) require that the lateral velocity gradient normal to the ice—bed interfacebe negative in order that negative strain rates be produced. There are at least twomechanisms that can generate these velocity proffles: extrusion and sheet flow. Sheet flowSheet flow is the flow of fluid between two parallel surfaces under the influence of alateral pressure gradient. In this case, the upper surface is the ice—bed interface andthe lower surface is the underlying bedrock or non-deforming sediment. We introduce acoordinate system with the origin at the bottom of the deforming layer and the x axispointing down-glacier parallel to the ice-bed interface. The z axis is positive upwardnormal to the bed. We assume that the subglacial material can be considered a fluidand that the deformation zone has infinite extent in the x— y plane.For an incompressible linear viscous isotropic fluid, the relationship between thedeviatoric shear stress tensor -r’ and the deviatoric strain rate tensor ‘ is= (3.19)where is the viscosity and where i and j denote the terms of interest. The deviatoricquantities may be expanded to give—=— kk6) (3.20)where Sjj is the Kronecker delta and a double subscript indicates summation overthe diagonal terms of the tensor. In the steady state and in the absence of bodyforces, momentum balance considerations give us the relation Or/Ox = 0 For anChapter 3. BED DEFORMATION: DATA ANALYSIS 78incompressible fluid, kk = 0. In addition, hydrostatic water pressure P = —Tkk/3. Ifwe restrict velocity and pressure fields to functions of the form u(z) and P(x), we candifferentiate Equation (3.20) to give82u OP--=— (3.21)For boundary conditions u(hb) = U, the velocity of the material at the top of thedeforming layer, and u(0) = 0, the solution to Equation (3.21) is/OP 1 Uzu(z) = -h-— —z(z— hi,) +---(3.22)andOu /OP1 U2e ==—) —(2z — hb) + — (3.23)where hb is the thickness of the deforming layer. The first term of Equation (3.22) represents the parabolic laminar flow solution (the planar analogue to Poiseuille’s equation)and the second term represents the linear simple shear solution.If we use values ofj = 2.0 x 1010 Pas, hb = lm, = +40yr, U = 5cmdaf’,and z = 0 (the location of maximum strain rate), then the required value of OP/Ox is—180 kPam1 or about—18m(H2o)m. This is an unreasonably large lateral pressure gradient. We have observed lateral pressure gradients as large as 100 kPam’sustained over a day, but we believe that maintaining a hydrostatic pressure gradientof —180 kPam’ over large distances and for long periods of time is unrealistic. Nevertheless, pressure gradient driven deformation could contribute to basal deformation. ExtrusionIf the ice—bed interface moves vertically then subglacial material tends to move laterally. This is much the same mechanism that forces toothpaste out of a tube. Rapidsurface elevation changes of several centimetres (which presumably reflects verticalChapter 3. BED DEFORMATION: DATA ANALYSIS 79motion of the ice—bed interface) have been observed on Trapridge Glacier. Iken andothers (1983) have observed similar changes on Unteraargletscher as have Iken andBindschadler (1986) on Findelengletscher and Kamb and Engelhardt (1987) on Variegated Glacier.If the entire glacier is permitted to rise and fall, we expect to see quantities ofbasal material sluicing in and out at the margins of the glacier. As this large-scaledisplacement of basal material is not observed, we assume that local changes in ice—bed interface elevation are compensated by elevation changes of opposite polarity inneighbouring areas, by compression and/or expansion of subglacial sediments, or bythe growth and decay of subglacial cavities and/or channels.In deriving a mathematical expression for this extrusive flow, we will make certainsimplifying assumptions: (1) Sediment flows only in the x direction. (2) The changein elevation is small compared to the thickness of the deforming layer. (3) Materialflow resulting from extrusion has a parabolic velocity profile between the upper andlower boundaries, just like the solution for sheet flow. (4) Subglacial sediments areincompressible.At first, we assume that the velocity U, as defined above, is zero. The velocitydistribution through the layer is thenu(x,z,t) = A(x,t)z(z— hi,) (3.24)where A is some function of x and time t. We can position the coordinate system atthe spreading centre of the extrusive flow such that u(O, z, t) = 0. The rate of volumechange, per unit width, of the deforming layer between the origin and a point x is thendefined asV(x,t) = w(t)x (3.25)Chapter 3. BED DEFORMATION: DATA ANALYSIS 80where w(t) is the velocity of the ice—bed interface along the z axis. Mass conservationfor an incompressible fluid requires that the change in volume be equal to the quantityof material entering or leaving the region. Thus,,.V(ct)=_J u(x,z,t)dz (3.26)0Combining Equations (3.24), (3,25), and (3.26) gives the solution6w(t)xzu(x,z,t)=(z—hb) (3.27)1We now superimpose the linear simple shear solution to give6w(t)xz Uu(x,z,t)= (z—hb)+---z (3.28)11bandãu 6w(t)a U2e ==(2z—he,) + -— (3.29)LiZFigure 3.11 shows the displacement in the z direction required to reproduce thestrain rate curve of the top sensor in 891178; the solution is for x = 10 m, Jib = 1 m,U = 5 cm day1, and z = hb (i.e. the top tilt sensor is at the top of the deforminglayer). The value of was chosen somewhat arbitrarily, but is of the same order of sizeas we believe connected zones to be. We note that the displacements in Figure 3.11are small compared to displacements that have been observed at Trapridge, whichsuggests that extrusion could easily account for all aspects of the observed strain rates.Unfortunately, we have no survey data with sufficient temporal or spatial resolution todetermine whether Figure 3.11 is representative of glacier surface motion.Chapter 3. BED DEFORMATION: DATA ANALYSIS 810—1d-2(mm)—3-413Fig. 3.11: The vertical displacement record required to reproduce thestrain rate record for the upper tilt cell in 891178 at a point 10 m from thespreading centre of the extrusion. See text for a complete explanation.Although the extrusion model seems to offer an explanation for large, and sometimes negative, strain rates, it does predict coherent deformation along the glacier bed,even if deformation instruments are located in different horizons within the bed. Anexamination of the records in Figures 3.8 and 3.9 reveals that there is little correlationbetween the movement of the tilt sensors in the two holes; the diurnal fluctuations arenot synchronized. Since the basal sediments cannot be considered homogeneous on ourscale of observation, we can reasonably argue that the smooth velocity profile definedby Equation (3.25) is not preserved over appreciable distances.Unfortunately, the extrusion model has one major drawback: Equation (3.28) doesnot take into account the viscosity of the subglacial material. For = 2.0 x 1010 Pas,the glacier must exert a local downward pressure far greater than its weight in orderto achieve the vertical motion described in Figure 3.11. Extrusion is therefore unableto explain the observed deformation.6Aug 1989Chapter 3. BED DEFORMATION: DATA ANALYSIS 823.5.2 Roller bearing modelsIn an attempt to derive a deformation model which more accurately describes theinhomogeneous nature of the subglacial material, we turn to models of granular deformation. There is evidence that deformation of subglacial material may result inthe formation of weak, dilated shear planes along which most of the deformation isacconunodated (Murray, 1990). If the layer of deforming material beneath TrapridgeGlacier exhibits this tendency, then we can imagine that the layer splits up into longblocks that extend across the direction of glacier flow. In this model, the blocks havepolygonal cross-sections and churn against each other in accordance with the frictional,translational, and rotational forces to which they are subjected. The blocks may breakup and coalesce, again in accordance with the rheological properties of the block andthe forces acting upon it.Herrmann and others (1990) examined the problem of packing cylinders togetherin such a way that all cylinders roll against their neighbours. They solved this problemusing a fractal distribution of cylinder sizes. Since subglacial sediments contain avariety of particle sizes, this fractal packing arrangement seems to offer a reasonabledeformation model. Unfortunately, rolling friction will exist between the cylinders;with an infinite number of cylinder-to-cylinder contacts, the fractal packing will seize.A simplified version of this model is illustrated in Figure 3.12. The prismaticblocks are represented by infinitely long cylinders of equal radius that are packed ina hexagonal fashion between two bounding plates (representing the ice above and thenon-deforming material below). Some variable friction coefficient function is assignedto the surface of each cylinder and the two boundaries. The upper plate is loadedvertically and moved horizontally at some velocity U. This will cause the cylinders tobegin rotating in response to the normal and tangential forces at their contacts withneighbouring objects. If the separation between the top and bottom plates is fixed andthe cylinders forming the lateral bounds of the packing are forced to maintain theirChapter 3. BED DEFORMATION: DATA ANALYSIS 83relative positions, then no rearrangement of the hexagonal packing is possible. Thishas two consequences: (1) The geometry of the packing will force some cylinders to slipagainst neighbouring objects. (2) The centres of all the cylinders will move horizontallyat velocity U/2.Fig. 3.12: The deforming layer modelled as a series of hexagonally-packedcylinders. Some cylinders will be forced to rotate in a reverse sense. A tiltsensor next to a cylinder rotating in reverse will experience a negative strainrate.Solving for the rotation direction of each cylinder and how this direction mightchange over time is a difficult problem. An equilibrium analysis of the packing quicklyshows that the system of dynamical equations describing the packing is under-determined. Nevertheless, it should be clear is that some cylinders will rotate in a normalsense (clockwise in Figure 3.12) and others will rotate in a reverse sense (anti-clockwise).In addition, the direction of rotation of a given cylinder will change with time asthe cylinder rotates and areas of different frictional coefficient are presented to itsneighbours. Clearly, a tilt sensor placed next to one of these cylinders can experienceboth positive and negative strain rates and any transition is strain rate sign will berapid.ICEBASEMENTChapter 3. BED DEFORMATION: DATA ANALYSIS 84If some method of solution were found, then this representation of the deformingglacier bed could perhaps be used to model basal shear stress and the space- and time-averaged properties of basal deformation. Unfortunately, the model has limitationswhich would render suspect any simulations: (1) Rearrangement of particles is notpossible. (2) All particles translate at the same speed. (3) All particle rotate at thesame speed. (4) Even if the cylinders are conglomerates, there is effectively only oneparticle size. (5) The interstices between cylinders are not dealt with by the model.3.5.3 The shadow box computerRather than attempting to solve the roller bearing problem numerically, Tavi Murray(personal communication) suggested that an analogue computer could be built to solvethe problem. An assemblage of two-dimensional particles cut out of 3 mm thick particleboard is placed in a gap between two verticle plexiglas sheets. A ruler is passed betweenthe plates and is used to apply normal and shear forces on the cut-outs. By projectinga bright light from behind the sheets, the silhouettes of the cut-outs can be recordedon video tape for viewing and digitizing at a later time. This apparatus is dubbed the“shadow box computer”.In addition to solving the roller bearing problem with alacrity, the shadow box hastwo great advantages over the numerical approach: (1) gravitational forces are includedand (2) a variety of particle shapes and sizes can be used.Figure 3.13 shows a pair of images recorded from the shadow box computer. Thesilhouettes clearly show particles rotating in the normal and reverse senses as well asparticles not rotating at all. Dilation of the particle packing is also evident.Chapter 3. BED DEFORMATION: DATA ANALYSIS 85Fig. 3.13: A pair of frames from a video record of the shadow box. Theupper panel shows the initial configuration of the blocks between the platesof the box. As the upper boundary, with the particle marked “A” attached,is moved to the right, the particle configuration in the lower panel results.Notice that particle “B” has experienced normal rotation, whereas particle“C” has experienced reverse rotation. Some dilation of the sediments is alsoevident.3.6 DiscussionBasal strain rate is observed to change polarity, and to fluctuate wildly. If a deformation rate of 6 cm day’ (surface velocity less sliding velocity as determined by theChapter 3. BED DEFORMATION: DATA ANALYSIS 86slidometers; internal ice deformation neglected) is associated with uniform shear deformation in a 1 m thick subglacial layer, then Equation (2.4) gives a basal strain rate ofabout 11 yr. The mean strain rates of 8—30 yr1 in Figure 3.9 are consistent withthis estimate, but the instantaneous strain rate often exceeds the mean value by anorder of magnitude and can be of opposite polarity. The coherence of the three tracesin Figure 3.9a suggests that this is not a consequence of local heterogeneity in thesediments. The erratic behaviour might be caused by rapid changes in the couplingbetween the glacier and the bed, or changes in the thickness of the actively deforminglayer.3.6.1 Boulton and Hindmarsh flow modelsOur in situ measurements of subglacial deformation demonstrate that the rheologicalrelations derived by Boulton and Hindmarsh (Equations (3.16) and (3.17)) are inappropriate for the sediments beneath Trapridge Glacier. Because the sediments beneathBreidamerkurjökull and Trapridge Glacier are without doubt different, this inapplicability is perhaps not surprising. Nevertheless, we have demonstrated that measurementsof net strain, such as those made by Boulton and Hindmarsh, are fundamentally flawedbecause they ignore the time-varying nature of subglacial deformation. In section 3.4.5,we showed that quite different values for mean strain rate can be obtained simply byconsidering deformation over different time intervals. We make no statements regardingthe utility of Equations (3.14) and (3.15) for describing subglacial deformation otherthan these: (1) the definition of effective pressure must be rigorous (i.e. Equation (3.6)is probably inadequate), (2) the effective pressure cannot be reliably inferred from measurements of either subglacial water pressure or pore pressure (see discussion below),and (3) a good understanding of the shear stress exerted on the sediments is required.ChapterS. BED DEFORMATION: DATA ANALYSIS 87Although we have demonstrated that the Boulton and Hindmarsh relations as notapplicable to sediments underlying Trapridge Glacier, we have not derived an alternative rheology. The reasons for this are two-fold: (1) our data demonstrate no correlationbetween effective pressure (as defined by Equation (3.6)) and strain rate and (2) wehave no measurements of local shear stress.3.6.2 Shear stress and normal stressThe use of shear stress in these experiments warrants some discussion. We have beenusing a mean shear stress calculated using Equation (3.4), but there is strong evidencethat the basal shear stress is far from uniform — this has obvious implications forvalues of effective viscosity computed with Equation (3d1) and is a problem faced byall glaciologists. Consider hole 891150. As we noted above, the pressure in this holeexceeds the nominal flotation pressure by Sm(H20). It is clear that the glacier doesnot accelerate upwards as a result of this apparent force imbalance, so the overburdenpressure at 891150 must be greater than the thickness of the ice suggests. Since theglacier, on a diurnal time scale and over distances less than the glacier thickness, is asomewhat rigid body, lateral transfer of loading on the bed is possible; the load canbe greater at some points and less at others, so long as the mean loading is equal tothe theoretical overburden pressure pgh. We must also bear in mind that the shearstress on the bed is not necessarily proportional to the normal loading on the bed, asEquation (3.4) implies. A sticky patch (with a high shear stress) can have a low normalload and a slippery patch (with a low shear stress) can have a high normal load; again,the stability requirement is only that the mean shear stress averaged over the glaciersole be equal to that computed by Equation (3.4).The folly of relying on subglacial water pressure to determine effective pressureand of relying on geometry to determine subglacial shear stress is revealed! If theoverburden pressure cannot be determined, then it is impossible to compute the effectiveChapter 3. BED DEFORMATION: DATA ANALYSIS 88pressure by any of Equations (3.6), (3.8), or (3.10). Similarly, a non-uniform shear stressdistribution makes theoretical shear stress calculations of dubious value.We expect that slippery patches are associated with connected zones and stickypatches are associated with unconnected zones. The characteristic overpressure conditions observed in unconnected boreholes (boreholes drilled into unconnected zones)demonstrate that the normal loading over unconnected zones is generally larger-than-mean (this is why over-pressurized water in unconnected holes does not simply force itsway out). The larger-than-mean normal loading implies a capacity to support larger-than-normal shear loading. We have, however, observed overpressure situations in connected zones (on occasion, we encounter artesian outflow conditions when completing aconnected borehole). Furthermore, the significant pressure gradient observed between891114 and 891178 indicates that a connected zone is not just a puddle of water underthe glacier; some resistance to hydraulic flow is present. This suggests that connectedzones consist of a basal layer that has been washed clean of fine particles, leaving anopen hydraulic aquifer capable of supporting both normal and shear loading.This sticky/slippery—connected/unconnected model of the bed provides an explanation for the diurnal strain rate fluctuations observed in 1989 when there were nocorresponding fluctuations in water pressure. Because hole 891150 is unconnected, thenormal loading at this point on the bed must be high. Holes 891178 and 891180, thelocations of the two deformation instruments, may also be experiencing a larger-than-mean normal loading. Although we have no measurements of local shear stress, we caninfer that this area is also capable of supporting larger-than-mean shear stress; thiscould result in greater deformation. The question arises: how does lateral transfer ofshear stress occur?In the weeks prior to the bed deformation experiments, pressure sensors installedat various locations near the deformation experiment site recorded dramatic diurnalfluctuations — each day during the afternoon and evening, the subglacial water pressureChapter 3. BED DEFORMATION: DATA ANALYSIS 89rose to a peak at about 21:OOh — but on 1 August, four days prior to the beginningof the deformation experiment, all these pressure sensors began reporting quiescentpressure levels. As we will show in Chapter 6, subglacial electrical phenomena inchcated that the subglacial environment was anything but quiescent; large fluctuations inapparent resistivity and natural potential ‘occurred until 10 August. Since meitwateris the only significant diurnal force acting on the subglacial environment (tidal effectsand thermal stress, both certainly negligible, are the only other candidates we canthink of), we strongly suspect that subglacial water pressures continued to rise duringthe afternoon between 1 August and 10 August, but that none of our pressure sensorshappened to be located where these pressure cycles could be observed. Increases inwater pressure elsewhere in the subglacial environment would, through processes discussed in Chapter 1, reduce the shear stress in those areas; the principle of constantmean shear stress then requires that the shear stress increase elsewhere. If the regionof the deformation experiment experienced a rise in shear stress, we could expect tosee diurnal cycling in deformation without any change in subglacial water pressure atthe strain measurement the latter part of the 1989 deformation experiment, the strain rate makes atransition from negative to positive late in the evening, a few hours after the presumedmaximum in subglacial water pressure; this correlation is consistent with the idea ofpositive strain rate indicating down-flow deformation of the sediments. As the waterpressure elsewhere in the system fails, the inferred enhanced shear stress on our sensorarray vanishes and we see a negative-going trend in strain rate. This trend is interpretedas being some form of sediment relaxation phenomena.3.6.3 Effective viscosityIn closing this chapter, we wish to reiterate our distrust of using effective viscosity indiscussions of sediment deformation. Although we have derived these values of effectiveChapter 3. BED DEFORMATION: DATA ANALYSIS 90viscosity for Trapridge Glacier sediment, we believe that effective viscosity is only usefulas a aid in visuaiizing the ability of subglacial material to resist deformation and glacierflow. There is considerable evidence that the rheology of subglacial sediments is nonlinear (e.g. Boulton and Hindmarsh, 1987; Murray, 1990; Kamb, 1991) and we havediscussed the difficulties in applying the very concept of viscosity to deformation ofgranular material.Chapter 4ELECTRICAL PHENOMENA— THEORY4.1 IntroductionSubglacial measurements of electrical phenomena are sensitive to the movement anddistribution of water within the glacier bed. This is because the water is in largemeasure the electrically active component of the subglacial environment. By makingmeasurements at intervals over a period of time, we can also monitor temporal changesin the subglacial hydraulic system. This chapter presents a theoretical foundation fordirect current (d.c.) electrical resistivity and natural potentials; we will give particular attention to electrokinetic phenomena. The objectives and justifications for, anddata from, our subglacial electrical measurements are discussed in Chapter 6. We begin we discussions of the microscopic structure of the rock—electrolyte interface andmechanisms for electrical conduction through rocks and sediments.4.2 The rock—electrolyte interfaceFluid filling pores within rocks and sediments can be divided into two componentswith distinctly different properties: bulk fluid and fluid associated with the interfacebetween the two phases. Since we will be using the term electrolyte in association withthe pore fluid, we call this interface the rock—electrolyte interface.Whenever two phases of dissimilar electrical properties are in contact, a separationof charge arises at the interface — this separation is called an electrical double layer because it consists, in crude terms, of two planar regions of opposite charge. The simplestconceptual model for the electrical double layer is usually attributed to Helmholtz: two91Chapter 4. ELECTRICAL PHENOMENA — THEORY 92layers of opposite charges are arranged in parallel to form a sort of molecular capacitor. Such a double layer can easily form between two solid phases, but it is difficultto imagine how a plane of charges could persist in a fluid electrolyte where thermaldiffusive forces compete with electrostatic forces for the control of ions. Indeed, theions will spread out into a diffuse layer.4.2.1 Gouy—Chapman modelThe theory for a diffuse double layer was developed independently by Gouy (1910)and Chapman (1913). Figure shows the ion and potential distributions associatedwith the Gouy—Chapman model. The significance of the potential will be discussed insection 4.6. The Gouy—Chapman model represents the solid surface as a uniform planarcharge distribution with potential o relative to the bulk electrolyte. The charges inthe pore fluid are modelled as point charges. The surface attracts ions of oppositecharge (counter-ions) and repels ions of like charge.To solve for the potential distribution, we begin with Poisson’s equation for thepotential ‘ arising from a volumetric charge distribution Pv= _ (4.1)where e0 is the permittivity of free space and D is the relative permittivity of the fluid(though D ‘s-’ 80 for bulk water, it may be significantly different close to an interface asa result of poorly understood interactions with the solid surface). At equilibrium, theelectrical and diffusional forces on each ion species i must balance, i.e.= —zeV (4.2)where z is the valence of the ion, e is the charge on an electron, and the chemicalpotential is defined asgui = gu + kBT1nn (4.3)Chapter 4. ELECTRICAL PHENOMENA — THEORYFig. 4.1: Two models for the rock—electrolyte interface. The diagramson the left indicate the ion distribution and those on the right indicate thepotential distribution. In this example, the surface potential o is negative.(a) The Gouy—Chapman or diffuse model. The potential at the hydrodynaniicshear plane (dashed line) is the potential. (b) Stern’s model. The outerlimit of the Helmholtz layer is indicated by dotted line (after Morgan andothers, 1989).93where T is the temperature, kB is Boltzmann’s constant, = is the number densityof the ion species, and 4 is the chemical potential for n = 1 (Keizer, 1987, P. 107).All ions, not just the counter-ions, are included in Equations (4.2) and (4.3) becausee ee:1 eee eeCbthey all contribute to the electric field structure.Chapter 4. ELECTRICAL PHENOMENA — THEORY 94Taking the gradient of Equation (4.3) and solving for rz (where 4 = 0 and n =in the bulk electrolyte) gives Boltzmann’s equation= exp (—::) (4.4)We notice thatp0 = >Zmjzje (4.5)and so the distribution obeys the Poisson—Boltzmann distribution= -Jnzjeexp (_-,) (4.6)where the permittivity is e = e0D.The analytic solution to Equation (4.6) involves hyperbolic trigonometric functions(these arise because ion species of like valency and opposite charge can usually be pairedoff), but the solution is essentially exponential in form (see Figure Stern modelThe Gouy—Chapman model is sufficient for explaining the origin of electrokinetic phenomena, but it is not the currently accepted model of the rock—electrolyte interface.The Gouy—Chapman model does not make allowance for adsorption of ions onto theinterface, the non-point nature of the ions, or for secondary effects such as the alignment of polar molecules in the electric field and the resulting dampening of the field(the most common polar molecule in a groundwater system is water). Stern (1924) wasthe first to propose that the ion distribution has two major components: (1) an inneror compact layer of ions (termed the Helmholtz layer) where the charge arrangementand potential distribution are controlled by short-range or adsorptive forces and bygeometrical restrictions; (2) an outer Gouy—Chapman or diffuse zone that follows aChapter. ELECTRICAL PHENOMENA— THEORY 95Poisson—Boltzmann distribution. The ion and charge distributions found in the Sternmodel are shown in Figure 4.2 shows a detail of the Stern model interface and the associated potentialdistribution. The inner Helmholtz plane (IHP) marks the limit of closest approach ofany adsorbed ions or molecules and the outer Helmholtz plane (OHP) marks the limitof influence of adsorptive forces. The diffuse layer extends outward from the OHP.Fig. 4.2: Detail of the Stern model rock—electrolyte interface. The IHPdemarcates the distance of closest approach of ions and molecules. The OHPis the limit of influence for adsorptive or short-range interaction between ionsand molecules. This example has negative surface charge 4 and negativeadsorbed ions. Notice the preferential orientation of water molecules near theinterface. The diffuse layer ion distribution has been omitted (after Bockrisand others, 1963; Hunter, 1981).C•— POSITIVE ION©c0 WATER MOLECULE— NEGATIVE IONChapter.4. ELECTRICAL PHENOMENA — THEORY 96The preferential orientation or polarization of water molecules near the interfaceis evident in Figure 4.2. Bockris and others (1963) note that the permittivity e of thewater drops from about 80 beyond the OHP to 32 between the IHP and OHP, and to6 within the IHP; since the dielectric properties of water depend in part on the freedomof the molecules to move about, this decrease in permittivity reflects the increasingorientation restrictions placed on the water molecules as the surface is approached.The advantage of the Stern model is that it permits changes in the magnitudeand polarity of the potential with changes in concentration and pH; it is hard toaccomplish this with the diffuse model alone since the surface charge o is relativelyconstant. In general, as the pH is lowered (more acidic conditions), the potentialincreases; the relationship between potential and ion concentration and species ismore complex (Hunter, 1981, Chapter 6). Under certain conditions, unusually high ionconcentration and low pH, the potential can be made equal to zero, and streamingpotentials vanish; this is known as the point of zero charge.The potential distribution in the Stern model is troublesome to describe mathematically because the double layer incorporates so many interacting processes; we willnot delve into the mathematics here (readers can refer to Hunter, 1981, Chapter 2), butthis omission is irrelevant since the conceptual model we have developed is sufficient tounderstand electrokinetic phenomena.4.3 Conduction mechanismsThere are three mechanisms by which electrical current can propagate in natural settings: electronic, dielectric, and electrolytic conduction.Electronic conduction requires the presence of free electrons, as found in metalsand suiphide minerals. These materials have very low resistivities (less than 10 1 m).Although suiphide minerals are found in the moraines surrounding Trapridge, the predominant rock types are carbonaceous and siliceous; the minerals composing these rocksChapter 4. ELECTRICAL PHENOMENA — THEORY 97have resistivities above 1010 m (Telford and others, 1990, p. 285—289). We do notexpect electronic conduction to be a significant contributor to subglacial conductivity.Dielectric conduction results from the slight displacement of bound electron relative to their nuclei when in the presence of an electric field. The displacement isnecessarily small, and the effect is not observable as a current except when the externalelectric field varies. The displacement current is analogous to current flowing througha capacitor. Since our measurements are made with static or near-static electric fields,we do not observe displacement currents and we do not measure dielectric propertiesof the subglacial material.The subglacial material beneath a glacier is a mixture of water and minerals. Thepore water is an electrolyte containing dissolved mineral salts that may or may not be inequilibrium with the surrounding minerals. Under the influence of an external electricfield, the ions in the water will move. Electrolytic conduction has two components: bulkconduction and surface conduction. Bulk conduction arises from the movement of ionsbeyond the hydrodynamic shear plane. Surface conduction is the enhanced conductivitywithin the hydrodynamic shear plane that arises because of the high concentrationof charge carriers. Although the ions themselves do not move, current is carried byprotons (Hj hopping from place to place, and perhaps also by H3O (Parks andothers, 1966; Anderson and Parks, 1968). Since most of the subglacial minerals arepoor conductors, subglacial conduction is dominated by electrolytic conduction; theresistivity of the subglacial material will vary with the mobility, concentration, anddegree of dissociation of the ions, as well as with the properties of the rock—electrolyteinterface.4.4 Electrical resistivityElectrical resistivity is a material property that quantifies the resistance of a materialto the flow of electrical current. This general empirical relationship, known as Ohm’sChapter 4. ELECTRICAL PHENOMENA — THEORY 98Law, is expressed as—VV = pJ (4.7)where p is the electrical resistivity tensor, V is the electrical potential field (a macroscopic potential field rather than a microscopic surface potential field), and J is theelectrical current density. Anisotropy in the resistivity tensor indicates current flow in adirection other than that of the applied electrical field B = —VV. For most substances,and for the analyses in this thesis, this effect is neglected and resistivity is treated as ascalar quantity p. Equation (4.7) is sometimes written asJ = —oVV (4.8)where o- = i/p is the electrical conductivity.As discussed in section 4.3, current flow through water-saturated sediment is carried by bulk conduction and surface conduction. If we neglect surface conduction(Morgan and others (1989) note that neglecting surface conduction in clay-rich sediments is risky) and remember that the minerals in the sediment are poor conductors,we can imagine that the electrical resistivity of the sediment will depend on the amountof water in the sediments and on how well pockets of water are connected together.Archie (1942) proposed the formulaan_mS_dpw (4.9)to describe the resistivity of porous rock; n is the fractional pore volume (porosity), Sis the fraction of the pores containing water (saturation), m is a cementation factor, dis the saturation exponent, and Pw is the resistivity of the pore water. The range ofvalues for the constants d, a, and m are: ci 2, 0.5 a < 2.5, and 1.3 m <2.5. Forwater-saturated subglacial sediments, S = 1. The value of m is roughly proportionalChapter 4. ELECTRICAL PHENOMENA — THEORY 99to the age of the rock and is about 1.3 for Tertiary sediments (Parasnis, 1986), but isprobably even smaller for deforming sediments with high porosity.4 .. 2 Measuring electrical resistivityTwo- or four-electrode methods may be used to measure electrical resistivity in thefield. Although two-electrode systems are logistically simple, they suffer from two imperfections: (1) It is difficult to ensure a good consistent electrical connection betweenthe electrodes and the ground; a significant contact potential, which is indistinguishablefrom the potential arising from current flow through the medium, often arises. (2) Ascurrent passes through the electrodes, charge carriers become depleted in the vicinity ofthe electrodes and the resistance of the circuit increases over time; this effect is knownas electrode polarization. In consequence, two-electrode systems are used rarely andonly for specialized purposes (e.g. Neftel and others, 1985).Both of the problems mentioned above are associated with the flow of electricalcurrent through electrodes. By separating the electrodes measuring potential fromthose delivering current, four-electrode systems avoid these problems entirely. Onepair of electrodes is used to inject a metered current into the ground using as muchvoltage as is required; a second pair is used to measure the resulting electrical field.4.2.1 Potential fieldsThe equation describing the potential distribution V resulting from a current source Ion the surface of a half-space isV (4.10)2irrwhere r is the distance from the current source (Telford and others, 1990). The singularity at r = 0 is of no importance since real current sources are not point sources.Chapter.4. ELECTRICAL PHENOMENA — THEORY 100Since Equation (4.7) is linear, the output of a four-electrode array with all fourelectrodes on the surface of a half-space can be computed by superimposing the voltagecontributions from multiple current sources. The two current electrodes are numbered 1and 2 and the two potential electrodes are numbered 3 and 4; current enters at electrode 1 and electrode 4 is at zero potential. The voltage V at electrode 3 is givenby(4.11)2ir \r r14 r23 r24jwhere rmn represents the scalar distance between electrodes m and m. Equation (4.11)is often written asv = (4.12)where(4.13)\r13 r14 r23 r24Jis a geometrical factor. We will use the term d.c. potential to describe the inducedpotential V; this term reflects the anthropogenic nature of the potential and its association with constant (direct) current levels. InterpretationIn practice, the Earth is not an isotropic homogeneous medium. Although this anisotropy introduces complications, a homogeneous medium would not be particularlyinteresting. Establishing the relationship between Pap and the true resistivity structureof a half-space is the burden placed on geophysicists.The resistivity computed from a pair of current and voltage observations is calledapparent resistivity; the apparent resistivity is equal to the homogeneous half-spaceChapter 4. ELECTRICAL PHENOMENA — THEORY 101resistivity that would yield the same voltage and current readings and from Equation (4.12) is given by2irVPap= IG(4.14)The simplest form of anisotropy is layering; the resistance encountered by currentflowing along the layers (so the layers act as resistors wired in parallel) may be quitedifferent from the resistance encountered by current flowing across the layers (so thelayers act as resistors wired in series). Analytic solutions for voltage response can be derived for simple Earth models (e.g. buried spheres, simple layered models), but for morecomplex models, other approaches are required. A basic characteristic of four-electroded.c. resistivity arrays is that the farther apart the current electrodes are, the deeperthe current penetrates into the ground. The apparent resistivity reflects some spatiallyaveraged resistivity near this “pseudo-depth”. For many years, geophysicists reliedon extensive sets of type curves compiled for various simple layered Earth structures(e.g. one, two, and three layers over a half-space) and standard electrode configurations(e.g. Wenner, Schiumberger, pole-dipole, etc.). With the advent of powerful computers has come the ability to infer more complex two- and three-dimensional subsurfacestructures using inverse techniques and forward modelling.In our subglacial experiments, the electrodes are not placed in standard, or evenrectilinear, configurations. This is not because we wanted disorderly arrays, but because the holes drilled to the bed of the glacier are rarely plumb. For this reason,we have developed a simplistic measure for the depth of observation, or pseudo-depth,for a generalized d.c. resistivity array. The method involves computing the apparentresistivity over a simple layered-Earth model.At an interface between two homogeneous regions of differing electrical resistivity,the tangential electric field must be continuous. This condition givesñxE1=ñxE2 (4.15)Chapter .4. ELECTRICAL PHENOMENA — THEORY 102The normal current flow must be continuous across the interface givingñ•E1 = ñE2 (4.16)P1 P2Analytic solutions to Equation (4.7) subject to Equations (4.15) and (4.16) are possible,but the solutions are unwieldy. Current density J falls of with an inverse-square relationas we move away from a current source. This relation is the same as for light and ithappens that the equations describing reflections between semi-transparent mirrors canbe applied to currents in the Earth. Telford and others (1990, Chapter 8) show thedevelopment of this method-of-images technique for solving d.c. resistivity problems.We chose a single layer over a half-space for our pseudo-depth calculations. Thismodel was chosen because the ice above the array can be considered infinitely resistive(section 6.1). For very thin layers, the array in question will respond primarily to thehalf-space, or basement, resistivity, and for a thick layer, the array will respond to thelayer resistivity. By calculating the apparent resistivity for a varying layer thickness,we can determine what layer thickness corresponds to transition from layer responseto basement response.The method-of-images equation for the potential distribution on the top of thelayer resulting from a current source I (also on the top of the layer) isI_________V=— +2 (4.17)2ir rmO./(2mb)2 + r2wherekP2I)1 (4.18)P2 + P1and where p’ is the layer resistivity, P2 is the half-space resistivity, b is the layer thickness, and r is the separation of the current source and voltage measurement point.The value of k is analogous to a reflection coefficient for the layer-basement interface:Chapter 4. ELECTRICAL PHENOMENA — THEORY 103k = —1 for a conductive basement and ic = +1 for a resistive basement. Equation (4.17)is analogous to Equation (4.10) for a homogeneous half-space. Following the development of Equation (4.14), but using Equation (4.17) in place of Equation (4.10), wederive an expression for the apparent resistivity of a generalized d.c. resistivity arrayover the layered Earth model:Pap [1+ (2b)2 +r300 k’-(2mb)2 + T3 + (2mb)2 + r }] (4.19)where G is defined as in Equation (4.13).Figure 4.1 shows the apparent resistivity curves for a Wenner array (ri2 = r23 =r34 = 1 m, electrodes all in a line). The solid line is the curve for a resistive layer andthe dotted line is the curve for a conductive layer. Since the resistive layer providesa sharper transition, we will use a resistive layer in our calculations of pseudo-depth.The pseudo-depth z, is defined as the point where, on a logarithmic scale, the apparentresistivity is midway between p’ and P2. In Figure 4.3, that point is at Pap = 3.16 1 mand = 0.480 m. Notice that z depends only on the electrode geometry — it isindependent of the resistivity values chosen. Current switchingAs will be discussed in section 4.5, natural electric potentials exist in the Earth; thesenatural potentials are added to the d.c. potentials. The standard method for removingthis natural potential offset is to measure the d.c. potential with the current beingdriven in both directions. Because Equation (4.10) is linear, changing the polarity ofthe current will change the polarity of the resulting potential. If the natural potentialChapter 4. ELECTRICAL PHENOMENA — THEORY 104101E100 I_________________io— 102 101 100 101 102LAYER THICKNESS (m)Fig. 4.3: The apparent resistivity for a layer over a half-space model asobserved by a Wenner array configuration with a 1 m electrode spacing. Thehorizontal axis shows the variation in the layer thickness. The solid line is thecurve for a resistive layer (p = 10 fm, p2 = 1 1 m), and the dotted line isthe curve for a conductive layer (p1 = 111 m, P2 = 10 Il m).remain unchanged, subtracting the potentials observed at each current polarity removesthe natural potential component.In order to make these measurements, a typical electrical resistivity apparatus usesthe current waveform illustrated in Figure 4.4. The forward biased current is appliedfor a period of time followed by a period of no current flow. Then the current supplyis reversed and the pattern is repeated. Transient effectsThe Earth’s voltage response to changes in applied electric current is not immediate.There are at least two processes that create a delay in establishing a stable voltage;electrode polarization and membrane polarization (see Marshall and Madden (1959);- —--------\\\\\I I IChapter . ELECTRICAL PHENOMENA — THEORY 105Fig. 4.4: The alternating, pseudo-static current waveform traditionallyused for making d.c. resistivity measurements.Keevil and Ward (1962) for concise overviews). These processes are known collectively as induced polarization (IP) effects and are often used in mineral exploration,principally in the search for disseminated suiphide minerals.Electrode polarization was mentioned in section 4.4.2 in connection with charge-carrier depletion at the current injection electrode; the same phenomenon occurs whencurrent passes between electrolytic conductors (e.g. pore water) and electronic conductors (e.g. graphite and sulphide minerals). We distinguish these two situations whereelectrode polarization arises by speaking of ore-body polarization when referring to thesource of IP effects.Ions drawn to a metallic interface under the influence of an external electric fieldexchange electrons with the metallic body. Since the metallic body does not interruptthe electric or current fields, but does present a physical barrier to ion movement, ionsaccumulate on the boundaries of the metallic body and cause a gradual increase in theobserved surface potential. After the electric current is turned off, a period of timeelapses before the ions return to their initial positions; the resulting potential decaycan be observed (Figure 4.5). Most time-domain IP exploration equipment measuresIzwDC.) TIMEChapter 4. ELECTRICAL PHENOMENA — THEORY 106the IP signal on its decaying limb after the current is shut off, either by integrationor sampling over time. The dependence of ore-body polarization on ion concentration,mineralogy, mineral dissemination, porosity, pore size, etc. is not fully understood.Fig. 4.5: The effect of induced polarization on the observed voltage response of the Earth. The deviation from the square wave in Figure 4.4 is theIP effect.Physical barriers to ion movement can also be generated by a pre-existing chargedishibutions in the ground. The electrical double layers discussed in section 4.2 canblock pores if the pore size is comparable to that of the double layer thickness; thiseffect is known as membrane polarization. Ore-body and membrane polarization areindistinguishable, but ore-body polarization effects are generally the larger of the two.In 1987, the first year we took d.c. resistivity measurements under TrapridgeGlacier, we made oscilloscope observations of potential waveforms measured at theglacier bed and found no evidence for IP potentials. Despite an apparent absence ofIP effects, we chose to measure d.c. potentials at least 1 s after turning on the current;this delay is large enough to eliminate all but the most persistent IP effects. Our caution was motivated by a concern that IP effects might develop during the course ofour experiments; owing to a lack of metallic minerals, we did not expect to observeLU0>TIMEChapter 4. ELECTRICAL PHENOMENA — THEORY 107ore-body polarization potentials, but the presence of streaming potentials (discussedin section 4.6) implies the presence of electrical double layers and suggests that themembrane polarization mechanism might be active. Because of instrument limitations,we made no attempt to measure IP effects.4.5 Natural PotentialsNaturally occurring potentials are often referred to as self potentials or spontaneouspotentials, but to avoid confusion of these terms, both having the acronym “SP”,with streaming potentials, we will use the term natural potentials to describe them.Sources of natural potentials include: streaming potentials, thermoelectric potentials,electrochemical potentials, and telluric potentials.Thermoelectric and diffusion potentials are phenomena related to the activity ofions in the pore fluid electrolyte. Under isothermal conditions, the differing mobility ofthe ions creates a liquid-junction or diffusion potential; as the ions diffuse in responseto the concentration gradient, the more mobile ions (usually the cations, as they arephysically smaller) will move faster; a charge separation develops.There are several mechanisms producing thermoelectric potentials. Since the mobility of ions is temperature dependent, temperature gradients under conditions ofconstant concentration will produce a thermoelectric potential. Thermoelectric potentials also appear when two dissimilar metals or metallic minerals come in contact;this effect, named after Seebeck, is the operational principle of thermocouples, but theSeebeck effect is not of concern here. In the ground, thermoelectric potentials created by differing ion mobility are typically 0.2—0.3 mV °C’ (Nourbehecht, 1963) andhave been used in geothermal exploration (e.g. Corwin and Hoover, 1979; Corwin andothers, 1981).Chapter (. ELECTRICAL PHENOMENA — THEORY 108Corrosion or mineralization potentials are produced by electrochemical cells formedbetween the pore fluid electrolyte and certain conductive minerals (principally sulphides, graphite, and some oxides). These potentials can be very large— hundreds ofmillivolts— and are analogous to the reactions occurring in primary and secondarybatteries. Typically, corrosion potentials arise over large ore bodies where the conditions of burial cause one part of the body to undergo oxidation and another to undergoreduction (Telford and others, 1990). Corrosion potentials are the principal justification for using metal/metal salt electrodes since corrosion potentials on an electrode canbe large compared to natural potentials (see section 5.3.4).Telluric potentials are caused by the electric currents induced in the Earth byfluctuations in the Earth’s magnetic field. The frequency of these fluctuations rangesfrom iO Hz up into the audio range and the potentials resulting from the telluriccurrents are of the order lOmVkm’ (Telford and others, 1990).There are other sources of higher frequency electric signals — for instance tnboelectric and seismoelectnic phenomena associated with rock fracture and transientstress (Butler, 1991) — but because we are making measurements at intervals over aperiod of many days, it is unlikely that these transient potentials could contaminateour data.Irreversible thermodynamicsStreaming potentials result from a cross-coupling between pressure gradients and electric potentials; several of the other sources of natural potential mentioned above alsoderive from similar cross-coupling mechanisms. To fully understand these coupled phenomena, we must delve into the thermodynamics of irreversible processes.In a pair of seminal papers, Onsager (1931a, 1931b) considered the interactionof two or more irreversible transport processes (e.g. heat conduction, diffusion, electrical conduction, and fluid flow) and laid out the framework for thermodynamics ofChapter 4. ELECTRICAL PHENOMENA — THEORY 109irreversible processes. Onsager’s first postulate of the thermodynamics of irreversibleprocesses is that, in a system where many flows and forces mingle, it is possible toexpress each flow J as a linear combination of all the extant forces X,. The resultingsystem of equations= L,X, (4.20)constitutes the thermodynamic equations of motion or the scalar phenomenologicalrelations. L3 are phenomenological coefficients. The phenomenological coefficients L3can be divided into two classes: where i j, L, are generalized conductivities andwhere i 0 j, are coupling coefficients between the different flows and forces. Theforces X2 are usually written as negative gradients of potential fields.Equation (4.20) is a low-order Taylor expansion of the general equation J =f2(X1,X2,X3,. . . , X,,j when we stipulate that no flow occurs in the absence of forcing.The linear relationship between forcing and flow is not preordained and must be testedfor each physical system, but there are many empirical relations of this form in commonuse. For instance,Jcurrent = —uVV Ohm’s Law (4.21a)Jheat = —K1VT Fourier’s Law (4.21b)Jflujd = —K2VP Darcy’s Law (4.21c)soiute = --DVC Fick’s Law (4.21d)where o, K1, K2, and D are appropriate conductivities, and VV, VT, VP, VC aregradients of electric potential, temperature, pressure potential (fluid pressure less hydrostatic pressure), and concentration. In general, linear relationships should hold forsmall displacements from equilibrium, but for larger excursions, higher order termsmust be introduced. For example, at high fluid flux Jfiuid, turbulence can introducenon-linearities in Darcy’s Law.Chapter 4. ELECTRICAL PHENOMENA — THEORY 110Onsager’s reciprocal relation states that the phenomenological coefficient matrixL is symmetrical:= (4.22)This relation was first proposed by Thomson (Lord Kelvin) (1853) in connection withthermoelectric phenomena and later by Helmholtz (1882) in connection with diffusionpotentials (diffusion potentials are sometimes referred to as Nernst potentials in honour of Nernst (1888) whose kinetic model derivation of Equation 4.22 corroboratedexperimental results).Onsager’s second postulate of irreversible thermodynamics is the principle of microscopic reversibility. Microscopic reversibility states that for conservative force fields(a conservative force field is one where the work required to move a particle from onelocation to another is independent of the path taken), the dynamical laws are alwaysreversible. That is to say that if the velocities of all the particles are reversed simultaneously, the particles will retrace their former trajectories, reversing the entire sequenceof prior states. Using the example of a chemical equilibrium, Onsager (1931a) showedhow Equation (4.22) can be derived from the principle of microscopic reversibility (seeHaase, 1969 and de Groot and Mazur, 1984, for discussions of microscopic reversibility).Nourbehecht (1963) presents a clear development of how irreversible thermodynamics can be applied to study a host of seemingly unrelated phenomena occurring inthe ground. Equation (4.20) does not determine which forces are relevant for a particular flow; these choices must be made experimentally. Table 4.1 lists some of theinterdependences between forces and flows that have been shown to exist. As has beendiscussed above, the linearity of the phenomenological relations must be verified.Chapter . ELECTRICAL PHENOMENA— THEORY 111__________FORCINGFLOW pressure electric temperature concentrationfluid Darcy’s Law electroosmotic thermo- normaleffect osmosis osmosiscurrent streaming Ohm’s Law Seebeck diffusionpotential potential potentialheat flow Isothermal Peltier Fourier’s Dufourheat transfer effect Law effectsolutes streaming electro- Soret Fick’s Lawcurrent phoresis effectTable 4.1: Some established phenomenological relationships between theindicated forcing fields and flows.It is appropriate to note that Onsager’s linear reciprocal relations are not universally accepted. Truesdell, a respected authority on thermodynamics, has coined theterms “Onsagerism” and “Onsagerist” to describe concepts and individuals making useof these principles; the tone of these words conveys Truesdell’s opinion! Truesdell (1984,p. 404) quotes Jaynes (1980) as follows:“This approach (Onsager’s work), therefore, reached a dead end. The logic ofusing equilibrium relations in nonequilibrium situations was hardly an advance overthat used by Thomson in 1854; indeed, we are unable to see wherein they differ atall. To make further progress beyond this point, it was necessary to go back to firstprincipals and reason things out all over again, much more carefully. The coup de graceand final benedictions were administered by Wei and Truesdell.”We must admit that we too are disturbed by the use of equilibrium relations innonequilibrium situations, but we cannot dispute the appearance of streaming potentials in our data (Chapter 6) — as many others do, we will apply Onsager’s reciprocalrelations to our observations. As we will show in Chapter 6, we do observe a linearrelationship between pressure gradient and streaming potential response.Chapter 4. ELECTRICAL PHENOMENA — THEORY 1124.6 Electrokinetic phenomenaThere are two electrokinetic phenomena that arise out of the phenomenological equations for electrical current flow and water flow. If we write Equation (4.20) for theseflows we getJ L1 £12 —VV= (4.23)q £21 L22 —VPwhere J is the current density, q is the fluid flow, V is the electric potential, andP is the pressure potential (the fluid pressure less hydrostatic pressure). ComparingEquation (4.23) with Ohm’s Law and Darcy’s Law, we see that is the electricalconductivity o-, and that £22 is the hydraulic conductivity.Caution is required in isolating a pair of flows as we have done in Equation (4.23)since the presence of any one gradient will generate, through coupling, other gradientsand flows. For example, Equation (4.23) neglects thermoelectric effects and the Nernsteffect. This separation is necessary, however, in order to simplify the analysis.By Onsager’s reciprocal relation, L12 = L21. If these two phenomenological coefficients are non-zero, then we will observe two coupled phenomena: (1) The electric fieldwill contribute to water flow. This effect is known as electro-osmosis. (2) The pressuregradients will contribute to the electric current. This contribution can be modelled asan additional electric potential field known as the streaming potential. We can writeJ = -(VV + GVP) (4.24)where C =L12/L1 is the streaming potential coefficient and V is the electric field.We will note here that some early work on streaming potentials suggested that thephenomenon is not linear (Wyllie, 1951), but this nonlinearity is probably associatedwith streaming potentials in turbulent flow (Boumans, 1957a, 1957b, 1957c, 1957d;Kurtz and others, 1976).Chapter . ELECTRICAL PHENOMENA — THEORY 113J,.6.1 Observing streaming potentialsThe dependence of streaming potentials on pressure gradients makes for some interesting consequences. If a pressure source is located within a homogeneous half-space,then streaming potentials will exist throughout the medium, but no streaming potential anomaly will be observed on the surface; this is because no tangential pressuregradients can exist on a free surface. Since most geophysicists are concerned with measurement of natural potentials on a free surface, the theory developed in the literatureconcentrates on how subterranean inhomogeneities can result in surface potentials. Werecognize that the subglacial environment does not have any extensive free surfaces(the closest approximation to a free surface might be a large, low pressure-gradientsubglacial conduit), yet it is instructive to follow the development of streaming potential theory as it applies to free-surface measurements. Indeed, the ice—bed interfacemay represent the ideal locale for making measurements of streaming potentials..6.2 A theoretical developmentFrom Equation (4.24), we can define a total electric potential=V+CP (4.25)such that the current flow is given byJ = —crV& (4.26)Equations (4.25) and (4.26) disregard all sources of electric fields except for the streaming potential V. Nourbehecht (1963) and Fitterman (1978, 1979a, 1979b, 1984) arguethat in the absence of electrical current sources, the current field is divergence-free(V . J = 0) and the governing equation for any homogeneous region becomes(4.27)Chapter . ELECTRICAL PHENOMENA — THEORY 114Note that pressure sources are not considered electrical current sources because weassume that the water injected is electrically neutral.It is important to realize that the potential b cannot be directly measured; electrodes placed in the medium will measure V. For instance, in a homogeneous mediumwith no current sources, we will observe streaming potentials, but J and & are everywhere identically zero since V = —CF.Equation (4.27) ensures that sources of can only be found at discontinuities inthe medium. At any interface, continuity of normal current flow requires that=0 (4.28)where 1 and 2 denote the two media on opposite sides of the interface. Continuity ofF and V at the interface requires thatL=cLF=s (4.29)where S is a generalized source function. Because the source function is a step changein potential, it can be modelled as a dipole charge distribution along the surface. Thisis indeed what Nourbehecht and Fitterman have done.The upshot is that for a surface anomaly to exist on a free surface, the necessaryand sufficient conditions are that (1) there exist a boundary separating regions ofdifferent streaming potentials coefficient, and (2) there exist a gradient of pressureparallel to this boundary (Nourbehecht, 1963; Fitterman, 1978). Because the ice—bedinterface is not a free surface, streaming potentials can be generated and observedat the interface, but this “free-surface” result reveals that contacts below the ice—bedinterface can give rise to additional streaming potentials. These additional streamingpotentials may also be observed at the ice—bed interface.Chapter 4. ELECTRICAL PHENOMENA — THEORY 1154.6.3 Hydrodynamics and boundary layersNext to the rock—electrolyte interface, there is a boundary layer within which no fluidmotion occurs. The limit of this boundary layer is indicated by the dashed line inFigure 4.1 and is known as the hydrodynamic slipping or shear plane. The shearplane does not necessarily coincide with the OHP discussed in section 4.2.2. Under theinfluence of pressure gradients, only the fluid beyond the shear plane will move, carryingwith it an excess of either positive or negative charges. This movement constitutes aconvective flow of electric current, and as we will discuss below, gives rise to streamingpotentials.By Onsager’s reciprocal relation, we know that if a system can produce streamingpotentials, then it can also produce electro-osmotic effects. If we apply an externalelectric field, the mobile ions outside the shear plane will begin to drift. Viscous dragwill cause the pore fluid to be entrained; the electric field causes fluid flow. Since thethe late nineteenth century, electro-osmosis has been used in geotechnical applicationsfor stabilising water-saturated embankments and other earthen structures (Adamsonand others, 1966; Lewis and others, 1975). Note that the electric field established bystreaming potentials will create an electro-osmotic effect that will tend to accelerateor retard fluid flow (depending on the predominant charge polarity found beyond theshear plane). Fortunately, this effect is very small and may be neglected.4.6.4 Zeta potentialsThe potential at the hydrodynamic shear plane is the potential. As we will show,it is this potential that controls the magnitude and polarity of streaming potentials.We note that the ( potential does not have to be equal to 4’ or even have the samesign; Stern (1924) attributes the first discussion of this relationship to Freundlich,although no reference is given. For a given solid surface, the value of will dependon temperature, ion concentration, ion species, and pH. For ion concentrations and pHChapter 4. ELECTRICAL PHENOMENA — THEORY 116values generally found in soils and rocks, is negative and ranges between —100 and—2OmV (Hunter, 1981; Ishido and Mizutani, 1981; Morgan and others, 1989).Consider the laminar flow through a single cylindrical capillary tube (similar developments may be found in: Levine and others (1975); Hunter (1981); Morgan andothers (1989) — this is Helmholtz-Smoluchowski theory). The axial velocity at a distance r from the axis of the tube is given by Poiseuille’s equation(R2—r)() (4.30)where R is the radius of the capillary at the hydrodynamic shear plane and is thedynamic viscosity of the fluid. The streaming current I is found by integrating theproduct of charge density and velocity over the cross-sectional area of the tube:I, = 2rJ rvz(r)pv(r)dr (4.31)0where pv(r) is the volumetric charge density distribution. We are only interested in thecharge distribution near the shear plane since the bulk fluid in the tube is electricallyneutral, so if we substitute the approximation (R2 — r2) 2R(R — r) for R r intoEquation (4.30) and substitute the result into Equation (4.31) while letting r’ = R — r(r is distance away from the shear plane) we getis = () j’(R — r’)r’p(r’)dr’ (4.32)Now, since pv(r’) is small for moderate and large values of r’, we can approximateEquation (4.32) withirR2 /0P\ RI =----) j r’p(r’)dr’ (4.33)Chapter (. ELECTRICAL PHENOMENA — THEORY 117Substituting Equation (4.1) into Equation (4.32) and integrating by parts givesirR2e /p\ R ,d24=— (%\) j r —dv (4.34)77 Oz o dr’2= 7rR2e (OP ‘ (ylR — 1R--dr’l (4.35)77 \Oz) (\ dr’Jr,_o J0 dv’ J= irR2e (4.36)77 \9ZJThe definite term vanishes because d4’/dz = 0 at r’ = R. Normalizing Equation (4.36)by the cross-sectional area of the pore gives= . (r (437)77 \OzJSurprisingly, this expression has no dependence on the surface area of the pore — theimplication is that grain size has no effect on the magnitude of the streaming potential.The Kozeny—Carman relation (e.g. Bear, 1972, section 5.10; Berryman and Blair,1987) is a microphysical model of permeability that is consistent with Darcy’s Law.As with the Helmholtz-Smoluchowski theory, the derivation of the Kozeny-Carmanrelation assumes a Poiseuille distribution of flow qp within a cylindrical pore; to arriveat the bulk fluid flow q, an empirical tortuosity correction Co is applied to the flowwithin the pore givingq = conqp (4.38)where n is the porosity and c0 is taken as 8/5.Making a similar adjustment for porosity and tortuosity in Equation (4.37) andassuming that preferred electrical current flow paths are parallel to the pressure gradient(since current flows by surface conduction and bulk conduction, this is equivalent toassuming that water flow paths are parallel to the pressure gradient), we getJs = coE(VP (4.39)Chapter 4. ELECTRICAL PHENOMENA — THEORY 118If the geometry and orientation of the fluid conduction paths force fluid flow oblique tothe pressure gradient, then anisotropic effects must be considered (see Section 4.6.6).Equation (4.39) indicates that it is difficult to measure the value of (, even in alaboratory setting, because measurements of J3 (or rather E) do not isolate fromgeometrical, dielectric, and viscosity effects. We have already discussed variability in e— it is also likely that the viscosity of a fluid in a narrow pore is different from that ofa bulk sample of the same fluid.It is interesting to note that the fundamental streaming potential proportionalityconstant L12=e</ in Equation (4.37) is the same as the value of L21 derived forelectro-osmosis, also using Helmholtz-Smoluchowski theory (Mitchell, 1976, Chapter 15— note that Mitchell uses c.g.s. electrostatic units). This agreement probably has novalue in confirming or refuting Onsager’s reciprocal relation, since both results arebased on the same initial equations and models, but it is comforting that the resultL12 = L21 is consistent with our electrostatic distribution model for the pore (seeSection 4.4.5).4.6.5 The reverse currentThe streaming current J3 does not generate an electric field; J3 is a simple transportof charged particles and has an effect no different from that of a stream of chargedparticles drifting through a vacuum in the absence of any force fields. The streamingpotential appears because the transport of charge gives rise to a separation of charge.We know that charge conservation principles must be satisfied over some spatialscale, yet streaming potentials seem to involve the movement of electrically chargedfluid; we need some mental image that reconciles these seemingly conflicting observations. There are at least two conceptual models for understanding how streamingpotentials are generated, but neither is particularly gratifying.Chapter . ELECTRICAL PHENOMENA — THEORY 1194.6.5.1 The charge accumulation modelConsider a system in equilibrium with no fluid flow and no electric field. When apressure field is impressed on the system, the fluid flow will begin carrying off theexcess charges found outside the hydrodynamic shear zone. The destination of thesecharges is not clear, but imagine that these charges eventually begin to accumulatesomewhere; as the flow progresses, an electric field resulting from the charge separationis established. This electric field produces an electric counter-current that eventuallygrows to balance the streaming current.If we neglect electronic and surface conduction mechanisms and if we assume ahomogeneous medium (which ensures that the return current flows parallel to the fluidflow), we can imagine that the convective streaming current and the ionic countercurrent cancel; the excess ions in the pore fluid will remain stationary and the waterwill flow past them; in other words, once the charge distribution is established, thecharge carriers stop moving. The charge conservation modelConsider a volume of material as a system through which pore water flows. Figure 4.6illustrates an equivalent electric circuit for the streaming potential phenomenon. Thestreaming current I acts as a current source, continuously moving charge carriers fromone end of the system to the other. We can reasonably impose charge conservationon the system: the net charge on the fluid entering the system must equal the netcharge leaving it. Charge conservation demands a counter-current I to balance I andelectrical resistance in the conduction paths available to I necessitates the existence ofan electric field.Chapter . ELECTRICAL PHENOMENA — THEORYFig. 4.6: An equivalent circuit for streaming potentials. The convectivestreaming current I, is balanced by a reverse current I which flows via bulkand surface conduction. A potential difference arises across the circuit.J.6.6 Calculating the streaming potential120We can compute the streaming potential V resulting from the streaming current definedin Equation (4.37) by appealing to Equation (4.8), Ohm’s Law. The resulting expressionis= PEHvp77(4.38)where p 1/u is the electrical resistivity of the material. Notice that Equation (4.30)implies that streaming potential gradients are parallel to pressure gradients. In general, anisotropy in the phenomenological coefficients o and K2 (Equations (4.21a)and (4.21c)) and the quantities derived from them (p and H respectively) producesstreaming potential gradients and pressure gradients that are not parallel.Is>0— RBULKRSURFACE <WrSTREAMING POTENTIALChapter 4. ELECTRICAL PHENOMENA — THEORY 1214.7 Electrical phenomena relevant to this studyAlthough streaming potentials can develop in any porous system (e.g. Ahmad, 1964;Bogoslovsky and Ogilvy, 1972, 1973; Korpi and Bruyn, 1972; Hunter, 1981; Ishido andMizutani, 1981), streaming potentials are greatest where clay is present (Hunter, 1981,p. 17—21). There are two reasons for this: (1) substitution of lower-valence metalatoms (typically magnesium for aluminium) in the crystal structure of clay mineralscreates a large negative surface charge o, which results in a large negative potential;(2) the specific surface area of clay-rich sediments is large so that the volume fractionof pore water under the influence of double layers is greater than for other sediments.The sediments beneath Trapridge Glacier are not particularly clay rich, but do contain quantities of calcite, kaolinite, illite, and montmorillonite (Tavi Murray, personalcommunication). Pressure gradients parallel to the ice—bed interface are a naturalconsequence of the down-slope orientation of the glacier, so we can expect streamingpotentials beneath this glacier. We also expect negative potentials.Natural potentials have been used for many years for mapping groundwater flowand transients events that cause changes in subterranean pressure fields such as earthquakes (Mizutani and others, 1976) and thermonuclear explosions (Nourbehecht, 1963).Discussions of natural potential field studies concentrate on streaming potentials (e.g.Bogoslovsky and Ogilvy, 1973; Ishido and others, 1983; Schiavone and Quarto, 1984)and thermoelectric potentials (e.g. Corwin and Hoover, 1979; Corwin and others, 1981;Ernstson and Scherer, 1986). The same holds true for theoretical treatments (Nourbehecht, 1963; Fitterman, 1978, 1979a, 1979b, 1984; Morgan and others, 1989). We havenot located any references to field studies of diffusion potentials, but conditions beneathTrapridge Glacier suggest that such potentials may be of concern. In the summertime,fresh solute-free surface melt water can enter the subglacial system beneath TrapridgeGlacier, often on a diurnal cycle. Until dissolution processes mineralize this fresh water, concentration gradients will exist and diffusion potentials along with them. SinceChapter 4. ELECTRICAL PHENOMENA— THEORY 122we have a very poor understanding of how surface water reaches the bed (i.e. is it adistributed source, line source, or point source), we cannot gauge how these diffusionpotentials will influence natural potential measurements. Data presented in Chapter 6will show that diffusion potentials are a small component of the total natural potential.The paucity of ore-bearing rock in the moraines surrounding Trapridge Glaciersuggest that there are no significant ore body outcrops under the glacier and thatcorrosion potentials are unlikely to contribute significantly to the natural potentialsignal. Even if corrosion potentials exist, they are not likely to be time-varying andwill therefore not disrupt our measurements.We can expect telluric currents since the cold glacier ice presents no significantobstacle to electromagnetic radiation at frequencies up to several hundred kllz. Monitoring changes in the Earth’s magnetic field allows us to attempt compensation for telluric currents, although as stated above, telluric potentials are of the order lOmVkm’(Telford and others, 1990) and are probably insignificant. Thermoelectric potentialsare unlikely to exists beneath the glacier since the ice—bed interface in the experimentalsite is at a constant temperature: the pressure melting point (Clarke and Blake, 1991).In addition, thermoelectric potentials would not be time-varying.Chapter 5ELECTRICAL PHENOMENA METHOD5.1 IntroductionThe electrical phenomena experiments carried out under Trapridge Glacier measuredd.c. resistivity and natural potentials at the glacier bed. These measurements wereperformed using four-electrode apparatus at a number of locations and over a periodof several years. For these experiments, we used two different measurement systems.The fundamental difference between the two systems concerns the nature of the electrodes: the 1987 apparatus employed the same electrodes for both current injection andpotential measurement whereas the apparatus used in later years employed dedicatedelectrodes for the two tasks.For all our experiments, data were collected on Campbell Scientific CR10 dataloggers. These data loggers have eight differential analogue input channels (each differential channel can also be configured as two single-ended channels), eight digitalinput/output channels, and some other interface channels not relevant here. Peripheral equipment, such as d.c. resistivity equipment, can be controlled using the digitaloutput channels and the data recorded through the analogue input channels. The dataloggers are programmed to execute a series of instructions at periodic intervals; theseinstructions may initiate measurements, change the status of the digital ports, insertdelays, or process data.5.2 The 1987 apparatusThe 1987 experiment was our initial foray into making long-term measurements oftemporal variations in subglacial d.c. resistivity. We also wanted to investigate the123Chapter 5. ELECTRICAL PHENOMENA— METHOD 124spatial variation in d.c. resistivity, both laterally and with depth. Because the electrodeswe place at the glacier bed are fixed in position and are not recoverable, this lastobjective requires that we have many electrodes at our disposal, of which we can choosefour as appropriate.Thus, our d.c. resistivity apparatus requires three major components: (1) a switch-able, current-limited, and polarity-reversible high voltage supply for injecting currentinto the subglacial material, (2) a way to measure the induced potential field, and(3) a mechanism for selecting which electrodes are “active” at any given time. Sincethe equipment is battery powered, we must also have a method for turning off theequipment.5.2.1 Electrode configurationsTo fulfil these requirements, we designed and built an instrument that uses eight fourpole/double throw relays to select four of eight identical electrodes for use as a d.c.resistivity array. Figure 5.1 lists the eight electrode configurations available. Thedesired configuration is selected by setting three digital output lines on the data loggerso that they represent, in binary code, the configuration number. A digital logic circuitdecodes the three binary inputs into signals to energize the various relays.If the electrodes are spaced evenly and sequentially in a line, then configurationsP1 through P5 represent Wenner arrays stepping along a profile and configurationsSi through S3 represent a depth sounding centred at the middle of the line. Noticethat configuration Si duplicates configuration P3.5.2.2 Electrode designThe eight electrodes were simply 15 cm sections of standard half-inch copper pipe soldered to a length of single-conductor stranded wire leading to the surface. The electrodes were placed 0.5 m above the bottom of the borehole.Chapter 5. ELECTRICAL PHENOMENA— METHOD 125P1 I V V IP2 IVVIP3 IVVP4 I VV IP5 IVVISi IVVLS2 I VV IS31 VV IFig. 5.1: The electrode configurations available with the 1987 d.c. resistivity apparatus. The eight electrodes are numbered 1 through 8 across thetop, and the eight configurations are numbered P1 through P5 and Si throughS3. An “I” indicates that the electrode is used for current injection and aindicates that the electrode is used to measure voltage.Copper electrodes corrode and are therefore not eminently suitable for makingvoltage measurements, but we did not deem this a problem since the alternating d.c.resistivity measurement cycle removes pseudo-static natural potentials, of which electrode corrosion potentials are one component.5.2.3 Voltage measurementThe two electrodes chosen as voltage electrodes are simply routed to a differential inputchannel on the data logger. During the 1987 field season, the only difficulty encounteredwas the large electrode polarization voltage that develops when an electrode is usedto inject current. This potential decays steeply after the current is interrupted; if thisSWITCHINGCIRCUIT7//Iz0DC,LI..0C)1 2 3456ELECTRODE78Chapter 5. ELECTRICAL PHENOMENA — METHOD 126same electrode is used to make voltage measurements a short time later, the polarizationpotential decays significantly during the d.c. resistivity measurement cycle and rendersthis alternating cycle ineffective in cancelling natural potentials.We eliminated these problems by judiciously selecting the sequence in which theeight electrode configurations were used. Measurements were made in three groupsseparated by 10 mm “rest” intervals to allow electrode polarization potentials to decay to near-constant levels. The configurations used within each group avoided usingelectrodes as both voltage and current elements. The groups were arranged as follows:(1) configurations P1 and P4; (2) configurations P2 and P5; (3) configurations Si, S2,and S3.5.24 Current source and current measuringRecalling the theoretical development in Chapter 4, we realize that the high voltagesupply (HV circuit) and potential measurement sections (LV circuit) must be electrically isolated— in other words, the potential of the current electrodes must be ableto float relative to that of the voltage electrodes. This isolation is also necessary toprotect the data logger from the high voltages present in the current supply.In our apparatus, the high voltage is generated using a d.c. to d.c. voltage converter.Voltage converters operate by chopping the incoming voltage, thereby creating a square-wave voltage signal. This pseudo-sinusoidal voltage is fed into the primary coil of astep-up or step-down transformer. The resulting secondary coil voltage is then rectifiedand filtered to produce the output voltage of the converter. A high chopping frequency(typically 10—20 kllz) permits the use of small transformers and reduces the physicalsize of the converter. The transformer also serves to electrically isolate the input andoutput voltages. The converter used in our apparatus produces a 250 V output from a12 V input. This high voltage is fed through a coarse two-terminal current limiter. TheChapter 5. ELECTRICAL PHENOMENA — METHOD 127circuit for the limiter is shown in Figure 5.2. In typical operation, voltages of 70-100 Vare required to inject 10 mA of current into subglacial sediments.Fig. 5.2: The two-terminal current limiter placed in series with the highvoltage current supply. Current flowing through Rb and the 2.5 V Zener diodeensures that the base of the NPN transistor is held at 2.5 V relative to point X.Current also flows into the base of the transistor causing collector-emitter current, but the voltage drop across Re will limit this current to about 1.9/Re mA(1.9 V equals 2.5 V less the approximate base-emitter voltage of the transistor).One technical hurdle remains and that is the measurement of the current beingdelivered to the current electrodes. in the HV circuit, the current is easily measured bymonitoring the voltage drop across a small resistor, but this voltage must be conveyedto the data logger while maintaining the electrical isolation mentioned above. Weaccomplish this by feeding the voltage drop representing the current into a voltage tofrequency converter. The resulting frequency is used to drive an opto-isolator circuitwhose detector is in the LV circuit; the voltage drop is recovered using a phase-lockedloop (PLL).Whenever a set of d.c. resistivity readings is made, a measurement of the outputof the PLL is measured before the high voltage current supply is turned on. The temperature stability of the isolation circuit can be determined by monitoring the changesCURRENT IN TIP5ORe180R CURRENT OUTRb250k1/2W2.5V ZENERLM285-2.5Chapter 5. ELECTRICAL PHENOMENA— METHOD 128in this zero-current PLL output as the seasons change; the temperature stability isestimated at 100 ppm °C’. This temperature drift is negligible compared to the uncertainty introduced by electrode position errors.The HV current sense circuitry is powered by a separate 12 V battery, whereas theLV circuit and the high voltage supply are powered from the same 12 V battery as thedata logger.5.2.5 Additional controlThree additional digital outputs from the data logger (for a total of six) are used tocomplete control of the d.c. resistivity apparatus. One control line is used to turn theentire apparatus on and off. A second control line is used to turn on the high voltagesupply. A third control line is used to change the polarity of the high voltage output(this function is implemented using a single double pole/double throw (DPDT) relay).The control line usage is summarized in Table 5.1.Control Line Function1 Apparatus on/off2 High voltage on/off3 High voltage polarity4 Configuration select line 15 Configuration select line 26 Configuration select line 3Table 5.1: A summary of the control line function for the 1987 d.c. resistivity apparatus.Light-emitting diodes (LED) on the front panel of the box indicate when the highvoltage supply is active, what current polarity is selected, and which electrodes arebeing used as voltage and current elements. An analogue voltage meter connected tothe output of the current regulator is also mounted on the panel. These indicatorsfacilitate monitoring the operation of the apparatus.Chapter 5. ELECTRICAL PHENOMENA— METHOD 1295.. 6 Technical specificationsThe entire d.c. resistivity apparatus fits into a plastic box having dimensions 6 cm by12 cm by 18 cm. The precision of voltage measurements is determined by the datalogger (better than 1 part in 7500). The precision of current measurements is limitedby the opto-isolator circuit calibration; we estimate the error at ±2 %.5.3 The 1988 apparatusFor the 1988 experiments, we wished to make accurate measurements of both d.c. resistivity and natural potentials. Because the metal/metal salt electrodes suitable forvoltage measurement are unsuitable for current injection, this task led to a naturalseparation of the current injection and voltage measurement functions. To replace the1987 apparatus, we designed and built several pairs of current and potential multiplexers capable of controlling 8 and 32 electrodes respectively.The primary functional difference between these multiplexers and the 1987 apparatus is that the electrode configurations cannot be set directly by the data logger.Rather, the data logger commands the multiplexer to step through one of a numberof programmed electrode configuration sequences; these configuration sequences arestored in erasable programmable read-only memory (EPROM) chips within the multiplexers. The multiplexers were constructed using wire-wrapping techniques, withsoldered discrete components in the HV circuit.5.3.1 EPROM programsBoth types of multiplexer use 27C64 EPROMs to store the configuration programs.Each 27C64 EPROM has 13 address lines and stores 8192 (213) bytes (a byte contains8 bits of digital on/off codes). The upper 4 address lines are tied to a set of switchesinside the multiplexer and the lower 9 address lines are connected to a binary counter.The counter is reset to zero as the multiplexer is turned on and advances when digitalChapter 5. ELECTRICAL PHENOMENA— METHOD 130clock pulses are received from a digital output on the data logger. When the multiplexeris on, the EPROM chip is continuously enabled; the data appearing at the output of theEPROM chip always reflects the data stored at the location indexed by the compositeaddress (made up the switch settings and counter value). Thus, the switches can beused to manually choose one of sixteen multiplexer “programs” (each program is 512steps long) and the data logger is used to step through the program; at each step, theeight digital outputs of the EPROM are used to select the active electrode(s).EPROM programs are a compromise between flexibility and simplicity of control.Although the data logger cannot select specific electrodes directly, inteffigent programming of the EPROM allows most, if not all, desired functions to be programmed intothe multiplexer. The characteristics of the multiplexer can be changed easily by simply reprogramming the EPROM. Control of this type of multiplexer is very simple:for simple electrode selection operations, only two control signals are required: thefirst signal both controls power to the multiplexer and resets the program counter; thesecond signal is used to step through the program.5.3.2 The current multiplexerThe 1988 high voltage apparatus uses the same circuits for generating the high voltageand for limiting, measuring, and reversing the current, but instead of using a digitalnumber provided by the data logger to select the electrode configuration (see Table 5.1),the output from an EPROM is used to designate which two electrodes are to be usedas current electrodes. Table 5.2 lists the control signals used to operate the currentmultiplexer.Chapter 5. ELECTRICAL PHENOMENA — METHOD 131Control Line Function1 Apparatus on/off and clock reset2 Clock input3 High voltage on/off4 High voltage polarityTable 5.2: The control line functions for the 1988 current multiplexer.Three of the output lines from the EPROM select the electrode connected to thepositive terminal of the high voltage supply and another three output lines select theelectrode connected to the negative terminal; the two remaining output lines are notused. The multiplexer uses 16 single pole/single throw relays (8 for each terminalof the high voltage supply) to link the electrodes. This arrangement allows each ofthe 8 electrodes to be connected to either terminal of the high voltage supply. Themultiplexer can even connect both terminals of the high voltage supply to the sameelectrode, effectively short circuiting the supply; of course, the EPROM should nevercontain such a program step!Notice that the current multiplexer does not need a full complement of eight currentelectrodes; it can operate with as few as two electrodes. For the 1988 experiments, theEPROM programs were designed to connect the attached electrodes to the high voltagesupply in all possible combinations. Table 5.3 lists the 16 EPROM programs used inthe current multiplexer.5.3.3 The potential multiplexerThe operation of the potential multiplexer is very similar to the current multiplexer,but because its capacity has been increased to 32 voltage electrodes, two EPROMs areneeded to designate the electrode pair. One EPROM is used to select which voltageelectrode is connected to the positive side of the data logger’s differential input channeland the other selects which electrode is connected to the negative side of the channel.Only seven bits of each EPROM are used to select one of the 32 electrodes (32 = 2).Chapter 5. ELECTRICAL PHENOMENA — METHOD 132Program number Function0 8 3 electrodes1 9 4 electrodes2 10 5 electrodes3 11 6 electrodes4 12 7 electrodes5 13 8 electrodes6 14 source sweep7 15 sink sweepTable 5.3: The EPROM programs for the 1988 current multiplexer. Thetwo sweep programs connect electrode 0 (as current source or sink) with eachof the other electrodes in sequence.Voltage electrodes are connected to the data logger, not with relays, but withsolid-state analogue switching circuits; the analogue switches used in our potentialmultiplexer are the same as those used to perform multiplexing tasks inside the datalogger. Table 5.4 lists the 16 EPROM programs used in the potential multiplexer. Aswith the current multiplexer, each program selects all combinations of electrode pairsfrom a set voltage electrodes, but unlike the current multiplexer, internal short-circuitsettings are included (the two sweep programs connect the first electrode to each ofthe others in sequence). The short-circuit settings should give a zero voltage reading;they are used to check the proper operation of the multiplexer and to estimate thenoise level introduced by the multiplexer. Instrument noise levels observed under fieldconditions were less than 30 V.Note that proper operation of the data logger requires that it be grounded; this isto ensure that the potentials reaching the apparatus do not exceed the common moderanges of the multiplexer (+7.5 V) and data logger (+5 V). We accomplish this bygrounding the apparatus to one of the electrodes, picked at random; this electrode iscalled the grounding electrode.Table 5.5 lists the control functions required for operation of the potential multiplexer.Chapter 5. ELECTRICAL PHENOMENA — METHOD 133Program number Function0 4 electrodes1 5 electrodes2 6 electrodes3 7 electrodes4 8 electrodes5 9 electrodes610 electrodes7 12 electrodes8 14 electrodes9 16 electrodes10 20 electrodes11 24 electrodes12 28 electrodes13 32 electrodes14 positive sweep15 negative sweepTable 5.4: The EPROM programs for the 1988 potential multiplexer.The two sweep programs connect electrode 0 (as positive or negative) witheach of the other electrodes in sequence.Control Line Function1 Apparatus on/off and clock reset2 Clock inputTable 5.5: The control line functions for the 1988 potential multiplexer.5.3.4 Electrode designThe current electrodes used in 1988 were the same copper pipe sections used in 1987.The potential electrodes were of a different design.As discussed in section 4.2, there are a number of reasons relating to electricalnoise for choosing metal/metal salt electrodes over metallic electrodes for voltage measurements. In a laboratory setting, inert metals such as gold and platinum may beused as electrodes, but fiscal restraint is required in a field setting where the electrodesare not recovered. The standard-issue electrode in geophysical exploration is the copper/copper sulphate porous pot electrode; a shallow porous ceramic pot is ifiled withChapter 5. ELECTRICAL PHENOMENA — METHOD 134copper sulphate crystals, some water, and a copper electrode. The pot is allowed toage for a period of days. During this time, the dissolved copper sulphate approachesequilibrium with the excess copper sulphate crystals and the copper electrode. In use,the outer surface of the pot is placed in contact with the ground and the potentialmeasuring instrument is connected to the copper electrode.Petiau and Dupis (1980) investigated the long-term stability of various metal/metal salt electrodes for making telluric measurements. In measuring the noise levelsproduced by various electrodes over a frequency range of 0.001—100 Hz, they foundnoise levels of 3 V at 0.01 Hz for Cu—CuSO4 electrodes. Ag—AgC1 and Pb—PbC12electrodes exhibited slightly lower noise levels whereas Cd—CdC12,steel, brass, andgraphite electrodes showed higher noise levels.Figure 5.3 shows the variation on the classic Cu—CuSO4porous pot design that wasused for our experiments. Since the electrode is to be immersed in the water presentat the bottom of every borehole, the porous pot must be sealed. We contracted with aVancouver potter to furnish a number of small porous vase-like pots (the porosity of thevessel is controlled by the kiln temperature). After coating the neck of each pot witha resin epoxy, we fed in a small piece of copper tubing soldered to the electrode lead.The pot was then completely filled with copper sulphate crystals and water, followedby an annular paper wad and a final plug of epoxy. Since the pot gets saturated withwater when being ifiled, the initial hydrophobic epoxy coating allows the epoxy plugto bond well to the pot.Chapter 5. ELECTRICAL PHENOMENA — METHOD 135COPPER ELECTRODE-COPPER SULPHATEPOROUSCERAMIC POTII1 cmFig. 5.3: A cutaway view of the Cu—CuSO4porous pot electrode used asthe potential electrode in the 1988 and later experiments.Data will be presented in Chapter 6 showing that our Cu—CuSO4electrodes introduce a voltage offset level of about 1 mV between the fluid and the electrode lead. Wehave no measure of the frequency-dependent noise potential of these electrodes.5.4 The 1989 apparatusIn 1989, the 1988 multiplexer designs were fabricated as printed circuit boards. Thiswas done to increase the reliability of the circuits, especially in cold winter operation.Slight modifications were made to the 1988 designs; these included: (1) rationalization of the logic circuits that translate the EPROM output into electrode selectionsignals, (2) using a second d.c. to d.c. converter to provide an isolated 12 V powersupply for the HV circuit, and (3) modification of the EPROM addressing scheme sothat larger-capacity EPROMs could be used in the multiplexers (27C128 and 27C256chips). Larger EPROMs allow for a greater number of programs or longer programs.The 1989 apparatus was employed during the 1989 and 1990 field seasons usingthe 1988 EPROM programs.ELECTRODE LEADChapter 6ELECTRICAL PHENOMENA DATA ANALYSIS6.1 IntroductionSome of the first glaciological applications of electrical resistivity techniques includedlocating ice-cored moraines (østrem, 1959, 1964) and inferring temperature and density distributions (Hochstein, 1967), but most researchers have used electrical resistivity sounding to determine glacier thickness (e.g. Röthlisberger and Vögtli, 1967;Vögtli, 1967; Fisch and others, 1977). Röthlisberger (1967) gives a good overview ofsuch work.The resistivity of glacier ice, as determined by the aforementioned workers, is inthe M1 m range, whereas the resistivity of water-saturated sediments and high porosityrocks is in the k m range (e.g. Telford and others, 1990). Assuming that the glaciersole is made of similar water-saturated material, we can expect a rather large resistivitycontrast between the ice and the bed. Electrical resistivity techniques are well suited fordetermining the thickness of a layer having a resistivity much larger than the materialbelow, but little information about the properties of the underlying material can beobtained.To study the material underlying the glacier, it is desirable to skirt the resistiveice overburden and to place the electrodes near the ice—bed interface. As far as weknow, Haeberli and Fisch (1984) and Brand and others (1987) have published the onlyaccounts of such experiments. Both reports discuss the spatial variation of subglacialelectrical resistivity; Brand and others also made two sets of measurements separatedby a period of two days and noted a temporal variation in electrical resistivity.136Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 137In Chapter 4 we discussed the importance of pore water in controlling the electrical properties of the glacier bed. Both streaming potential and electrical resistivitymeasurements made over time should provide clues concerning the development andevolution of the subglacial drainage system. The importance of this drainage systemin relation to glacier surging was discussed in Chapter 1.Glaciologists investigating subglacial processes are faced with the problem of maximizing the amount of information they obtain from each borehole within the constraintsof available resources. The Trapridge Glacier research group is lucky in that large numbers of boreholes are drilled every summer (the glacier is relatively thin and our hotwater drill is efficient), but fiscal constraints prevent us from instrumenting each borehole as we would like. For example, a network of 50 or more pressure sensors wouldbe unprecedented in glaciological research and would certainly provide invaluable datarelating to the nature and evolution of subglacial drainage systems, but the sensor costis too great. One of our aspirations for the subglacial streaming potential experimentsis that they will prove an inexpensive method for expanding the coverage of a modestpressure sensor network: the cost to instrument a borehole with a pressure sensor ismore than twenty times the cost for a potential electrode (‘.$15).The data presented in this chapter represent the first electrical resistivity measurements made beneath a surge-type glacier and the first streaming potential measurements made beneath any glacier. This work also represents the first attempt to makedetailed long-term measurements of subglacial electrical phenomena.6.2 Predicted forcing/response relationshipsBased on arguments presented in Chapter 4, we can speculate on a number of electricalresponses to hydraulic forcing. These forcing phenomena can be divided into diurnaland episodic events.Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 1386.2.1 Diurnal phenomenaDuring the summer, given appropriate meteorological conditions, there is a diurnalmelting of surface snow, firn, and ice. Solar noon, the time of maximum solar energyinput, occurs at 21:2OhUCT (Universal Coordinated Time) because the longitude ofTrapridge Glacier is 140.3°W, but summer daylight savings time and unruly time zonesconspire to place chronological noon at 19:OOhUCT. This means that solar noon occursat 14:20 h local time. Peak melt, then, occurs sometime in the afternoon.Surface meitwater can percolate down to the bed of the glacier, probably throughdeep crevasses and englacial channels (e.g. Seaberg and others, 1988; Hooke and others, 1988); this water should reach the bed in the afternoon or evening. Surface meltwater from glaciers is quite resistive; since it originates as precipitation, it has almost nodissolved minerals. In contrast, water that has spent time in contact with minerals atthe glacier bed has a considerable dissolved ion content and will be relatively conductive. Therefore, we expect the resistivity of the fluid at the ice—bed interface to increasein the afternoon as the resistive meitwater reaches the bed, and then to decrease againduring the night as this meltwater mineralizes or mixes with conductive water deeper inthe subglacial sediments. These changes in fluid resistivity may affect measurements ofapparent resistivity. The input of meitwater will also cause the concentration of chargecarriers to decrease and we might expect a corresponding increase in ( potential. Thiscould result in larger streaming potentials during the afternoon and evening.Evidence presented in Chapter 3 suggests that deformation rates can have diurnalfluctuations. In section 4.4, we discussed the dependence of electrical resistivity onporosity. Since porosity is affected by deformation, we might expect diurnal fluctuationsin electrical resistivity.The diurnal input of water can also cause diurnal changes in the subglacial water pressure, which might change the lateral pressure gradients. The largest pressureChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 139gradient changes will occur at the margins of unconnected zones in the bed (see section 2.5.2); the water pressure in the connected zone will change quickly whereas thepressure in the unconnected zone will not. A large pressure gradient can result, with acorresponding increase in streaming potential.6..2 Episodic phenomenaIn addition to diurnal cycling, pressure records from beneath Trapridge Glacier sometimes exhibit large sudden increases — and less often, sudden decreases — in subglacialwater pressure. The pressure change resulting from these events typically decays exponentially over a period of a day or so. We will show examples of such pressure pulsesbelow.When we are drilling a borehole, we sometimes record a pressure pulse in neighboring holes when the drill reaches the glacier bed. During drilling, the borehole isusually completely filled with water. If the borehole reaches a connected zone at thebed, the water usually drains rapidly into the subglacial hydraulic system, causing arise in local water pressure. If the borehole reaches an unconnected zone, the watercolumn in the borehole drains much more slowly into the glacier bed and pressure disturbances are small or non-existent. In either case, rapid changes in lateral pressuregradient can develop with corresponding sudden changes in streaming potential. Wehave also observed naturally-occuring pressure pulses that, depending on the distribution of subglacial pressure gradients, could also cause streaming potential variations.6.3 AssumptionsIn analyzing the results from all our electrical experiments, we have made two primaryassumptions: (1) the resistivity of the glacier ice is very high compared to that of thesediments; (2) although each electrode is placed about 0.5 m above the glacier bed inChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 140order to protect it from being torn apart, the electrode is effectively at the ice—bedinterface.The resistivity of the glacier ice can be tested by making d.c. resistivity measurements in truncated boreholes that terminate within the glacier. This was done in 1987.Four holes were drilled to a depth of 34 m and electrodes were installed. After waitingfor the boreholes to freeze shut (so as to eliminate waterborne conduction paths), theresistivity of the glacier ice was measured as 2.0 + 0.5 x 1010 f m. This value comparesfavourably with laboratory measurements on clean, cold glacier ice by Cagniard (1959)and østrem (1967) and is an order of magnitude lower than the values obtained byQueille-Lefèvre and others (1959). The resistivity of ice varies over several orders ofmagnitude depending on its temperature and purity, so our value of resistivity is in theexpected range.Close to the ice-bed interface, the resistivity of the ice might decrease because ofenglacial debris, increasing temperature, and/or increasing water content; we have notmade measurements of ice resistivity variations with depth, but since the bed beneathour study site is at the pressure melting point (Clarke and Blake, 1991), the ice thereis melting continuously and it is unlikely that a high concentration of englacial debrisexists. We will assume that the electrical resistivity contrast between the glacier and itsbed is large enough that the glacier can be treated as an infinitely resistive half-space.Our assumption concerning effective position of the electrode also requires thatthe ice be resistive compared to the basal water. If this is the case, then the waterin the borehole forms an extension to the electrode that reaches the glacier bed at apoint directly beneath the electrode. The borehole itself freezes shut overnight, sealingthe electrode wire into a chamber having only one good electrical connnection to itssurroundings: the bottom of the borehole. Judging from our experiences with redrillingboreholes after they have frozen shut, the first parts of the borehole to freeze shut are10—20 m from the surface — this is where the glacier is coldest. Because the bottomChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 141of the glacier is at the pressure melting point, we expect that the borehole around theelectrode itself does not freeze shut during the summer field season. Creep closure ofthe borehole will take several months to close the ice in around the electrode.6.4 The 198T experimentsIn the summer of 1987, we made our first attempts to measure subglacial electricalresistivity using the apparatus described in section 5.2. We had several objectives forthis first field season: (1) to test the apparatus, (2) to test the practicability of makingsubglacial measurements of electrical resistivity, and (3) to determine if our boreholesare drilled vertically. This was also the field season during which we discovered time-varying natural potentials.64.1 Forefield operational testOn 25 July 1987, we placed a rectilinear eight-electrode array across the terminus ofTrapridge Glacier, about 10 m in front of the ice. The exposed sediments in this areawere completely water saturated by meitwater seeping off the glacier and by ongoingprecipitation; there was standing water on the surface. Since 1987, the sediments atthe test site have been overridden by the advancing glacier.The electrodes, 7 cm long zinc-plated wood screws, were spaced at 4 m intervals.Figure 6.1 shows the d.c. resistivity values obtained for the seven distinct electrodeconfigurations (see Fig. 5.1) in “pseudo-section” form. The horizontal axis indicatesthe position of the midpoint of the array and the vertical axis represents the pseudodepth calculated using the method described in section 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 142ORDNATE (m)-8 -4 Q +4 +8I I I1940 1660 1730 1660 1600 L93 iI-2020 3.472260 5.00Fig. 6.1: The d.c. resistivity pseudo-section through forefield sediments.The values of resistivity are in IZ m.A cursory interpretation of this pseudo-section reveals a fairly uniform, moderatelyconductive surface underlain by progressively more resistive material. The reportedresistivity values are comparable to reported values for subglacial sediment (Haeberliand Fisch, 1984). The apparatus was functioning as expected.64.2 Experimental designEight boreholes, numbered 1 through 8 were drilled through the glacier at a spacingof 4m in a line parallel to the glacier flow direction. A copper electrode was placedin each of these holes. Two duplicate holes, numbered 2B and 7B were drilled next tothe 2nd and 7th boreholes respectively; electrodes were also placed in these holes. Wehad no borehole inclinometer with which to measure the position of the electrodes.The experiment ran from 20 July 1987 to 14 August 1987. During the course ofthe experiment, only eight of the ten available electrodes were iii use at any moment,but substitutions of electrode 2 for 2B and electrode 7 for 7B were made occasionally.The duplicate electrodes were drilled to help determine whether the boreholes wereplumb — our rationale was that if the apparent resistivity reported by an electrodeChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 143configuration involving either the 2nd or 7th electrode positions was independent ofwhich duplicate electrode was used, then the electrodes must be in the same position.Unfortunately, it became clear during the data analysis that the boreholes were farfrom plumb and that we had little idea of where the electrodes were located at theglacier bed — the duplicate electrodes served no purpose other than to confuse thedata records.Figure 6.2 shows the apparent resistivity data record for electrode configuration P1(see Fig. 5.1) calculated according to Equation (4.14) assuming that the relative electrode positions at the ice—bed interface are the same as those at the surface. Theresistivity values are lower than those found in the forefield test, but they are notunreasonable for wet sediment. The reader will notice large sudden jumps in electrical resistivity from time to time; these jumps result from switching between duplicateelectrodes as indicated on the figure. Recall that configuration P1 selects electrodepositions 1, 2, 3 and 4; the jumps occur whenever electrode 2B is substituted for electrode 2. It is clear that electrode 2 is nowhere near electrode 2B; the boreholes are notplumb.-2,7 2,7B2B,7.,%_.._ —F,-—2B,7Biii IE10AUG11111 I I IllillIllill20 30 5JULY 1987Fig. 6.2: The apparent resistivity record for electrode configuration P1.The gaps and sudden jumps are explained in the text.300E0Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 144The error introduced by an imperfect knowledge of electrode location is entirelyone of scaling; the error is in computing the geometric factor G. For this reason, wecannot determine absolute values of apparent resistivity (or attempt to recover trueresistivity), but we can discuss temporal variations.64.3 Diurnal cycling of d.c. resistivityFigure 6.2 shows a clear diurnal fluctuation. The resistivity readings increase in theafternoon and decrease around midnight. These oscillations correlate well with thepredicted meitwater-driven cycling discussed in section 6.2.1, but diurnal fluctuationsPap resulting from meitwater input are unlikely because the d.c. resistivity arrayprobably has a pseudo-depth on the order of 1 m (for an evenly-spaced linear electrodearray with 4 m spacing, the pseudo-depth for the P1 configuration is 1.93 m) and istherefore relatively insensitive to the fluid conductivity at the ice—bed interface. Insection 3.3.1, we showed that the penetration of water down into the basal material isa very slow process; it is possible that we are observing the diurnal migration of resistivewater deep into subglacial sediments, but such rapid water transport runs counter toour understanding of the hydraulic permeability of the bed.Another mechanism which could produce diurnal fluctuations in resistivity is deformation. At the conclusion of Chapter 3, we showed that diurnal cycling of deformation occurs beneath Trapridge Glacier (the 1989 data showed this best). Recall thatthe deformation rates tended to be positive in the early morning and negative in theearly afternoon. Although no significant pressure fluctuations were observed duringthe deformation experiments, we postulated that pressure fluctuations must have beenoccurring nearby; increases in subglacial pressure elsewhere can cause translocation ofbasal shear stress to other areas, such as our experimental site. We speculated thatthe observed down-flow deformation (indicated by positive strain rate) occurs in response to such an increase in shear stress. We also postulated that negative strainChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 145rates represent a relaxation or consolidation phenomenon that occurs when the shearstress drops. Murray (1990) discusses the increase in porosity and permeability associated with dilation — an increase in porosity should also cause an decrease in resistivity(Equation 4.9). If we assume that positive deformation is driven by glacier flow andnegative deformation is a relaxation process, then we would expect lower resistivitywhen deformation occurs and higher resistivity when relaxation occurs.644 Geometrical correctionsWe attempted to determine the location of the 1987 electrodes using iterative forward-modelling techniques. Using a model of a conductive layer between two half-spaces (theice above, with a known high resistivity, and the basement below), we tried to fit theapparent resistivity observed by each electrode configuration to data from one set ofobservations. For a given set of eight electrodes, the model had 27 degrees of freedom(resistivity of the layer and basement; thickness of the layer; and the three coordinates ofeach electrode). We tried to solve the problem using non-linear minimization techniques(Press and others, 1986, p. 289) and Monte-Carlo simulations, but all solutions obtainedwere unable to obtain satisfactory predictions of observed apparent resistivity values.We conclude that the resistivity structure of the subglacial material is more complicated than a three-layer model can realistically depict. Morphological changes resultingin alteration of conduction paths and inhomogeneity of the sediments are two possibleexplanations for our failure to divine the electrode positions.64.5 Polarity reversalsWe have evidence for time-varying changes in electrical conduction paths. Figure 6.3shows the apparent resistivity record for configuration P2. As in Figure 6.2, the verticalscale is incorrect because we go not know the correct value of G. The substitution ofduplicate electrodes is indicated on the figure.Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 146+150c-2B,7B-AJ2,7 ,7B—150— i i i i i20 30 5 10JULY 1987 AUGFig. 6.3: The apparent resistivity record for electrode configuration P2.Notice the diurnal changes in resistivity polarity in early August.In early August, polarity reversals are evident. We are confident that these reversals are not caused by equipment malfunction because (1) the transitions from positiveto negative resistivity are smooth and (2) no modifications to the apparatus or electrode array were made during this time. For a homogeneous, isotropic medium, polarityreversals are impossible.Figure 6.4 shows the plan view of a cross-shaped electrode configuration that willproduce polarity reversals for slight changes in the path taken by the electric current.If the current flows as indicated by the solid line, a positive resistivity will be recordedsince, as measured along the current flow line, the positive voltage electrode is closerto the positive current electrode; the reverse holds true if the current flows along thedashed line. Unfortunately, we do not have any information on absolute electrodeposition, so we cannot describe precisely how the water flow paths are changing. Inall three years of data collection, this apparent resistivity record, as well as a similarrecord from configuration P4, are the only observed instances of polarity reversal.Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 147.+1 .-I.-vFig. 6.4: An electrode configuration that can produce polarity reversalsin apparent resistivity. If the current flows along the solid line, a positivepotential difference will be recorded; if the current flows along the dashedline, the potential difference will be negative.6.4.6 Streaming potentialsBetween 25 July 1987 and 27 July 1987, the d.c. resistivity apparatus was removedfrom the glacier for testing in the forefield; this hiatus is evident in Figures 6.2 and 6.3.The longer data gap beginning on 3 August 1987 was caused by a malfunction of theapparatus — a loose wire prevented the high voltage current supply from activating.Five days passed before the fault was noticed, but during this time, the apparatuscontinued making voltage measurements. When these data were plotted, we noticedthat rather large temporal variations in voltage were occurring.Figure 6.5 shows the potential record for configuration P2 during this period. Notethat variations on several time scales (our sampling interval is one hour) and polaritychanges are evident.Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 14825>0E25—50vV///1v/wI I I I I I I4 5 6 .7 9AUG 1987Fig. 6.5: The natural potential record from configuration P2 during thetime the high voltage supply failed to operate.Because the electrodes are pieces of copper metal, it is possible that these time-varying potentials were produced by electrode corrosion, but because the electrodes hadalready been in place for more than 14 days, we expect that corrosion potentials wouldhave stabilized somewhat by the time these measurements were made. We decided tostudy these natural potential variations in more detail the following year.64.7 Recapping the 1987 field seasonThe results of the 1987 field season demonstrated that measurements of subglacial electrical phenomena are possible. We realized that the subglacial sediments are electricallyinhomogeneous and that our boreholes are not plumb. These realizations, together withour observations of time-varying natural potentials, suggested several modifications toour experimental design: (1) the use of dedicated current and potential electrodes,(2) the use of non-rectilinear electrode configurations, and (3) the use of a boreholeinclinometer to determine electrode location.l-t8Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 1496.5 Dedicated electrode arraysIn 1988 and 1989, we installed electrode arrays having separate electrodes for makingpotential measurements and for injecting current. Different apparatus were also used(see Chapter 5).Figure 6.6 shows the relative locations of the four electrode arrays installed during1988 and 1989. Note that the group of electrodes marked 88DCO1B are 1989 additionsto the 88DC01 array; between 1988 (when the positions indicated for the 88DC01electrodes were determined) and 1989, the 88DC01 electrode array had moved down-glacier by 33m so in 1989, the 88DC01 and 88DCO1B arrays were coincident. The flowdirection of the glacier is marked, as are the surface locations of all the electrodes inthe arrays. Notice that each array has a characteristic “L” shape.This b-shaped pattern is designed to allow examination of anisotropic characteristics of the glacier bed. Figure 6.7 shows a detail of the electrode pattern. Currentelectrodes (marked with a circle) are interspersed between potential electrodes (markedwith a cross) in such a way that electrical resistivity and natural potentials can be measured along the perpendicular arms of the array.Electrode arrays may contain a greater or lesser number of electrodes dependingon logistical considerations at the time the boreholes were drilled.6.6 The 1988 ExperimentsThe apparatus used in 1988 employed separate potential and current electrodes (seesection 5.3). By separating the electrode functions, we could make measurements ofnatural potentials without worrying about interference from contemporaneous electricalresistivity measurements.In 1988, we had an inclinometer on loan from the U.S. Geological Survey for measuring the trajectory of boreholes. Unfortunately, this instrument was not as accurateas the UBC fluxgate inclinometer (discussed in Appendix A) used in 1989. BasedChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 15089DC02No’N89DC01 ; . 88DC02‘BBDCOlB88DC01 m 50Fig. 6.6: The relative locations of the four electrode arrays installed in1988 and 1989. The dots indicate the surface locations of the boreholes containing electrodes. The designations 88DC01, etc. indicate the year and ordinal number of the given electrode array.on comparisons of the multiple inclinometer measurements performed for each borehole, we estimate that the error on 1988 electrode locations is 1—1.5m; some electrodepositions have even larger error.6.6.1 Experimental designTwo separate arrays of electrodes were installed in 1988. Figures 6.8 and 6.9 showthe electrode distributions for the 88DC01 and 88DC02 arrays. Both the surface positions of the borehole collars and the subsurface location of the electrodes are indicated.Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 151++++— potential electrode.+ — current electrodeI I +0 mFig. 6.7: Electrode array template for the 1988 and 1989 electrode arrays. Current electrodes are marked with a circle, and potential electrodeswith a cross. The “L” shape is designed to allow observation of anisotropiccharacteristics of the glacier bed.The locations of other subglacial sensors (e.g. pressure sensors and deformation instruments) are also marked. Notice that borehole deflections have completely obliteratedthe intended spatial arrangement of the electrodes.Streaming potential and d.c. resistivity records were collected during the summerand through the 1988—89 winter; records were collected for all possible electrode combinations. Although there are periods where the apparatus malfunctioned, the data setis large. We will not show all the data collected from these two electrode arrays, butwill instead show selected records from the full data set.Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 152Fig. 6.8: The 88DC01 electrode array. Current electrodes are markedwith a circle and potential electrodes with a cross. The locations of othersubglacial sensors are marked with triangles. The heavy markings indicatethe subglacial locations of the electrodes and the light markings the surfacelocations.88H04/N88H0588H1 I88H0988H120r088H56I10m88H1 488H1 888H188H17Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 153—oE0)CDNCDCDFig. 6.9: The 88DC02 electrode array. Current electrodes are markedwith a circle and potential electrodes with a cross. The heavy markings indicate the subglacial locations of the electrodes and the light markings thesurface locations. Note that no inclinometry data is available for 881134.C)CD—oC)N2:N CD2: CDCDCDCDN N= CT)2:tO0N2:CDCDNNz CDCDChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 1546.6.2 Telluric noiseIn Chapter 4, we indicated that telluric currents were a source of natural potentials andthat these telluric potentials might be indistinguishable from streaming potentials. Inorder to gauge the effects of telluric current noise on our measurements, and possiblyto remove the noise should it prove severe, we installed a telluric monitoring stationat our field camp in the summer of 1988. The telluric monitoring station was locatedto the north of the glacier on a south facing slope and consisted of a three-componentfluxgate magnetometer on loan from Dr. T. Watanabe and a ground potential array(see Figure 3.1). The ground potential array consisted of three Cu—CuSO4electrodesarranged at the vertices of a right triangle; the electrodes were of the same type asthose used under the ice. It is likely that the ground potential array recorded streamingpotentials resulting from down-slope groundwater flow on the slope, but the telluricnoise we were most concerned with was that at higher frequencies: those above oursampling Nyquist frequency (the streaming potential sampling interval ranged from1—10 mm during the summer). Recordings of the three magnetic field components andtwo ground potential components were made at 2s intervals throughout the summer,yielding a voluminous data set (the telluric station was also installed in the summerof 1989 and another data set was collected for that summer).Figure 6.10 shows three panels of data. Panel (a) shows the variations of the threecomponents of the Earth’s magnetic field during a 2.5 d period. The x axis of themagnetometer points towards magnetic north (a compass bearing of about 27°) andthe z axis points downward. The vertical scale indicates only the deviations of themagnetic field components; it is not an absolute scale. Panel (b) shows the potentialsrecorded along the two perpendicular axes of the ground potential array over the sameperiod of time; as in panel (a), the vertical scale shows oniy relative fluctuations inpotential. The a axis points roughly towards magnetic north. Panel (c) shows thenatural potential recorded between three pairs of electrodes in the 88DC01 array. ForChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 155all three natural potential traces, electrode 881108 is used as a reference potential; thetop trace is the potential at 881106 (an electrode 93 m north of 881108 — this electrodeis not shown in Figure 6.8), the middle trace is the potential at 881103 (an electrode3.5 m to the north of electrode 881108), and the bottom trace is the potential at 881113(an electrode 6.3 m upstream of electrode 881108). Recall that our estimated error onelectrode position is 1—1.5 m. A minor geomagnetic storm is indicated by the arrow.We find significant noise correlated with magnetic field fluctuations only on electrode881106 (i.e. along the 93m baseline) and even then the noise is only a few millivolts.Electrodes 881103 and 881113 (electrodes much closer to the reference electrode 881108)show insignificant disturbance. During the storm, the telluric potential between 881106and 881108 amounts to about 7OmVkm’; we conclude that, for natural potentialmeasurements within the spread of a single electrode array, telluric potentials are nota concern.The ground potential records, particularly for the a axis, show bursts of highfrequency noise — we strongly suspect that these bursts are produced by the electric generator used at the base camp. Fortunately, the generator does not appear tointroduce noise at the subglacial electrode arrays.6.6.3 Potential errorFigure 6.10 gives us the first indications that streaming potentials vary with time. Before discussing these fluctuations, a brief discussion of measurement error is timely.In Chapter 5, we noted that the sum of multiplexer noise (section 5.3.3) and Cu—CuSO4 electrode noise (Petiau and Dupis, 1980) should be about 30 pV. One methodfor estimating the error introduced by the entire electrical apparatus (i.e. data logger,multiplexer, wires, and potential electrodes) is to measure the potential difference between two seasoned electrodes located in the same borehole. Figure 6.11 shows such arecord from hole 891115 (recorded in 1989). The potential difference is about 0± 1 mV;Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 156Fig. 6.10: The effect of telluric noise on natural potential measurements.The arrow indicates the location of a magnetic storm. (a) The three components of the Earth’s magnetic field measured at 2 s intervals. (b) Voltagesmeasured along the two axes of the ground potential array. (c) Natural potentials relative to 881108 at the indicated boreholes. Only hole 881106, 93 mremoved from 881108 do we find significant telluric noise. Measurements weremade on the 88DC01 array.the error on our natural potential measurements is, for whatever reason, much larger1000I10->E0-+15E—15abC26 27JULY 198828than expected. Nevertheless, a potential error of ±1 mV is tolerable.Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 157By seasoned electrodes, we mean electrodes that have been placed in a borehole forseveral days. Since the electrodes are assembled on the glacier surface just before beinglowered down the hole, the copper electrode and Cu504 solution are not initially atequilibrium. New potential electrodes typically show an initial potential of 50—200 mVrelative to neighboring electrodes; this potential decays over 1—3 days towards a stable,near-zero potential with time-varying fluctuations superimposed.+2.5E°—2.5Fig. 6.11: The potential difference recorded between two seasoned potential electrodes located in the same borehole.6.64 Potential gradientsEquation (4.38) shows that for a negative potential, the pressure potential and streaming potential gradients are of opposite sign; the other coefficients (p, e, H, and ) areall positive. Since we expect a negative potential under Trapridge Glacier (see section 4.6.4), the streaming potential component of the natural potential should increasewith decreasing pressure (i.e. as one moves down-flow).The lowest trace in Figure 6.lOc begins with a reasonably steady negative potentialof about —3 mV; electrode 881108, the down-glacier electrode, is at a higher potentialthan electrode 881113. Since it is reasonable to assume that subglacial water is flowing down-glacier, this natural potential polarity is consistent with that arising fromAUG’ SEPT1989OCT NOVChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 158a streaming current. Late on 26 July, the potential drops quickly by about 6 mV toestablish a new level.If this sudden change in natural potential is caused by a change in streaming potential, we would expect a corresponding increase in the down-glacier pressure gradient. Inthis case, neither of the two nearby pressure sensors noted any sudden changes in pressure, but the pressure sensor in unconnected hole 881135, 1.1 m upstream from 881113,changed from a decreasing pressure regime to an increasing one, so some change in thebasal pressure distribution may have occurred.6.6.5 Hole connectionsBecause the completion of a borehole introduces a sudden increase in subglacial waterpressure, whether or not the hole connects, we expect that the completion event shouldcause changes in pressure gradients and hence changes in streaming potential.Figure 6.12 shows an assemblage of pressure, natural potential, and apparent resistivity data for an 18 day period in late summer, 1988; the data were collected onelectrode array 88DC01. The “L” on the plan map at the right shows the surface location of the electrode array; the “+“ (881106) and “—“ (881105) indicate the locationsof the natural potential electrodes. The trapezoid indicates the d.c. resistivity array(the dots are the current electrodes, 881109 and 881117, and the ticks are the potentialelectrodes, 881112 and 881116). The hatched line indicates the approximate boundarybetween an unconnected basal patch and a connected patch; at the time the boreholeswere drilled, all holes north of (above) the line connected to the subglacial drainagesystem whereas all the holes south of (below) the line did not connect.Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSISE>EE50—40 —25101.91.62311111111111111311 10JULY AUGFig. 6.12: Pressure, natural potential, and apparent resistivity data fromelectrode array 88DC01. The plan map at the right shows the surface locationof the electrode array. The “+“ and “—“indicate the locations of the naturalpotential electrodes. The location of the pressure sensor is indicated by a“P”. The trapezoid indicates the d.c. resistivity array (the dots are the current electrodes and the ticks are the potential electrodes). The hatched lineindicates the boundary between an unconnected bed and a connected bed.159We can see a number of features in these records: (1) Beginning on 25 July, thereare a series of pressure pulses that are correlated with pulses in the natural potentialrecord. (2) Beginning on 4 August, a series of diurnal fluctuations in natural potentialand pressure are evident. (3) The apparent resistivity is slowly dropping over theperiod. The decrease in resistivity may indicate a slow migration of water down intosubglacial sediments.Figure 6.13 shows the earlier part of the record in more detail. Note that data froma different d.c. resistivity array is shown in this figure (the dots are the current electrodes, 881109 and 881117, and the ticks are the potential electrodes, 881113 and 881114).Two boreholes were drilled through the glacier during this four day period. Hole 881135was completed at 18:12 h on 24 July; this hole was unconnected and reached the bedFJPRESSURURE1111111 I I I liii IIIlull I 11111 Il I I I IChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 1601.1 m from potential electrode 881113, which formed part of the apparent resistivityarray. The completion time is indicated by the first arrow on the time scale.55 -________________________________________E4020>E15E 1.00c. 0.95I I I24 JULY 28Fig. 6.13: Pressure, natural potential, and apparent resistivity data froma drilling episode. The “C” indicates the location of the connected hole 88H36and the “U” indicates the location of the unconnected hole 881135. See Figure 6.12 for an explanation of other features on the plan map.The apparent resistivity of the bed was being measured at 5 mm intervals; between 18:05h and 18:10h, a 4% drop in apparent resistivity was noted. The apparentresistivity record does not recover after this drop; some permanent change in the bed,perhaps only in the vicinity of 881113 occurred.Since an unconnected borehole does not drain quickly, the bed experiences a significant over-pressure; in other words, the water pressure at the bottom of the boreholeis significantly greater than the overburden pressure of the ice. Many unconnectedholes begin to drain slowly shortly after completion; we conclude that a borehole over-pressure condition can establish a temporary basal drainage system near the borehole.In this instance, at 20:04h, the pressure sensor notes a modest increase in pressure andPRESSUREI I II I IRESISTIVITYChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 161the natural potential exhibits a small pulse; these events may be associated with theslow outward progression of a water pressure wave from the unconnected borehole.On 25 July, at the time indicated by the second arrow, hole 881136 connected to thebasal drainage system as it was being drilled. The hole connected at 13:58 h (accordingto the driller’s wristwatch); sharp increases in water pressure and natural potentialwere recorded at 14:OOh (according to the data logger clock). Within the resolution ofour clock synchronization, estimated at 2 mm, the response to the hole completion wasimmediate. Notice that for more than a day after the hole was completed, the naturalpotential response was almost identical to the pressure. Even small details at the peakof the pressure pulse are reproduced in the natural potential record.The quick response of the pressure sensor to the completion of 881136 indicates thatthe connected patch is very efficient at distributing pressure at the bed; we suspect thatthe increase in water pressure also appears at the reference (negative) electrode for thenatural potential measurement since this electrode was also placed in a connected hole.Because the positive electrode is in the unconnected patch, the pressure rise will createa large pressure gradient between the electrodes; the polarity and magnitude of thenatural potential fluctuations are consistent with a streaming current.Figure 6.14 shows the diurnal cycling period in more detail. We notice that apparent resistivity peaks in the early afternoon, just as in 1987. Natural potential peaksa few hours later at about midnight. The pressure sensor shows no clear diurnal signals, but records from several other pressure sensors suggest that when diurnal cyclingin subglacial pressure occurs, the peak pressures usually happen between 21:00 h andmidnight. Since the natural potential variations in Figure 6.14 are consistent withstreaming potentials arising from diurnal cycling of pressure in the connected zone, wespeculate that morphological changes in the bed may have disabled the hydraulic connection between the pressure sensor and the 88DC01 array. Note that in Figure 6.12,the pressure record changes from a sawtooth pattern to a jittery, bumpy pattern onChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 1621 August; perhaps this change indicates the hydraulic separation of the pressure sensorand electrode array.I I I I I IPOTENTIALI I I I I IRESISTIVITY45-E4025-> -E -10-E 1.7-c.1.6—i i i i3 10AUGFig. 6.14: Pressure, natural potential, and apparent resistivity data froma diurnal cycling episode. See Figure 6.12 for an explanation of the plan map.6.6.6 Overwintering eventsOverwintering of electrical resistivity arrays is a difficult proposition, primarily becausethe electrodes do not last through the winter. Failure of an apparent resistivity configuration is characterized by a month-long period where the current delivered by thehigh voltage supply slowly drops to zero and where asymmetry in the d.c. potentialcurve (see Fig. 4.5) increases dramatically; these characteristics probably result frombreakage of the wires leading to the electrodes. Figure 6.15 shows an overwinteringrecord from the 88DC01 array; on 11 August, the sampling interval was changed fromonce per hour to one daily reading at noon (the reduced sampling rate is necessitatedby limited data storage capacity for the 11 month-long overwintering period). NoticeChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 163that prior to 11 August, the diurnal cycling of apparent resistivity, with a minimumaround midnight and a maximum in the afternoon, is evident. The choice of noon asthe sampling time during the winter was unfortunate, since before 1 August, the noonreadings appear erratic. In retrospect, midnight readings, when the resistivity is at aminimum, would have been a better choice.20001500Ec 10005001988Fig. 6.15: A record showing the overwintering character of an apparentresistivity record and the eventual failure of the array. The arrow indicatesthe time at which the d.c. potential record begins to become asymmetrical.The failure of natural potential electrode configurations is characterized by theonset of rapid oscillations in the observed potential. Figure 6.16 shows the continuationof the record shown in the middle trace of Figure 6.lOc (electrodes 881103 and 881108)through the winter. The failure of the electrodes on 17 June, 1989 is evident. Otherelectrode pairs failed on other days, so this is not a failure of the apparatus. The flat,near-zero natural potential shown in Figure 6.16 is typical of the winter period; if theprimary source of natural potentials is streaming potentials, then this indicates lowpressure gradients in the winter.AUG SEPTChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 164>EFig. 6.16: A record showing the overwintering character of the naturalpotential and the eventual failure of the array.We do not understand why the current and potential electrodes appear to breakso easily. As noted in Chapter 3, pressure sensors regularly survive through the winterwithout having their cables cut. Pressure sensors use a four-conductor cable; perhapsthe single-conductor cable connecting an electrode to the surface is not strong enoughor stretchy enough to endure the internal stresses of the glacier.Overwintering data from the 88DC02 array is similar in character, but the batteriessupplying the apparatus expired in late October, 1988, so the data record is not as longas for array 88DC01.6.7 1989 Experimental designTwo new electrode arrays were installed in 1989. Some additional electrodes werealso added to the 88DC01 array (Fig. 6.6) for the purposes of some manipulationexperiments. Unfortunately, persistent equipment malfunctions rendered worthless alldata from the 89DC02 array and much of the data from the 89DC01 array. Figure 6.17shows a plan map of the experimental site.J’A’S’O’N’DIJ ‘F’M’A’M1988 1989Chapter 6. ELECTRICAL PHENOMENA— DATA ANALYSIS 165EFig. 6.17: The 89DC01 electrode array. Current electrodes are markedwith a circle and potential electrodes with a cross. The locations of othersubglacial sensors are marked with triangles. The heavy markings indicatethe subglacial locations of the electrodes and the light markings the surfacelocations. A pressure sensor is located in hole 891114 and a conductivity sensorin hole 891127.1 -oWI U,0)a,0)a,Chapter 6. ELECTRICAL PHENOMENA— DATA ANALYSIS 1666.7.1 Fall shutdownFor the 1988 through 1991 field seasons, we have noticed a dramatic pressure-relatedevent occurring near the end of July; within a few days of each other, pressure sensorsthat recorded magnificent diurnal fluctuations during the summer months suddenly indicate that subglacial pressures are quiescent. We interpret this cessation of activity asa shutdown of the subglacial hydraulic system. Diurnal pressure cycling often resumestemporarily in late August and early September, probably driven by water input frommelting early-winter snowfall, but we have not seen any evidence of these fluctuationsin our natural potential records.Figure 6.18a shows the pressure records for holes 891106 and 891114 for a 30 day period beginning on 29 July 1989. Hole 891106 is located 66 m upstream from hole 891114.The shutdown phenomenon is evident on 2 August, especially in 891106. Unfortunately,difficulties with the new 1989 apparatus, constructed on printed circuit boards, werenot properly resolved until 1 August, so we were unable to monitor electrical phenomena through this transition. Figure 6.18b shows the natural potential recorded betweenelectrodes 891115 and 891117. Between 1 August and 10 August, the natural potentialshows a weak diurnal signal; this period is followed by the characteristic winter regimediscussed in section 6.6.7. The data gap between 10 August and 11 August results fromrewiring and reprogramming activities relating to preparations for overwintering.It is frustrating not having electrical data prior to 1 August so that changes inelectrical phenomena as the pressure fluctuations ceased could have been monitored,but we note with surprise that large diurnal natural potential fluctuations are presentwhen the subglacial pressure is so calm. In the absence of pressure variations, we mustseek another source for streaming potential variations. Perhaps subglacial deformation creates pressure gradients and zones of differing streaming potential coefficient Cwithin the subglacial sediments and it is these effects we are observing. Deformation,if accompanied by sediment dilation, generates strong internal pressure gradients asChapter 6. ELECTRICAL PHENOMENA— DATA ANAL YSIS0Cl)AUG 1989Fig. 6.18: Subglacial conditions at the end of the summer. (a) Subglacialpressures recorded by pressure sensors 891106 and 891114. (b) The natural potential record at 891168 relative to 891114. (c) An apparent resistivity record.891125 and 891128 are the current electrodes; 891121 and 891168 are the potential electrodes. (d) A conductivity record from hole 891127.16770E0>503200E1200abCdI I I -I I I I I I I I I I I I I I I I I I I I I I I I I I6-4-27IlIlIllIllIllIll III1 10111111111 I20pore fluid is moved to fill the void spaces being created; these pressure gradients canChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 168generate streaming potentials.Figure 6.18c shows that the apparent resistivity is also experiencing dramatic diurnal variations. The apparent resistivity fluctuations are peaking several hours earlierthan we have seen in other records; sometimes the peak is before noon. The natural potentials, which we have previously observed peaking at about midnight are nowpeaking in early morning, about half a day out of phase with the apparent resistivity.Although no pressure fluctuations are evident, water supply is the only significant diurnal forcing acting on the glacier bed — if we do not observe pressure fluctuations inthe area of our experiment, it must mean that our study site has become disconnectedfrom the still-active subglacial drainage system.Figure 6.18d shows the record from a conductivity cell located in 891127. Notethat no diurnal signal in fluid conductivity is discernable either before or after 1 August. This is our first evidence that changes in subglacial fluid conductivity (whichcan generate diffusion potentials) are probably not responsible for the natural potential fluctuations we observe. The calibration of the conductivity cell probably has anincorrect scaling factor, so the rather low conductivity values reported should not betrusted.6.8 Manipulation experimentsIn Chapter 4, we discussed the dependence of streaming potentials on ( potentialsvalue. The potential can be manipulated in at least three ways: (1) changing pH;(2) changing ion concentration; (3) adding surfactant. The first two possibilities werediscussed in Chapter 4, but the last method begs attention; our discussion of surfactantsis based on Hunter (1981, Chapters 6 and 8).In the mineral processing industry, one of the common methods for separating pureminerals from their ores is flotation. The surfactants used in flotation processes arehighly active hydrocarbon chains having a charged terminal group. These chemicalsChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 169are added to the ore/mineral slurry and displace the adsorbed ions on the minerals.When enough hydrophobic surfactant molecules are attached, the mineral grains floatsto the surface of the slurry and can be skimmed off. Common mineral processingsurfactants include dodecylammonium chloride, sodium dodecyl sulphate, and sodiumdodecyl suiphonate; the first is used in basic solutions with negative ( potentials and thelast two are used in acidic solutions with positive potentials. For acid surfactants,the potential becomes more negative as the concentration of surfactant increases(Hunter, 1981, Fig. 8.3).The surfactants used in household cleansing operate on a similar principle, exceptthe surfactant molecule has one end that displaces adsorbed molecules and anotherthat is hydrophillic. When washing clothing or kitchenware, adsorbed particles of dirtare surrounded by surfactant molecules and carried off into the wash water. Althoughwe do not know the effect of household surfactants on potentials, we were certainthat some change in streaming potential would result.In early August, 1989, we executed two manipulation experiments at the old88DC01 site; a salt injection and a surfactant injection. We had also planned an alkaliinjection experiment using quicklime (CaO), but this experiment was not performed.CaO turns to CaOH, a powerful base; had the experiment been carried through, wewould have expected a more strongly negative streaming potential (see section 4.2.2).Since most, if not all, of the electrodes in the 88DC01 array were cut off, we installeda series of new electrodes in the array. Figure 6.19 shows the locations of the injectionelectrode array electrodes.Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 170Fig. 6.19: Location map of 88DCO1B electrode array. Electrode and sensor designations are the same as in Figures 6.8 and 6.9. Hole 891165 containsa conductivity sensor.Between 18:38 h and 19:00 h on 4 August, approximately 3.5 kg of table salt wasflushed down into the bottom of hole 891176. The quantity of salt is uncertain sincesome spillage occurred on the surface. The salt solution was delivered to the bottomof the borehole using a special apparatus constructed for that purpose. One end of asealed section of 6 inch plastic plumbing pipe was attached to the pressurized water linefrom the hot water drill and the other end to a section of hose reaching to the bottom.By filling the pipe section with the salt solution and starting the pump, the solutionwas pumped to the bottom.89H6589H67I0 10mChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 171Five days later, between 21:26h and 22:17h on 9 August, approximately 4kg ofdishwasher detergent, also dissolved in hot water, was injected into the bottom of hole891182. Again, the quantity of detergent injected is uncertain because of spillage. Dishwasher detergent was chosen because it contains a high concentration of surfactants;no foaming agents, such as those used in hand dish-washing detergents, are used. Thedetergent used in our injection contained metasilicates, tripolyphosphate, and sodiumcarbonate as active ingredients.Both injections were made into poorly connected holes. Ideally, the injectionswould have been made into connected holes so that the salt and detergent could travelfreely through the subglacial drainage system. In retrospect, we should have made theinjections at one of the other two main electrode arrays, but at the time the injectionswere made, we wanted to avoid contamination of these two arrays.Figure 6.20a shows one of the natural potential records from the 88DC01 array(the potential of 891172 relative to 891161). The injection times for the salt andfactant are indicated. Unfortunately, an error in collecting information from the datalogger resulted in the ioss of data between 4 August at 19:29 h and 5 August at 17:40 h.Nevertheless, no response to either the salt or surfactant injections is evident. The sudden rise in potential on 9 August occurs at 17:39h, 4h before the surfactant injection,but the rise may have been associated with the driffing of 891182 between 16:03 h and18:05 h. We note that electrode 891172, the down-glacier electrode, is at a potentialhigher than that of electrode 891161. Once more, this is consistent with a negativedown-glacier pressure potential gradient and a negative < potential.Chapter 6. ELECTRICAL PHENOMENA— DATA ANALYSIS 17220>10E0______________________________— I I I I I I I I I I3000LI)250________— I I I I I I I—5AUG 1989Fig. 6.20: (a) A sample natural potential record from the manipulationexperiments (891172 relative to 891161). The increase in apparent noise levellate on 3 August is caused by setting the data logger to a wider input range;the digital resolution has dropped to 0.33 mV from 33 V. (b) The subglacialconductivity cell record from array 88DCO1B.Figure 6.20b shows the conductivity record from 891165 (as in Figure 6.18d, thescaling of the fluid conductivity is uncertain). Some ambiguous diurnal fluctuations arevisible, with maxima around noon, but the vertical scale has been expanded so muchso that the resolution limit of the data logger is visible; these changes are small and donot correlate with either injection.Although electrode array 88DCO1B contains current electrodes, the HV supply(1988 vintage) did not function properly, so no apparent resistivity data are availablefor the injection experiments.aSALT1bI I10Chapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 173We belive that the absence of any observable response to our manipulations reflectspoor hydraulic conductivity in the bed rather than a failure of the electrical phenomenato respond— we believe that the added chemicals did not reach the electrodes duringthe course of our measurements.6.9 ConclusionsWe have demonstrated the practicability of long-term monitoring of subglacial electricalmeasurements. Our measurements have revealed diurnal and episodic fluctuations inelectrical properties of the glacier bed; we have been able to correlate these events withchanges in other subglacial processes.6.9.1 Apparent resistivityApparent resistivity measurements indicate that the glacier bed is electrically inhomogeneous; this inhomogeneity is not on the scale of particles in the bed, but on a scale ofat least several metres. We have also observed time-varying changes in electrical flowpaths within the bed.We are not confident about the values of apparent resistivity calculated from ourelectrode arrays. Even in 1989, when we used a relatively accurate inclinometer to locate the electrodes, we still computed unreasonably large or small (sometimes even negative) apparent resistivities. Apparent resistivity records seem to fall into three classes:(1) 1000 <Pap <2000gm, (2) Pap > 10000 tim, and (3) —200 <Pap < 200am. Twoor more electrode configurations encompassing the same subglacial sediments often indicated very different values of Pap — we suspect that electrode positioning error andbasal inhomogeneity are both responsible for these variations.Because the terms of Equation (4.11) are linear, any fractional error in electrodeseparation will generate a corresponding fractional error in Pap. In 1988, the positioningerror was of the same order of magnitude as the electrode spacing, so errors in Pap wereChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 174extreme. In 1989, errors of 10—30% are expected. We estimate, with much reservation,that the resistivity of the sediments beneath Trapridge Glacier lies in the range of1000—2000 f m. This range is consistent with some of our subglacial records, withthe forefield measurements in 1987, and with measurements by others (Haeberli andFisch, 1984; Brand and others, 1987).Brand and others (1987) installed a rectilinear electrical resistivity array of eightelectrodes on Storglaciären, a non-surge-type glacier in northern Sweden. The surfacespacings they used for their linear electrode array were similar to those used for ourexperiments. They used a half-Schlumberger electrode arrangement where the twopotential electrodes were placed close together in holes separated by 1 m at the surface.One current electrode was placed in the lateral moraine of the glacier, 300 m from theother electrodes; the other current electrode was moved between the remaining six holes.They report values of pp ranging between 3 m for small pseudo-depth and 1O f mfor large pseudo-depth; the lowest value of Pap is unreasonably low. We suspect that thetwo holes containing their potential electrodes were not separated by 1 m at the ice—bedinterface; if our experience with borehole deviation is any indication, their holes werenot plumb (Storglacliiren is 120 m thick, about 50% thicker than Trapridge Glacier).Nevertheless, there can be no doubt that the two data sets they collected two days apartdemonstrate that time-varying electrical resistivity occurs beneath Storglaciären.We believe that the most interesting aspect of our apparent resistivity measurements is the correlation between sediment deformation and electrical resistivity, andthe inferred correlation with subglacial water pressure.6.9.2 Streaming potentialsWe have shown that the natural potential variations observed beneath Trapridge Glacier are caused by streaming potentials and not telluric or diffusion potentials. TheChapter 6. ELECTRICAL PHENOMENA — DATA ANALYSIS 175polarity of the observed streaming potentials indicate that the potential of the subglacial sediments is negative, as expected. Unfortunately, we do not have enoughinformation to estimate the value of the potential for the sediments or how it mayvary with time.The hole connection data from 1988 indicate that streaming potentials can be usedto supplement pressure sensors, but it is clear that there are other subglacial processesthat cause changes in streaming potentials; sediment deformation may be one of these.Chapter 7CONCLUSIONS7.1 General commentsThis thesis documents the development of two novel methods for exploring the subglacial environment: measurements of subglacial deformation and electrical phenomena. Using these techniques, we have recorded several heretofore unobserved subglacialprocesses. Foremost among these is evidence for time-varying lateral transfer of normal and shear stress beneath the glacier; these transfers of stress are controlled by themechanical properties of the glacier bed and the behaviour of the subglacial hydraulicsystem. Such lateral variations in basal stress have not been properly addressed bycurrent subglacial rheology hypotheses; if the study of subglacial rheology is to providea greater understanding of glacier surge mechanisms, rather than localized stress—strainresponses, then these lateral variations must be considered.At first glance, subglacial deformation and subglacial electrical phenomena appearunrelated, but we have shown that both these phenomena may be interpreted independently, or in complementary fashion, to investigate the deformation and hydraulicproperties of the glacier substrate.7.1.1 Shear stress and normal stressGlaciologists have traditionally used mean values of stress calculated from the geometryand flow patterns of glaciers to describe the forces acting on the glacier bed, but wehave demonstrated that such calculations risk being inaccurate, at least for TrapridgeGlacier. This is of great concern since both of the variables fundamental to sedimentrheology — effective pressure and shear stress— are obtained from these calculations.176Chapter 7. CONCLUSIONS 177We caution those wishing to derive sediment rheologies based on in situ deformationexperiments to ensure that, in addition to measurements of subglacial water pressureor pore pressure, they have some method for monitoring the normal and shear stressesat the bed. In particular, we wish to emphasize the uncertainty in computing any measure of effective pressure (Equations (3.6), (3.8), or (3.10)) based solely on subglacialpressure measurements.As discussed in Chapter 3, we have noted subglacial water pressures in excess ofthe nominal flotation pressure in several unconnected holes. It is clear that the glacierdoes not accelerate upwards as a result of this apparent force imbalance, so the localoverburden pressure must exceed that suggested by the ice thickness (if it did not, thepressurized water would quickly leak out into the lower-pressure subglacial drainagesystem). Since the glacier is somewhat rigid on a diurnal time scale over spatial scalessmaller than its thickness, lateral transfer of normal loading on the bed is possible; theload can be greater at some points and less at others, so long as the mean loading is equalto the theoretical overburden pressure. In addition, the shear stress on the bed is notnecessarily proportional to the normal loading on the bed, as implied by Equation (3.4).A sticky patch (with a high shear stress) can have a low normal load and a slippery patch(with a low shear stress) can have a high normal load; again, the stability requirement isonly that the mean shear stress averaged over the glacier sole be equal to that computedby Equation (3.4). We expect that slippery patches are associated with connected zonesand sticky patches are associated with unconnected zones. We have, however, observedoverpressure situations in connected zones (on occasion, we encounter artesian outflowconditions when completing a connected borehole).Evidence for uneven time-varying distribution of subglacial shear stress is foundin our data for 1989. In the weeks prior to our experiments, pressure sensors installedat various locations in the study site recorded dramatic diurnal fluctuations, but on1 August, four days prior to the beginning of the deformation experiment, all theseChapter 7. CONCLUSIONS 178pressure sensors began reporting quiescent pressure levels. As shown in Figures 3.9and 6.18, other subglacial sensors indicated that the subglacial environment was anything but quiescent; large fluctuations in deformation rate, apparent resistivity, andnatural potential were observed. Since meitwater is the only significant diurnal forceacting on the subglacial environment, we strongly suspect that diurnal cycling of subglacial water pressure continued after 1 August, but that none of our pressure sensorswas located where these pressure cycles could be observed. Increases in water pressureelsewhere in the subglacial environment would reduce the shear stress in those areas;the principle of constant mean shear stress then requires that the shear stress increaseelsewhere. If the region containing our subglacial experiments experienced such a risein shear stress, we could expect to see diurnal fluctuations in strain rate, streamingpotentials and resistivity; corresponding fluctuations in local subglacial water pressure(or computed effective pressure) do not necessarily occur. The need for direct localmeasurements of shear stress and overburden pressure becomes clear; subglacial waterpressure and geometry constitute insufficient data to compute local shear stress andeffective pressure.T.2 Electrical phenomena7.2.1 Streaming potentialsWe have shown that streaming potentials are the primary source of natural potential variations in the subglacial environment. The streaming potential measurementshave proved useful in measuring pressure gradients at the glacier bed, although morework needs to be done to discover how other subglacial processes, such as sedimentdeformation, affect these potentials.The dramatic correlation between subglacial water pressure gradients and streaming potentials suggests a number of experiments to further investigate this cross-coupledChapter 7. CONCLUSIONS 179phenomenon. We envisage an array with the electrodes arranged on a rectangular grid.At locations along the periphery of the grid, pressure sensors would be installed togetherwith the potential electrodes. This arrangement of electrodes would allow a detailedanalysis of how streaming potentials can be used to interpolate pressure gradients andmight also permit an analysis of time-varying hydraulic and electric conduction paths.It would also be interesting to try to place a string of electrodes vertically though thedeforming sediments; such an array could, for several days, monitor changes in thevertical pore pressure distribution. By measuring the diffusion rates of pressure downinto the sediments, we could infer the hydraulic diffusivity of the sediments.7.2.2 Electrical resistivityMeasurement of subglacial electrical resistivity presented the most difficulty. We haveobserved exciting changes in polarity and magnitude in subglacial resistivity; these results suggest that not only is the subglacial environment electrically inhomogeneous,but that this inhomogeneity changes with time. Absolute measurements of electrical resistivity suffered from insufficiently accurate electrode position information; withbetter position control, we are certain that much could be learned about subglacialdeforming sediments using this technique.7.3 Basal deformation7.3.1 RheologyWe have demonstrated that the rheological relations derived by Boulton and Hindmarsh(Equations (3.16) and (3.17)) are inappropriate for the sediments beneath TrapridgeGlacier, but we have not developed an alternative rheology for the substrate of thisglacier because we do not have the necessary local shear stress and effective pressureChapter 7. CONCLUSIONS 180data— this is a problem faced by all glaciologists investigating subglacial deformation,Boulton and Hindmarsh included.Were we to synthesize mean strain rate values from our continuous records of sheardeformation, in order to mimic the type of data collected by Boulton and Hindmarsh,we still would be unable to develop a rheology. As discussed in section 3.1, we haveonly one theoretical shear stress value and a small range of effective pressure valuesinferred from subglacial water pressure; we cannot fit mean strain rates to conceivablerheological functions. Again, we stress that local measurements of subglacial shearstress and overburden pressure must be available.7.3.2 Effective viscosityWe have calculated effective viscosity values for the sediments beneath Trapridge Glacier, but we are not convinced that effective viscosity values are meaningful since theyare derived assuming a linear rheology and, in our case, by averaging strain rate over anarbitrary length of time (the length of our data set). Having shown that (1) mean strainrate of almost any value desired, both positive and negative, can be computed from acontinuous strain record and (2) basal shear stress cannot be calculated reliably, we areloath to suggest that the effective viscosity values given in Chapter 3 are significant.Our deformation data do indicate that experiments measuring net strain over aperiod of time are vulnerable to large errors because of the time-varying nature of basaldeformation. We have shown how negative net strain rates can be synthesized fromour data.There is a considerable amount of work to be invested in developing bed deformation techniques further. It is possible, with today’s miniature radio telemetry technology, to build tilt sensors that do not require a cable leading to the surface; such deviceswould remove all worries about inter-cell wires interfering with deformation measurements. Gregory and Stubbs (1983) built an electromagnetic positioning system for useChapter 7. 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Petrol.Eng., 13(1), 5—11.Appendix AINCLINOMETER DESIGN AND DATA PROCESSINGBorehole inclinometers are standard equipment for field glaciologists and are commonly used for investigating the flow law of ice and for measuring the spatial position ofenglacial and subsurface sensors. The recent development, at the University of BritishColumbia (UBC), of a prototype inclinometer that employs a three-component fluxgatemagnetometer to obtain a compass bearing has stimulated our interest in borehole indinometry. Following a review of various approaches to glacier incinometry, we presenta unified theory of data interpretation that can be applied to all inclinometers, discussthe application of the theory to the UBC inclinometer, and discuss the sensitivity ofthe theory to error in the data.The substance of this appendix has been published previously in the Journal ofGlaciology (Blake and Clarke, 1991b).A.1 IntroductionWhenever a deep hole is drilled into the surface of the Earth, there is uncertaintyabout its trajectory. Asymmetries in the drill, layering in the material being drilled,and shortcomings in driffing technique may cause the borehole to stray from its intendedpath (e.g. Walker, 1986). Incinometry tools remove spatial uncertainty by mapping thedeviation of the borehole. Typically, two pieces of information are collected at a seriesof stations, or locations, along the length of the hole: the tilt of the inclinometer fromvertical and the azimuth of that tilt in a geographical coordinate system. Of the manytechniques available for making these measurements, we present a brief overview ofthose that have been used for glaciological work. Sources for some of this inclinometryequipment are given.197Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING 198In this appendix, we attempt to formalize some aspects of inclinometry data analysis and, at the same time, introduce a new indinometry tool that was recently assembledat the University of British Columbia (UBC). We describe the calibration procedurefor the UBC inclinometer, formulate a general theory for interpretation of inclinometerdata, and examine the sensitivity of this theory to error in the data.A.2 Historical overviewInclinometry of glacier boreholes has long been associated with field investigations ofthe flow law for ice (Perutz, 1947, 1949, 1950; Gerrard and others, 1952; Sharp, 1953a,1953b; Nye, 1952, 1953, 1957; Miller, 1957; Savage and Paterson, 1963; Paterson andSavage, 1963a, 1963b; Shreve and Sharp, 1970; Raymond, 1971a, 1971b; Hooke, 1973,1981; Garfield and Ueda, 1976; Paterson, 1983; Hooke and Hanson, 1986; Hooke andothers 1987; Van der Veen and Whillans, 1990). Many of these researchers used incinometry data to infer the internal velocity field of the glacier and thereby deduceparameters of the flow law. Increasing interest in basal processes has created a new application for incinometry, that of accurately determining the position of englacial andsubglacial sensors. As an example, the measurement of subglacial resistivity (Brandand others, 1987) requires knowledge of the spatial position of current and voltageelectrodes within adjacent boreholes.Glaciologists are fortunate in that drilling through ice is relatively inexpensiveand easy when compared with drilling through other crustal materials. Hydro-thermaldrilling is likely the most common method in use today, but our experience on TrapridgeGlacier has shown that boreholes drilled with hot water are often far from plumb.A.2.1 Basic dip and azimuth measurementsA rudimentary fluid-level inclinometer can be built by partially ifihing a sealed glassvial with a solution of hydrofluoric acid. If the vial is kept at a fixed tilt angle forAppendix A. INCLINOMETER DESIGN AND DATA PROCESSING 199an appropriate time, the acid will etch the glass, leaving a record of the fluid surface.If several vials are positioned along the length of a borehole while the etching occurs,they provide a discrete record of the borehole tilt (e.g. Savage and Paterson, 1963,p. 4522). An exotic variation of the above approach substitutes a warm solution ofgelatin (“Jello” is a suitable commercial product) for the acid; the vials are left inplace until the gelatin hardens and tilt is preserved in the surface of the gelatin (Shreveand Sharp, 1970, p. 71). Such fluid levels have three major drawbacks: (1) The vialsmay freeze to the walls of the borehole. (2) The precision of the tilt measurements ispoor. (3) The vials do not measure tilt azimuth, although in the case of the gelatinlevel, a magnetic compass needle suspended within the gelatin can provide a magneticbearing.“Floating-compass” inclinometers improve on fluid-level instruments by providingazimuth information, but they are laborious to use. Sperry-Sun Corporation (Houston,Texas) and Parsons Survey Company (South Gate, California) manufacture floating-compass inclinometers that employ a miniature camera. The camera in the Sperry-Suninstrument is focused on a weighted spherical magnetic compass ball that has azimuthand tilt markings, much like lines of latitude and longitude. At predetermined timeintervals, the camera takes a photograph of the illuminated compass ball; it is left tothe operator to ensure that the instrument is at the proper position along the boreholeat the time each picture is taken. The camera in the Parsons instrument photographsan illuminated plumb bob against a compass graticule. The single-frame version ofthis instrument must be brought to the surface after each picture has been taken (e.g.Shreve, 1958; Savage and Paterson, 1963), but an improved multi-frame version useselectrical signals from the surface to advance the film (Garfield and Ueda, 1976). Withboth the Sperry-Sun and Parsons instruments, azimuth and tilt readings are transcribedfrom the developed film images, an obvious impediment to in-field data acquisition.Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING200Clamping the compass so that it can be examined once the inclinometer is broughtto the surface is an alternative to ifim recording of compass position. The inclinometerbuilt by Pajari Instruments Ltd. (Orillia, Ontario) uses a gimbal-mounted horizontalcompass needle. After a predetermined interval of time has passed, a locking mechanismarrests the gimbal rings and the compass. The operator sets the timer, lowerstheinstrument to the desired location, and waits for the timer to lock the compass. Theinclinometer is then brought to the surface, the gimbal mount is removed from thepressure casing, and azimuth and tilt are read from graduated scales in the unit. Thislaborious procedure is repeated for each station within the borehole. R. M. Koerner(personal communication, 1987) has used this instrument on the Agassiz Icecap.A.3 Electronic inclinometryin recent years, advances in electronic technology have enabled the developmentofinclinometry systems that transmit position information directly to the surface. Compared with earlier techniques, the time and labour savings of electronic inclinometry areappreciable; with appropriate equipment in place at the borehole site, the trajectoryof the borehole can even be computed and displayed as the data is collected.A.3.1 Measuring tiltElectronic tilt transducers allow precise and repeated measurements of tilt to be madefrom a glacier surface. The Fredericks Company and Applied Geomechanics inc. (SantaCruz, California) manufacture electrolytic tilt transducers that are suitable for borehole inclinometry. General Oceanics (Miami, Florida) and Slope Indicator Company(Seattle, Washington) manufacture force-balance tilt transducers.A.3.2 Measuring azimuthTo reconstruct the borehole trajectory, each set of tilt measurements must be combinedwith a measurement of instrument azimuth. Rigidly-coupled inclinometers, trackedAppendix A. INCLINOMETER DESIGN AND DATA PROCESSING 201inclinometers, magnetically-oriented inclinometers, and gyroscopically-oriented inchnometers are the four design categories known to us.A .3.2.1 External azimuth controlThe orientation of a rigidly-coupled inclinometer is controlled by a rod leading fromthe surface. This method is suitable for shallow holes, but as the distance betweenthe instrument and the operator increases, azimuth control deteriorates and logisticalproblems grow (Gerrard and others, 1952).Tracked incinometers require that a special grooved casing be installed in theborehole before incinometry begins. Spring-loaded wheels or pins keep the inclinometeraligned with the grooves in the casing as the instrument moves through the hole. Forshallow holes, twisting of the casing is assumed to be negligible; it follows that theazimuth of the inclinometer is constant and equal to the azimuth of the casing at thetop of the hole. For deeper holes, a torsion tool can be used to measure the twist in thecasing. This tool consists of two sections, each of which has a set of tracking wheels,that are connected by a sensitive torsion transducer. Any twisting of the casing isrecorded by the transducer as the tool is moved through the hole. If a torsion tool isnot available, the presence or absence of casing twist must be inferred from the data.Although the effort and expense of installing an incinometry casing is an inconvenience,the casing provides a uniform borehole geometry for the inclinometer, serves to smoothout small perturbations in the borehole trajectory, and allows the operator to cancelany offset error in the instrument by executing precisely reversed runs through thehole; great accuracy can be achieved. Cased holes are also advantageous in long-termdeformation studies because they are not prone to closure from creep and freezing.The biaxial inclinometer manufactured by Slope Indicator Co. (Seattle, Washington) is an excellent example of a tracked inclinometer. Hooke and Hanson (1986) andAppendiz A. INCLINOMETER DESIGN AND DATA PROCESSING 202Hooke and others (1987) have made precise measurements of glacier ice deformationusing this instrument, but we are certain that they have cursed the need for a casing.A.3.2.2 Internal azimuth controlGyroscopically and magnetically-oriented inclinometers do not require a casing, hencetheir appeal for glacier work. A gyroscopically-oriented inclinometer was built by theresearch team that made the historic measurements of deformation within Jungfraufirn(Gerrard and others, 1952). Their paper contains a brief description of this instrument,but we have been unable to locate the detailed instrumentation paper to which theyrefer (Broad, Jason, and Perutz, 1952, J. Sci. Instrum., designated “in press”) andsuspect it remains unpublished.Indinometers that use a magnetic compass to determine azimuth and that reportthe azimuth electrically are a vast improvement over the Sperry-Sun, Parsons, and PaJan designs; the output from the tilt sensors and the compass can be recorded while theinclinometer is on station. The primary disadvantage of magnetic orientation is thatthe compass is sensitive to geomagnetic disturbances, as well as magnetic field variations caused by remnant magnetization and magnetic susceptibility in nearby materials.For ice, which has no magnetic signature, only geomagnetic disturbances are of consequence. Glaciologists have used several designs for magnetically-oriented incinometers.Raymond (1971c) describes an instrument that was used to measure the internal yelocity structure of Athabasca Glacier (Raymond 1971b). Philip Taylor (Hydro-Tech,Seattle, Washington) has built several magnetically-oriented inclinometers specificallyfor use in glaciers. These instruments employ two orthogonal tilt sensors and a gimbalmounted compass that produces a voltage signal proportional to the magnetic bearing.Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING 203Further improvements in reliability can be gained by using a non-mechanical compass, such as a magnetometer. Levanto (1959) used a downhole three-component flux-gate magnetometer to measure the in situ magnetic susceptibility of rock. The instrument did contain a tilt sensor, but the sensor was used to maintain the attitudeof the magnetometer rather than measure inclination. With minor modification, thisinstrument would have served well as an inclinometer.A.4 The UBC inclinometerIn the spring of 1989, the glaciology group at the University of British Columbia, incollaboration with Slope Indicator Canada Limited (Richmond, British Columbia) andNarod Geophysics (Vancouver, British Columbia), developed a prototype inclinometer.This instrument is similar to the Hydro-Tech instrument in that it uses the Earth’smagnetic field as an azimuth reference, but the field sensing device is a fluxgate magnetometer rather than a mechanical compass. Two force-balance tilt transducers andthe magnetometer are enclosed within a non-magnetic stainless steel pressure tube(Fig. A.1). Brass centring springs help keep the inclinometer centred in the boreholeand prevent the instrument from spinning when on station. The tube is 1.5 m longand has an outside diameter of 2.54cm (1.00 in). The tilt transducers are Slope Indicator devices, identical to those used in their tracked incinometers. The magnetometeris a Narod Geophysics miniature ring-core fiuxgate magnetometer. (Wyckoff (1948)gives an overview of the principals of operation for a fluxgate magnetometer; moredetailed treatments related to this magnetometer are found in: United States Patent3800213 (26 March 1974), UK Patent 2044460 (16 Oct 1979), Russell and others(1983), Narod and Russell (1984), and Narod and others (1985)). The sensing elementof this three-component magnetometer is a 12.7 mm (0.50 in) cube. As the tilt sensorscontain permanent magnets, the magnetometer is mounted as far from the tilt sensorsAppendia A. INCLINOMETER DESIGN AND DATA PROCESSING 204as possible. Ideally, the magnetometer would be placed at the bottom of the instrument, far from any secondary magnetic fields, but the tilt sensors completely ifil theinterior of the pressure tube so that no wires can pass by them; thus the tilt sensorsmust be positioned at the bottom and the magnetometer at the top. The space betweenthese sensors is occupied by the electronics package. The instrument cable attaches tothe top of the instrument. This cable carries power to the instrument and analoguedata to the surface, and has a steel strength member. The magnetic bias created bythe steel cable and the electric currents flowing in the cable is compensated by themagnetometer calibration.Fig. A.l: A block diagram of the peripheral devices attached to the TJBCinclinometer (inclinometer not drawn to scale). The power module providespower to the instrument from a small 6 Ah sealed lead-acid battery. Analoguesignals are fed to the data logger which, on command from the handheldcontrol box, records data from the tilt sensors and magnetometer into thestorage module. The operator keys in depth information on the control box.Appendiz A. INCLINOMETER DESIGN AND DATA PROCESSING 205Power for the inclinometer is provided by a 6 Ah sealed lead-acid battery thatcan be connected to a solar panel for charging. The five analogue data signals (twofrom the tilt sensors and three from the magnetometer) are wired to data logger inputchannels. The inclinometer operator holds a small weather-tight control box whichis connected to the data logger by a 5 m cable. The control box contains two SPSTswitches with indicator lights, a momentary switch, a 100—position push button digitalpotentiometer, an acknowledgement lamp, and a piezoelectric buzzer.For each hole, the inclinometer is stepped down and up at one metre intervals,resulting in a two-fold redundancy in the measurements. The depth of the inclinometerat each station, as marked on the cable, is coded manually on a digital potentiometerthat is mounted in a handheld control box. After the operator closes a momentaryswitch, indicating that the inclinometer is on station, the data logger records the time,the digital values of the tilt and magnetometer signals, the voltage on the potentiometer,the state of the two SPST switches, and the voltage of the inclinometer battery. Thisdata is copied immediately to a solid state storage module connected to the data logger.The acknowledgement lamp and buzzer are then activated briefly to signal the operatorthat the instrument can be moved. The two SPST switches are used to indicate specialconditions, such as the bottom of the hole (which often does not fail on an integralnumber of metres) or a calibration reading. At the end of the day, the storage moduleis carried to our field camp where the data are transferred to a computer for analysis.This inclinometer system is very efficient. A single operator can make two inchnometry passes through a 70 m hole at 1 m depth intervals in 25 minutes.A.5 Coordinate systemsThree right-handed Cartesian coordinate systems are required to process data from atracked or gyroscopically-oriented inclinometer. Two additional coordinate systems associated with the Earth’s magnetic field and the magnetometer are required to processAppemc1i A. INCLINOMETER DESIGN AND DATA PROCESSING 206data from a magnetically-oriented inclinometer (Figure A.2). The coordinate systemintroduced by Nye (1957), and used in several subsequent papers by Nyc and others,is aligned with the flow direction of the glacier. This is natural for studies of internal deformation, but because our primary concern is locating sensors placed withinboreholes, we choose a system related to map coordinates.H HFig. A.2: The five right-handed inclinometer coordinate systems displayed as a stereo pair. Top: the geographical coordinate system U and thegeomagnetic coordinate system 7-1. The angle “d” between the grid north axisYu and the magnetic north axis X is the magnetic declination. Bottom: thetilt sensor (T), case (C), and magnetometer (B) coordinate systems share acommon z axis pointing downward along the body of the inclinometer.The geographical coordinate system U is tied to the Universal Transverse Mercator (UTM) coordinate system used on Canadian topographic maps. The u, Yu, andz axes point east, north, and upward, respectively. The geomagnetic coordinate system (labelled 7-1 after the standard designation for magnetic field) is tied to the localdeclination of the Earth’s magnetic field. The z axis is positive downward and theXH axis points along the magnetic declination. This definition follows the internationalAppendix A. INCLINOMETER DESIGN AND DATA PROCESSING 207convention for geomagnetic coordinate systems and results in positive ZH readings in thenorthern magnetic hemisphere. The remaining three coordinate systems have z axesthat point downward along the axis of the inclinometer. The tilt sensor coordinatesystem T has its XT and YT axes aligned with the sensitive axes of the tilt sensors. Thecase coordinate system C has the x0 axis aligned with some feature on the inclinometer.In the case of a tracked inclinometer, this would be the alignment mechanism. TheUBC instrument has its x0 axis aligned with a machined facet on the inclinometercase (the use of this facet is discussed below). The magnetometer coordinate system(labelled B after the standard designation for magnetic flux density) represents thethree axes of sensitivity of the fluxgate magnetometer. For the UBC inclinometer, theinstrument-based coordinate systems are distinct, and the rotational angles betweenthe XT, x0, and x axes must be known in order to process data. For other instrumentsthese three systems may or may not coincide.A.6 Data analysisThe analysis of inclinometer data proceeds in three distinct steps: (1) The instrumentcalibrations are used to compute the two tilt angles and the azimuth at each station.(2) The tilt and azimuth values are transformed into vectors representing the orientationof the inclinometer at each station. (3) An inversion scheme computes the continuoustrajectory of the borehole based on the discrete set of orientations.In the following two sections, we discuss in detail the calibration and transformation procedures for the UBC inclinometer. Although these procedures vary from instrument to instrument, all share the common goal of determining the vertical unit vector iiand the orientation unit vector ih in an inclinometer coordinate system (Fig. A.3). Fora magnetically-oriented inclinometer, ili represents the Earth’s magnetic field vectorand for a tracked inclinometer, th represents the orientation of the track grooves. TheUBC inclinometer yields the true magnetic vector, whereas an instrument that uses aAppendix A. INCLINOMETER DESIGN AND DATA PROCESSING 208gimbal-mounted compass yields the magnetic vector projected onto the gravitationallyhorizontal plane.A.6.1 CalibrationThe tilt sensors are calibrated using a calibration frame that allows positioning of theinclinometer at precise tilt angles (to within 1 mm of arc) throughout its operatingrange of +30° of tilt from vertical. Repeated calibration of the instrument betweenfield seasons indicates negligible drift in the calibration. The output voltage from thetilt sensors is proportional to the sine of the tilt angle from vertical. In the T system,the components of fl are given by= sinI= sinI,I n=1(n2+n2)where I and I are the two tilt angles.The calculation of the orientation vector ii would be trivial if the three axes ofthe magnetometer were to have zero offset and equal sensitivity, but in practice, then nYTXTZTFig. A.3: A stereogram showing the two unit vectors measured by aninclinometer and the tilt coordinate system T in which they are resolved.The tilt sensors yield ii and the orientation apparatus yields ih.m} (A.1)Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING 209design of the magnetometer sensor makes it very difficult to realize these characteristics,particularly with regard to voltage offsets. A calibration is required.If a series of measurements of the Earth’s magnetic field are taken with the magnetometer positioned at many orientations within the field, then the measured pointswill map out a surface in the 13 system. If we assume that the axes of sensitivityof the magnetometer (with their non-zero offsets and unequal sensitivities) are mutually orthogonal, then the surface is a translated ellipsoid. The calibration function weseek transforms this ellipsoid into a unit sphere centred on the origin; applying thecalibration to any given point will produce the corresponding orientation vector.The calibration data are best collected on the glacier surface where the magneticenvironment can be controlled. The operator, stripped of magnetic clothing, holds theinclinometer at a succession of orientations. At each position, the inclinometer is heldsteady while the three magnetometer outputs are recorded. The orientations are chosensuch that the Earth’s magnetic field intersects the magnetometer sensor from as manydirections as the patience of the operator allows. The specific calibration orientationsare not important, so a hand-held calibration can be used. In addition to instrumenterror, reading errors result from fluctuations in the Earth’s magnetic field during thecalibration and movement of the inclinometer as data is recorded (the bandwidth ofthe magnetometer is 5 Hz and the three components are measured sequentially withina span of ll4ms).Typically, about 200—500 calibration triplets are collected. A calibration transformation that maps a data point M(B) on this ellipsoid onto a unit sphere centred on theorigin is given bym(B) S 0 0 M,(B)+TZm(3) = 0 S, 0 M(B) + T (A.2)mZ(B) 0 0 MZ(B) + TAppendia A. INCLINOMETER DESIGN AND DATA PROCESSING 210where T is an offset vector, S is a scaling matrix, and m(B) is the transformed point.The six free parameters S, S,, T, T, T) are determined by minimizing theleast squares objective function=[m + m(B) + m(B) — 1j2for the n transformed calibration triplets S, and S are non-negative). An algebraic approach to solving the minimization is algebraically horrific, if not intractable, sowe use an iterative six-dimensional simplex algorithm (Press and others, 1986, p.289).The expanded objective function is given by=n+S+ (E + mT + 4T z +4T3 +— 2S ( + nT) — 2S (y + nT) — 2S (z + ThTz2)+xy +Ty +Tx +2Txy+ 2T yjx + 2TT x + 2TT y + 4TT XiYi)+2SSL(TT + + T + T + 2T+ 2T zjy + 2TT yj + 2TT z + 4TT YiZi)+2SS(T:T: +yz + T + T + 2T+2TzxAppendix A. INCLINOMETER DESIGN AND DATA PROCESSING 211— 4ST > x — 4S,T — 4ST (A.4)Note that the summation terms are constants for a given data set.For the UBC inclinometer, the transformed calibration data fit a unit sphere towithin 2%. Removing the constraint on magnetometer axis orthogonality by allowingnon-zero off-diagonal terms in S results in a negligible improvement in fit, at the expenseof greatly increasing the geometrical complexity of data analysis.A.6. 2 TransformationsIn this section, we discuss the transformations that are applied to data from the UBCinclinometer. Some of these transformations are generic in nature and can be used withall inclinometers.In order to simplify the analysis, the scaled magnetic vector m(B) must be rotatedabout the inclinometer axis from the B system into the T system (Fig. A.2). Theangle 1, measured from the XB axis to the XT axis, is calculated as the sum of twointermediate angles. The angle between the XT and the x axes can be measuredaccurately on the tilt calibration frame. The angle between the x and XB axes is not aswell constrained. Repeated sightings on known geographical reference points are madethrough a telescope clamped to the machined facet on the body of the inclinometer.By reconciling the magnetic declination published on the 1:50 000 topographic map(Energy, Mines and Resources, Canada map sheet 115 F/i, edition 1, 1987) with thecoordinates of the reference points and the coordinates of the observation site, the valueof this angle can be computed to within ±0.5°. Weather permitting, these referencesightings are made before and after each hole is logged. This allows correction fortemporal variations in magnetic declination caused by geomagnetic disturbances; thesedisturbances can cause several degree of variation in declination, especially in polarregions where the Earth’s magnetic field is near-vertical.Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING 212A.6.2.1 NormalizationThe calibrated magnetic vector m(B), from Equation (A.2), is nominally a unit vector,but errors in the data will result in a slightly erroneous vector magnitude. If themagnitude is not corrected, then the components of m(B), and vectors derived from it,cannot be used as direction cosines. Therefore, the unit magnetic vector th is definedby rotating and normalizing m(B) using the equationm cos sin1 0 m(B)m = — sin 1 cos 0 m(B) (A.5)m 0 0 1 mZ(B)where Im(B)I denotes the scalar length of vector m(B).Examination of Figure A.3 reveals that, in the case where iii is the full magneticfield vector, there is redundancy in the data. The three degrees of freedom representingthe orientation of the inclinometer body are constrained by the four degrees of freedom embodied in the components of th and fl, which in turn are derived from the fiveinstrument readings (I,,, Is,, MZ(B), M(B), and MZ(B)). The two additional degrees offreedom governed by the instrument readings are the magnitude of the magnetic fieldvector (lost in normalizing iii) and the spin orientation of the inclinometer on its axis(in our analysis, this information is discarded). For the UBC inclinometer, the valueof 11 is far better determined than th, so we use ii to define the inclination angle ofthe ZT axis. Vector th is used to constrain the azimuth of the ZT axis about vector ñ.The functions of ih and ñ can be exchanged, but were th used as the inclination reference, accuracy would be lost. Note that data from magnetically-oriented inclinometersbecome indeterminate when th and ñ are parallel or anti-parallel.Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING 213A.6.2.2 Eulerian AnglesThe relationship between two arbitrarily-rotated coordinate systems that share thesame origin can be uniquely described using Eulerian angles. Figure A.4 shows how atriplet of Eulerian angles (4’, 0, 4,) defines the transformation between any two right-handed Cartesian coordinate systems. The transformation is achieved by three successive right-handed rotations about specified axes. Defined as a matrix operation, thetransformation from the unprimed coordinate system to the primed isx’=Ax (A.6)where x and x’ are coordinates of a point in and A is the transformation matrix( cos cos 4’ — cos 8 sin 4’ sin & cos b sin 4’ + cos 8 cos 4, sin & sin b sin 0A = — sin b cos 4, — cos 8 sin 4, cos — sin & sin 4’ + cos 8 cos 4, cos cos sin 8sin 8 sin 4, — sin8 cos 4, cos 0(A.7)Because A is an orthogonal matrix, A1 = AT, so the inverse transformation x =A’x’ is simplyx = ATxI (A.8)The inverse matrix A—’ satisfies AA’ = I, where I is the identity matrix, and thetransposed matrix AT is obtained by writing the rows of A, in order, as columns.Appendix A. INCLINOMETER DESIGN AND DATA PROCESSINGFig. A.4: A set of Eulerian angles describes the transformation betweentwo right-handed coordinate systems sharing a common origin. The transformation from an unprimed to a primed coordinate system is defined as a setof three right-handed rotations about specified axes. The three rotations are:(1) A rotation by angle g about the z axis. (2) A rotation by angle 0 aboutthe intermediate x axis (dotted line). (3) A rotation by angle about thez’ axis.214This level of complexity is only necessary when moving from an instrument-basedcoordinate system to a map coordinate system. Transformations between coordinatesystems within these two groups is accomplished by simple rotation about the commonzI::::::::::::::::::::::::z axis (9 = = 0).Appencli A. INCLINOMETER DESIGN AND DATA PROCESSING 215A.6.2.3 Eulerian TransformationOur objective now is find a set of Eulerian transformation angles q, 8, and b whichwill map Iii, as expressed in the T system, onto the U system such that the horizontalprojection of th has the proper magnetic declination d. By casting the transformationin this way, we avoid directly implicating the 7( system. The transformation must alsomap ñ, as expressed in the T system, onto the vertical in the U system. We will treatthe U system as unprimed and the T system as primed (Equation (A.7)).The angle 9 defines the net tilt of the ZT axis with respect to the vertical z,., axis.This net tilt is simply9 = cos1n (A.9)The angle ‘ controls the relative contribution of the two tilt angles to the net tilt,defined as= tan’’ (A.1O)\\ny IThe angle rotates the T system about the z axis; in other words, q is the Eulerianangle that controls the declination of the transformed magnetic vector. Given 9 and‘ç&, the value of ç must satisfyFsind mFcosd = AT m, (A.11)/[ZF2 mwhere 0 <F < 1 is the magnitude of iii when projected onto the horizontal (Usystem)plane and d is the magnetic declination east of UTM north. This system of equationshas an explicit solution for 4 and F, but the algebra can be simplified considerably byAppendix A. INCLINOMETER DESIGN AND DATA PROCESSING 216first solving for the case where d = 0 and subsequently adjusting the solution. This trickresults in an indirect association with the 71 system. The equation to solve becomes0F = AT m (A.12)/1_F2 mand the solution is1q=tan . . (13)(m sin b + m cos cos 9 — m sin 9F = [(mi cos& — m sinb)2+ cos2 9 (mi, cos — m sin &)2+ m sin 29 (m cos — m sin ‘v’)1/2+ m sin2 9] (A.14)The non-zero declination ci is reinstated by modifying Equation (A.13) to give—1 m,cos&—msin& 1ç=tan . . i—d 1A.15(m sin b + m cos Ø) cos 0 — m sin OjEquations (A.13) and (A.15) are curiously insensitive to the value of m in the sensethat a value of iui derived from a gimbal-mounted compass (i.e. th is horizontal) willgive the same value for as a value of th representing the full Earth field, but this insensitivity is expected. Both gimbal-mounted and full-field magnetometers are equallyadept at determining magnetic bearing.As the final step in the data transformation procedure, the orientation vector ofthe inclinometer (which corresponds to the ZT axis), as expressed in the U system, iscomputed by the transformationt 0 sin9sing= AT 0 = —sin9cosq (A.16)1 cosOAppendiz A. INCLINOMETER DESIGN AND DATA PROCESSING 217The Eulerian transformation matrix A is defined by Equations (A.7), (A.9), (A.1O),and (A.15). For this analysis the value of F is irrelevant.A .6.2.4 Universal application of transformationsNote that Equations (A.7), (A.9), (A.1O), (A.13), and (A.16) apply to any incinometrydata that can be expressed in terms of a tilt vector ii and an orientation vector ill.This includes data from tracked and gyroscopically-oriented inclinometers.A.6.3 Inverse problemIn this section, we develop a general theory, applicable to all inclinometers, for computing a continuous borehole trajectory based on discrete measurements of boreholedepth and inclinometer orientation. Before considering the details of this step, it isappropriate to consider the relationship between the inclinometer orientation and theborehole axis.The simplest assumed relationship is that the inclinometer is always parallel tothe borehole axis. Although this presumption is erroneous, without additional datadescribing the borehole geometry and centring device geometry, it is difficult to improveupon. At any given station along the borehole, the inclinometer axis lies along a straightline drawn between the centring devices. If the two centring devices operate identically,the borehole has smooth walls of constant diameter, and the curvature of the boreholeis a circular arc, then the inclinometer axis will lie parallel to the borehole axis ata point midway between the centring devices. In practice, these three conditions arerarely satisfied. It is here that tracked inclinometers exhibit improved performance overinstruments such as the UBC tool; the groove-tracking wheels can accurately positionthe inclinometer in the centre of the casing and the casing provides both smooth wallsand regular borehole geometry. The trajectory computation method described belowAppendix A. INCLINOMETER DESIGN AND DATA PROCESSING218will assume that the borehole and inclinometer axes coincide; the sensitivity of themethod to positioning error will be discussed at the end of the appendix.The task of computing a continuous borehole trajectory from a set of discretetilt and azimuth measurements constitutes an inverse problem. An inffnite number ofpossible trajectory solutions exist that will satisfy a given data set exactly. An infinitenumber of additional solutions exists that do not satisfy the data exactly, but whichdo fall within the error bounds of the data. A single solution of the first type can beisolated by placing appropriate geometrical constraints on the trajectory; the solutionto the inverse problem becomes an interpolation scheme. All treatises ofwhich weare aware, this development included, use geometrical constraints to choose an exactsolution that will hopefully reflect the true trajectory. This method for solving aninverse problem differs from traditional methods in that an a priori structuralmodel issolely responsible for isolating one of the possible solutions. Traditional methods canfind solutions of the second type by setting an objective function, such as minimizingthe rate of change of borehole tilt, and by allowing for assessment of error inthe data.Data smoothing is another possible approach to finding inexact trajectory solutions.Unlike interpolation between fixed points in space, interpolation of orientationvectors can lead to cumulative error in position; in reconstructing the trajectory of thehole from the surface downward, the error in determining the relative position betweentwo measurement stations accumulates.Many methods for interpolating slope angles are discussed in the petroleumindustry literature. The “terminal angle tangential method” is often mentioned as thebest known interpolation method: over any station interval, the orientation of theinclinometer axis is assumed to be equal to that of the lower station. In effect, nointerpolation is performed and the slope of the computed trajectory is discontinuous atevery station. Because of its faulty representation of inclinometer motion, this methodresults in appreciable error. Waistrom and others (1972) discuss five different methodsAppendia A. INCLINOMETER DESIGN AND DATA PROCESSING 219for interpolating the tilt angle and azimuth between stations. The underlying premiseof many of their mathematical models is that tilt angle and azimuth can be treatedas independent quantities. This premise does not always hold true, as is revealedby considering the radius-of-curvature method developed by Wilson (1968) and laterexpanded by Rivero (1971). Their method maps the borehole trajectory onto the surface of a vertical cylinder such that the desired azimuth and tilt are preserved at theendpoints of the spiral segment. The resulting trajectory is not independent of thecoordinate system in which the projection is made, indicating that in this case the tiltangle and azimuth cannot be treated separately. Angle interpolation methods are alsocluttered with special treatment for the multivalued nature of the inverse trigonometricfunctions.We assume that the path of the inclinometer between stations can be describedby a series of circular arc sections (Fig. A.5). Figure A.6 shows one of these arcs indetail. The endpoints of the arc (P1 , P2) are tangential to the normalized orientationvectors (t1,2), and the length of the arc L equals the measured depth increment onthe cable. Successive arcs join in a smooth manner with no discontinuities in slope,although discontinuities in the direction of slope change are present. If the position ofP1 is known, then the position of P2 or any intermediate point P3 along the arc can becomputed. This method has been discussed by Zaremba (1973), but Zaremba derivesan unnecessarily complicated solution.Appendix A. INCLINOMETER DESIGN AND DATA PROCESSINGFig. A.5: A perspective view of the circular-arc interpolation model. Between pairs of measurement stations, the borehole is assumed to follow acircular trajectory. The arc length is constrained by the measured distancebetween stations, and the plane of the arc is defined by the instrument orientation vectors at the endpoints of the arc. These orientation vectors aretangent to the arcA.7 Interpolation Scheme220We derive solutions for P2 and P3 given the initial position P1 and the unit tangentvectors to the circular arc and 2 (Fig. A.6). Consider the general problem of solvingfor the intermediate coplanar vector between two non-parallel unit vectors (Fig. A.7).The partitioning factor e ranges between 0 and 1. The unit vector ia is defined by thelinear combination= k11 + k2ix(A.17)Appendiz A. INCLINOMETER DESIGN AND DATA PROCESSING 221Fig. A.6: A detailed view of one circular arc in the borehole trajectory.Knowing the position of the starting point P1, the tangent vectors I and 2,and the length of the arc L, the position of the endpoint P2 can be computed.The position of any intermediate point P3 on the arc can also be computed.The angle 7 is defined by and t2.where k1 and k2 are appropriate scalar values. The solutions for k1 and k2 must satisfy= COS(E7) (A.18)= cos [(1—yj (A.19)Pe P3P2tiwherecos-y=ti.t2 (A.20)Equations (A.18) and (A.19) ensure that is a unit vector coplanar with i and 2•The solutions are= cos(E-y) — cos-ycos [(1— (A.21)sin 7sin(E7)k2 —Sin 7(A.22)Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING 222In the special case of bisection (e = 0.5),Yk1 = k2 =2 cos(7/2)t2(A.23)Fig. A.7: The intermediate coplanar unitnon-parallel unit vectors and 2 is shown.the position of the intermediate vector.vector t3 between two arbitraryThe value of 0 < e < 1 definesC=2()sin()and Equations B.1 and B.7 yield the unit vector along the chordtl+t2C— 2cos(7/2)so that we can write the solution for P2 as£ /7’\P2=Pi+—tan(—) [1+2]tiC t3In Figure A.6, the chord drawn between Pi and P2 bisects the two vectors andt2. The length of the chord is given by(A.24)(A.25)(A.26)Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING 223The position at any intermediate position P3, located at some fractional distance 0e 1. along the arc from P1 is given by a similar construction asP3 = P1 + tan () [(k1 + 1) + k22](A.27)where-y, k1, and k2 are defined by Equations (A.20), (A.21), and (A.22).In the special case where t2 —* £ (the anto-parallel case is unlikely), the circulararc will have infinite radius, and the solutions for P2 and P3 areP2=P1+L (A.28)3=Pi+eLi (A.29)The trajectory computed by this circular-arc method is independent of the coordinate system. The trajectory of the borehole is reconstructed by successively applyingEquation (A.26) or (A.28), as appropriate, beginning at the glacier surface, and workingdownward.A.8 SensitivityThe trajectory of the borehole as determined by the circular-arc method, or by any otherinterpolation method, is inherently incorrect — even if the borehole orientation dataare error-free. This is because the continuous borehole trajectory is sampled at a finitenumber of points. In the case of the UBC inclinometer, we do not believe that samplingdensity is a major source of error because our 1 m sampling interval is short comparedto the length of the inclinometer and to the length of the drill stem used to drill theholes; we do not expect perturbations in borehole trajectory on a scale smaller than1 m (this could be checked by excluding every second data point from the analysis).For this dense sampling interval, and given error-free borehole orientation data, thedivergence between the true trajectory and the computed trajectory is expected to beAppendix A. INCLINOMETER DESIGN AND DATA PROCESSING 224at most a few centimetres. Problems arise because the borehole orientation data is noterror-free.The UBC inclinometer has logged 125 boreholes during the 1989 and 1990 fieldseasons at Trapridge Glacier, Yukon. Each hole was logged at least twice. By comparingthe inclinometry results, we estimate that the instrument locates the bottom of a 70 mhole to within 20—30 cm. The error tends to be largest in the azimuthal sense; theradial deviation of a borehole is determined to within 15 cm. Based on this evidence,we identify two major sources of error: random positioning error of the inclinometer inthe borehole and systematic orientation error. Superimposed on these error terms arethe measurement errors of the tilt sensors and magnetometer.Accurate positioning of a non-tracked inclinometer within a borehole is dependenton the texture of the borehole wall and the performance of the centring devices. Glacierborehole walls are not necessary smooth and the centring devices allow the inclinometerto cant relative to the borehole axis. The discrepancy between the axis of the boreholeand the axis of the inclinometer is estimated at +0.5°. This estimate is based onthe diameter of the borehole and that of the inclinometer with the centring devicesfully compressed. It is also likely that the inclinometer tends to underestimate theborehole tilt since tension in the cable and the instrument weight work to force theinstrument towards a vertical orientation. As this effect depends on the lay of thecable and the local hole geometry, it is impractical to quantify; we satisfy ourselveswith the +0.5° positioning error. The tilt sensor error (less than one part in 10000) isneglected. The magnetometer error is appreciable and amounts to (±2%) on the XBand YB axes, and (+3%) on the ZB axis. These estimates are based on the calibrationfit and the specifications for the magnetometer. Neglecting stretch in the cable, theerror in positioning along the borehole is estimated at 1 cm for each 1 m interval.The effects of the random errors can be modelled using Monte Carlo techniques.Synthetic data are generated for a sequence of stations with error terms superimposedAppendix A. INCLINOMETER DESIGN AND DATA PROCESSING 225on the “correct” data. The standard deviation of the Gaussian-distributed error for eachreading is given above. Relative contributions to the net deviation can be examinedby selectively removing the error terms. Figure A.8 shows the results from a series ofMonte Carlo tests on a 70 station hole (69 m long). The figure shows horizontal mapsof the computed bottom locations for each Monte Carlo test. The underlying model isa linear hole dipping 100 to the south with the top at the origin. Each test contains1000 runs through the hole. Figure A.8a shows the effects of applying all error termssimultaneously. The cross hair indicates the location of the error-free hole bottom at11.98m south. Note the slight stretching of the “data cloud” in the azimuthal (eastwest) sense and the offset towards the north (origin). Both of these effects are primarilydue to error introduced by the magnetometer, as can be demonstrated by removing themagnetometer and depth error (Fig. A.8b). This results in a more symmetric cloud,with less noticeable offset. Figure A.8c shows how the magnetometer introduces thesetwo distortions. The east-west lineation is actually a short section of a circle with itscentre at the origin. This is easily seen in Figure A.8d where gross errors (±50%) in themagnetometer over one station interval result in a circular scattering envelope. Again,the cross hair indicates the error-free solution. The foreshortening (drawing in towardsthe origin) observed in Figures A.8a, A.8c, and A.8d results from the fact that errorin azimuth will always cause the radius of the interpolating circular arc to decrease;this results in a horizontal projection that is always smaller than, or equal to, thetrue projection. On average, error in tilt measurements also produces foreshortening(Fig. A.8b). Note that for holes of this depth, the foreshortening amounts to no morethan a few centimetres. In Figure A.8a, the net depth error amounts to no more than±2cm.Appendix A. INCLINOMETER DESIGN AND DATA PROCESSING0)a)a)E0a,a,EFig. A.8: Monte Carlo modelling of the sensitivity of the inclinometer toerror in its sensors is shown as a series of scatter plots. Panels (a), (b), and(c) show the horizontal deviation of the computed hole bottom from its trueposition for 1000 Monte Carlo runs down a 69 m hole dipping southward at10 (station interval of 1 m). The hole top is at the origin and the true bottomposition is indicated by the cross hair. (a) The combined effect of a +0.5error in tilt, a 2% error in m and m, a 3% error in m, and a 1 cmm errorin depth control. (b) The effect of tilt error alone. (c) The magnetometer erroralone. (d) Scatter produced by allowing gross (50%) error in the componentsof ih over one station interval.226—11.8U,a)a)E...:i—“C,..&. ‘T.!lI_L—12.—6 —.4 —.2 .0 .2 .4metres.2C.10.00—.10=20 —.10 .00metres.10 .20Systematic error is caused by the non-random orientation of the inclinometer asAppendix A. INCLINOMETER DESIGN AND DATA PROCESSING 227it moves through the borehole. The orientation of the inclinometer does not changegreatly as it moves from station to station, but in separate passes through the hole,the general orientation of the inclinometer may be quite different. It follows that anyoffset error in determining the angle 1 between the XT and XB axes will accumulatedifferently in different inclinometer passes, and this will result in additional azimuthalerror. By adjusting the value of Il such that the azimuthal error is minimized for allholes, we can remove much of the systematic error.Our estimated positioning error of 20—30 cm, with slightly better radial control, isconsistent with the results of the Monte Carlo test in Figure A.8a. This leads us toconclude that we have a good understanding of the factors influencing the performanceof our inclinometer.A.9 DiscussionOur experience with the UB C inclinometer has shown that it is an efficient and accurateinstrument for glaciological work. Surveys are performed quickly and do not require anyspecial attention to the instrument or preparation of the borehole. For these reasons,we expect that magnetically-oriented inclinometers using fluxgate magnetometers willgain popularity with glaciologists. For rigorous applications, some researchers maywish to combine the better tilt accuracy of a tracked inclinometer with the additionalcheck on orientation provided by a magnetometer.The principal shortcoming of these instruments is that they become confused atvery high magnetic latitudes. We estimate that the UBC inclinometer will not operatereliably where the dip of the magnetic field exceeds 85°. This constraint probably excludes its use throughout most of the Canadian arctic archipelago, and the Wilkes Coastand George V Land in Antarctica. We know that the accuracy of the magnetometeris being degraded by its close proximity to electrical currents inside the pressure tube.Redesigning the internal configuration of the instrument so as to avoid this magneticAppendix A. INCLINOMETER DESIGN AND DATA PROCESSING 228noise would reduce the error observed in Figure A.8c, and might allow the instrumentto be used closer to the Earth’s magnetic poles.These restrictions aside, the error in determining the position of subglacial and en-glacial sensors with the UB C inclinometer is small enough for all but the most exactingexperiments. We are seeking to improve the performance of the instrument, primarilyby redesigning the centring springs. If this results in reduced error in the tilt readings,then the instrument might approach the sensitivity represented by Figure A.8c.The circular-arc interpolation scheme is simple to apply and easy to visualize. Ona 12 Mhz IBM-PC compatible computer equipped with a numerical co-processor, thetime required to calibrate, process, and plot the data from a 70 m borehole takes afraction of a second. The foreshortening effect of the circular-arc method is the onlydrawback of which we are aware, but this effect is not unique to this interpolationmethod.We expect that improvements in the accuracy of inclinometry data inversion canbe obtained in at least three ways: (1) Incorporate additional borehole geometry andcentring device geometry. (2) Study the effects of centring device spacing relative tostation spacing. (3) Introduce full inversion techniques that produce trajectories havingan imperfect fit, within error, to the data. Data smoothing is one strategy for findingthese imperfectly-fitting trajectories, but it is important that the smoothing processdoes not result in violation of the error bounds on the data. We invite other interestedparties to consider these approaches.Appendix BSUBGLACIAL WATER AND SEDIMENT SAMPLERSThe current focus of glaciological research on basal processes and hydrology makesthe acquisition of samples from the subglacial environment a vital enterprise. In thisletter, we describe two devices for obtaining samples of basal water and sediment withinthe confines of a narrow borehole. The samplers have been operated at depths rangingfrom 70—300 m. They are lightweight and require only a single operator.The substance of this appendix has been published previously in the Journal ofGlaciology (Blake and Clarke, 1991a).B.1 Niskin samplerCollecting water at depth has been a concern of oceanographers for centuries (seeMcConnell, 1982); designs for sampling bottles abound and are slowly modified bygenerations of researchers. The modern Niskin sample bottle consists of an open-endedtube which can be closed on command by a pair of stoppers. A Niskin bottle, attachedto a wire rope, is lowered into the water and when the bottle reaches the desired depth,a messenger block is dropped along the rope. The block strikes the Niskin bottle andtrips the sampling mechanism.We have designed a modified Niskin sampler having a trigger mechanism thatoperates axially. This action allows the device to operate in a narrow borehole. Figure B.1 shows the sampler in its open, cocked position. The sampler consists of fourmajor units that move relative to one another: (1) The lower stopper is fixed to ahollow central rod. The central wire rope upon which the sampler is suspended passesthrough the rod and is held by a crimp at the bottom. (2) Two perforated brass disksare fixed within a Plexiglas sampling tube. The disks slide on the central rod and the229Appendiz B. SUB GLA CIAL WATER AND SEDIMENT SAMPLERS 230perforations allow water to move through the tube when the stoppers are open. (3) Thehead block, with the upper stopper attached, is free to slide on the central rod, but twospring-loaded catches hold the block in a cocked position at the top of the rod. Twolengths of fine wire rope suspend the Plexiglas tube below the head block. The wireropes are attached to small eye-hooks on the block and upper disk (for clarity, thesefixtures are not shown in Figure B.1). (4) A brass messenger block slides along thecentral wire rope.The sampler, in a cocked position, is lowered into position at the borehole bottomand the messenger block is dropped along the wire rope. When the messenger strikesthe catches, they spread apart and release their grip on the central rod; the head blockfalls against the sampling tube, the tube falls against the lower stopper, and a 220 mLwater sample is trapped inside the device. The weight of the upper block ensures awatertight seal between the two stoppers and the tube. The sampler is opened, and thesample collected, by pushing on the top of the rod. Experience has demonstrated thatwire rope(‘-‘-i0.5 mm diameter) must be used to connect the block and disk becauseweaker materials (such as fine chain or string) will break.The modified Niskin sampler is simple to operate and performs reliably; accidentaltriggering of the mechanism is unusual. Since the introduction of the Niskin samplerto our field program in 1986, we have obtained basal water samples from more than300 boreholes of 70 m depth and one sample from a 300 m borehole. Samples from theglacier bed often contain significant quantities of fine particulate matter, and sometimessubglacial material is found adhering to the lower stopper. The axial design allows thedevice to take samples in a borehole as narrow as 31.8 mm (1.25 in) in diameter.B.2 Subglacial vacuum samplerThe modified Niskin sampler excludes the sand, pebble, and gravel fractions of the bed.In order to collect these larger-sized particles, we built an active vacuum sampler thatAppendix B. SUBGLACIAL WATER AND SEDIMENT SAMPLERS 231*E REcArd-Es1-EAD BLOOCSTCPFIE REF9FORATED XKW4TRAL RODSAfl1 REETOFFFig. B.1: The modified Niskin sampler in a cocked position. As themessenger block strikes the catches, the head block is released from the top ofthe central rod and slides down to seal the top of the sample tube. The sampletube falls against the lower stopper; this seals the bottom of the sample tubeand traps a water sample inside the powered by the high pressure pump on our hot-water drill. Because it works like avacuum cleaner, we dubbed this device the “Hoover”.The Hoover design was inspired by the airlifts used to clear sediment from shallowsubmarine archaeological sites. According to Throckmorton (1969, p. 175), airlifts wereES5ENGO OG(cRtpinvented at the turn of the century for clearing mine sumps and for mud-pumping inAppendiz B. SUB GLA CIAL WATER AND SEDIMENT SAMPLERS 232harbours. The first archaeological application was by Jacques Cousteau in the early1950s at the Mediterranean site of Grand Congloué near Marseilles, France. An airliftconsists of a large diameter flexible pipe (perhaps 15—20 cm in diameter) leading fromthe archaeological site at the sea floor to the deck of a support vessel. Injection nozzlesmounted inside the pipe force air into the pipe. The presence of air into the pipe hastwo consequences of interest: (1) Viscous drag between the rising air bubbles and thewater in the pipe causes upward flow. (2) The bulk density of the mixture in the pipeis lowered to such a degree that buoyancy forces lift the mixture up the pipe. Buoyancyforcing is the stronger of the two effects. Given sufficient air content, water in the pipecan be lifted well above the free water surface and onto the deck of the vessel. Artifactssmall enough to be entrained by the water flow can then be collected in a sieve. Anadvantage of this system is that no impellers are required to drive water through thepipe; artifacts are not damaged and the pipe is unlikely to clog.The subglacial Hoover uses viscous drag from water jets (rather than air bubbles)to move water up through a Plexiglas tube that has a one-way valve at the bottom.Figure B.2 shows the design of the Hoover with an exploded view of the valve assembly. Cold water from our hot-water drilling system is fed into the 37°JIC swivel hosefitting. The water travels down the central feed pipe to the nozzle assembly wherethe direction of flow is reversed as the water emerges through two 1.59 mm (0.0625 in)nozzles. Vibration of the nozzle assembly is prevented by three pins that reach outto meet the inner walls of the Plexiglas sample tube. The pressure drop across thenozzles is about 2 MPa at a flow rate of 18 L miri’. The valve consists of a cylindricalrip-stop nylon fabric sock. The lower edge of the sock is sewn to a sock cage; the cageconsists of a brass ring that is brazed to a tetrahedral wire frame. The three wires inthe cage prevent eversion of the sock, but also reduce the maximum particle size thatcan be admitted by the valve. A threaded retainer ring holds the valve assembly ontothe valve seat. The bulkhead at the top of the Hoover has eight 11.1 mm (0.4375 in)Appendix B. SUBGLACIAL WATER AND SEDIMENT SAMPLERS 233holes drilled around its periphery; these holes provide an outlet for the injected andentrained water.HOSE RTTtIGBLLKI-EADSAMPLEVALVE SEATVALVE SOOCOil_______RETMERFig. B.2: An exploded view of the subglacial Hoover. Upward- directedwater jets are created by forcing water through the feed pipe and nozzle assembly. Viscous drag between the jets and the water in the sample tube createsa vigorous upward flow of water through the one-way sock valve. Sedimententrained by the flow is trapped inside the tube when the jets are turned off.When the Hoover is operated in air, a noticeable suction develops at the inlet.This suction is much greater when the device is operated in a water-ifiled borehole;C0C”1U’FEED PFENOZZLEASSEtLY0we have found clasts wedged into the bulkhead outlet holes, indicating that clasts ofAppendix B. SUBGLACIAL WATER AND SEDIMENT SAMPLERS 234considerable size are being driven forcibly through the sample tube. The largest claststhat we have collected have a typical grain diameter of —20—25 mm.Samples are taken using the following procedure. With the Hoover at the bottomof the borehole, the water supply is turned on and the pressure hose is gently movedup and down causing the Hoover to move against the glacier bed. After 20—40 s, thewater supply is turned off, and the sampler is hauled to the surface. Examination of theclosed valve assembly suggests that the valve does not close suddeiily as the water jetsstop, but rather that the valve is slowly pinched shut by sediment falling between thevalve sock and the sample tube. The sediment is released from the Hoover by removingthe retaining ring and valve assembly. Samples as large as 500 g have been obtained(for a representative sample, see Clarke and Blake, 1991, Fig. 11).B.3 ConsiderationsThe hot-water drilling system used at our field site on Trapridge Glacier, Yukon Territory, Canada, alters the glacier bed as each borehole is completed; surface water isintroduced to the subglacial environment and some fine material may be flushed away.In addition, the geometry of the both sampler inlets further biases the sampling of solidmatter from the bed. These effects must be considered during sample analysis.The Hoover tends to eject finer particles through the water outlet, although oursamples do contain a significant amount of coarse and fine sand. A screen placed overthe outlet might improve the retention of finer material.Appendix CSUBGLACIAL DRAG SPOOLC.1 IntroductionOne of the problems discussed in Chapters 2 and 3 is the partitioning of ice motionat the base of the glacier between basal sliding and deformation. The instrumentsdescribed in Chapter 2 make measurement of basal deformation, but the analysis ofthe data from these sensors cannot be complete without some measure of basal sliding(Figure 3.2).To our knowledge, only one prior attempt to measure basal sliding has been reported in the literature: Boulton and Hindmarsh (1987) screwed an auger-like anchorinto the sediments beneath the terminus of Breidamerkurjökull. The anchor was inserted through a sealed hole in the floor of an artificial ice tunnel and connected bystring to a chart recorder; as the anchor string payed out, the recorder took note. Theauthor is also aware of measurements made in a natural ice cave beneath the terminusof Isfallsglacinren, near the well-studied Storglaciren in Swedish Lapland. At Isfallsglaciären, an ice screw was fixed in the moving ceiling of the cave; a string connectedthe ice screw to a chart recorder on the floor of the cave. In this case, the glacier wassliding on a bedrock base; there was no subglacial deformation.C.2 The “Slide-O-Meter” or “drag spool”The instruments we designed for use under Trapridge Glacier is similar to the Boultonand Hindmarsh design only in that an anchor is placed in the bed and the amount ofstring payed out is measured. Our instrument can be installed in situ at the bottom of235Appendix C. SUBGLACIAL DRAG SPOOL 236a deep borehole and has been honoured with two names: the “Slide-O-Meter” and the“drag spool”. This last term is the more dignified of the two and will be used here.Figure C.1 shows a schematic diagram of the drag spool instrument. The percussion hammer (described in Chapter 2), fitting with a 0.635 cm (0.25 in) dowel attachment, is used to hammer an anchor into the glacier bed. A 0.635 cm socket is drilledinto the back of the anchor tip; the dowel fits into this socket and the drag spool casingis loosely attached to the dowel. After inserting the anchor, the hammer and dowelare withdrawn, leaving the anchor and the drag spool at the bottom of the borehole.The drag spool cable is then pulled tight. As the cable freezes into the borehole, thedrag spool becomes fixed in the ice; as sliding occurs, the anchor will distance itselffrom the glacier borehole and the string spool will turn as the string pays out. A5 k1 potentiometer is connected to the spool so that turns of the spool can be registered electrically from the glacier surface. Approximately 2 m of thin nylon string arestored on the spool — enough to last about four weeks at anticipated sliding rates of0—l0cmday’.It should be noted that the drag spool will place only an upper limit on glaciersliding because the spool anchors are placed within the deforming sediments and someof the observed relative motion between the anchor and the ice can be caused bydeformation of the intervening sediment.Appendix C. SUBGLACIAL DRAG SPOOL 237CABLECASESPOOLPOTENTIOMETERSTRINGANCHORFig. C.1: A schematic diagram of the drag spool. As the string attachedto the anchor is payed out, the potentiometer screw is turned and the resistance change can be measured.II1 cm


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