Finite element modelling of the creep behaviour of a small glacier under low stresses F. Azizi," S. Jim^ & W.B. Whalley" o/ o/ ABSTRACT Knowledge of the creep properties of ice or ice-rich materials is important in a variety of geological features. Although the behaviour of large ice masses such as glaciers and ice sheets has been considered in great detail in the past decade or so and is thought to be reasonably well understood, the deformation (or rate of deformation) generated by low stresses are yet to be properly investigated. Such conditions may occur where snow patches are preserved under debris covers and may remain for many years, undergoing a slow deformation over that period of time. Similarly, the long term creep behaviour of ice-debris mixtures is little understood. This problem occurs especially where ice content of granular materials is well below saturation level, or when the debris is dispersed in thin ice masses. The paper presents a contribution to the on-going debate related to the nature of behaviour under the above stated conditions. It includes a geomorphological analysis of the problems and, more importantly, presents a finite element modelling of the creep behaviour of a small glacier underlying a thin layer of debris. The numerical analysis is based on a power law creep type equation for which parameters are adjusted to take into account the debris-ice interaction. The paper also presents a brief analysis of the specific case of the effect of size and position of an ice mass on the stability of a debris slope. GEOMORPHOLOGICAL ASPECTS OF ICE-DEBRIS CREEP Monitoring the long term behaviour of ice creep in the laboratory may be complemented by field observations which might be used to extend observations over many years. Extended creep tests are likely to be of use in certain engineering cases where ice or ice-rock mixtures are concerned. An example might be building piled structures in permafrost areas.
366 Marine, Offshore and Ice Technology Normal glaciers can give important creep data but, because of their thickness, will still deform relatively rapidly (of the order of tens of meters per year). Very thin ice lenses, say <5m, occur in a variety of geomorphological forms (Washburn [7]). Under observation periods of a few years, most permafrost (permanently frozen ground below an "active layer" of seasonally thawing soil) show no creep. Thin ice bodies (a few tens of meters) would be a wedge of ice covered by a debris layer such that ice ablation is essentially nil. It would be possible for such a body to be at pressure melting point (isothermal) conditions, a situation that is difficult to achieve in the laboratory for sustained periods. Similarly, natural laboratories of this kind could be used to evaluate creep of mixed ice and rock or mixed debris, about which there is still little published work. However, there are fundamental difficulties about this approach. Knowledge of ice purity or ice/rock mixture ratios and of the thermal regime are very uncertain. These natural conditions are variable and many different solutions could be applied (Whalley and Martin [9], Whalley and Azizi [8]). Creep rate is especially significant when past climate change has to be taken into account. Geophysical exploration (e.g. use of seismic or resistivity properties to determine thickness) is also unable to assist with accurate thickness determinations because of the need to apply an appropriate mixture model (Whalley and Azizi [8]). This is unfortunate as it is one of the very things we would wish to know in a "field laboratory". Although the possibility of using a geomorphological feature as a test bed for long-term creep tests is fraught with problems, modelling ice wedge/lens behaviour does suggest some ways of resolving this problem. Conversely, knowledge of creep behaviour could help solve some geomorphologic problems associated with buried ice and ice-rock mixtures (Whalley and Martin [9]). A variety of models have been proposed but the simplest situation is that of a wedge of ice covered by an insulating debris layer. Such a condition might be in a permafrost regime (mean annual air temperature [m.a.a.t] <-2*C) and a correction factor for temperature would have to be applied. Yet, it could also operate where the m.a.a.t. is about 0 C. These models have different environmental interpretations, although these need not concern us here. Features such as depicted in Fig.l (protalus lobes) have low creep rates. The only accurate observations taken on this kind of feature so far come from Svalbard (Sollid and S0rbel [5]). Figure 1. Protalus lobes (arrows) in Northern Norway.
Marine, Offshore and Ice Technology 367 Surface velocities, measured by electronic theodolite over the period 1986-1990, show 10 cm per year near the front and 3 to 5 cm per year away from the front. Other features show no movement at all during this measurement period. Although it has been postulated that the creep reflects the movement of an ice rock mixture (of unknown composition), it is possible that a buried thin lens of ice could deform to give such velocities. This is what the finite element model shows in this paper. FINITE ELEMENT MODELLING. A variety of rheological models have been proposed for the features known as rock glaciers and protalus lobes which have surface velocities of up to 1m per year (see for example Whalley and Martin [9], Whalley and Azizi [8]). These different models have different environmental implications (which are being explored elsewhere) in both climate regime and as indicators of formational environment. There remains however a pressing need to compare the behaviour predicted by the models to the actual movement characteristics measured in the field. Finite element modelling provides a link between possible models and actual behaviour over various periods of time. In particular, the thermal conditions (e.g. permafrost with mean annual air temperature < -2 C) can be applied as boundary conditions affecting the flow law. Actual velocities of protalus lobes have relatively small values ranging from zero to 0.2m per year. In this paper, a simple geometrical model is used (Fig.2) to depict creep of a solid ice body subjected to a loading generated by a thin debris layer which acts as an insulating layer at the same time. As well as the velocity, it may also be possible to judge the veracity of the model by comparing the topographic landforms (once creep deformations have occurred) with those computed with the finite element model. 70 m Figure 2. Geometrical model used for finite element modelling. FORMULATION OF THE CREEP BEHAVIOUR. The finite element method (FEM) has been accepted as a general powerful tool for solving a wide range of physical problems. Its strength lies in its capability to be applied to complicated problem geometry, boundary conditions and any non-linearity in material behaviour. Over the last two decades, the FEM was
368 Marine, Offshore and Ice Technology used to model a wide variety of problems related to the flow of glaciers (Hooke et al. [3], Hodge [2], Stolle and Killeavy [6]). The method adopted in this paper is based on a solid mechanics approach using displacements as unknowns. The formulation allows for large strains and displacements to be taken in due account. The physical problem presented here relates to the dynamic behaviour of a small glacier as shown on Fig.2. For modelling purposes, the glacier was divided into six different slices (a to/ ). Zones a and/ were in turn divided into 15 (three nodes) triangular elements, whereas 36 (four nodes) quadrilateral elements were used for zones b,c,d and e (Fig.3). The loading at the top was simulated by a 0.5m thick layer of debris having a density of 2 Mg/nP. The pressure created by this loading was transmitted to the ice mass through the nodes as shown on Fig.3. Along the slip surface, non-slip conditions were imposed (conditions represented by small circles at the bottom of the mesh in Fig.3). Figure 3. Finite element mesh and boundary conditions. The creep is modelled with a power law linking the effective strain rate and the effective shear stress T (Paterson [4]): where 6 = (1) (2) (3) A = Aoexp[-Q/RT] (4) AQ = fluidity coefficient Q = activation energy (= 6 xlotf/mol for T < -10*C, = 1.39x105 J/mol for T > -10'C) R = universal gas constant (= 8.314 J/mol/K) T = absolute temperature in K
Marine, Offshore and Ice Technology 369 The computation is based on a standard finite element formulation. An implicit equilibrium enforced at the end of time-step scheme was adopted for time marching to ensure stability while maintaining efficiency. The time increment size was automatically adjusted depending on the creep rate. Neglecting the initial elastic deformations, the creep response of the glacier to the applied loading is, in part, dependent on the value of the coefficient A in eql which in turn is temperature dependent. To take into account the effect of temperature variations, three values of A were used as shown on Table 1. Table 1. Effect of temperature on fluidity coefficient (Paterson [4]). I/o/^i\ ( C) i /r& Q 1 \ A (Pa-3.s-^ 0-10 5.3x10-24 5.2x10-25 -25 10-25 The creep results presented on Fig.4 correspond to both horizontal (6%) and vertical ( y) displacements under the above mentioned conditions, of point C (represented by a solid circle on Fig.3). The figure shows that for T= O'C, a steady state is reached after about 2,000 years whereas a change of temperature to -15*C under the same loading conditions leads to a steady state after a time of around 10,000 years (a very short time by geological standard) has elapsed. A further decrease in temperature to -25*C has a dramatic effect on the creep behaviour in that the deformation occurs at practically the same slope in both x and y directions at an approximate rate of 0.15 mm per year. T - -25 C T - -15 C Pigure 4. Lateral and vertical displacements at C under different temperature conditions
370 Marine, Offshore and Ice Technology Around the melting point, the deformed mesh of the glacier is shown on Figs.Sa to e. As indicated on Fig.4, the bulk of displacement in this case (0*C) happens within 2,000 years. Deformation at Year Deformation at Year 500 Deformation at Year 100J Deformation at Year 1500 Deformation at Year 200 Figure 5. Deformed shape of the glacier with time.
Marine, Offshore and Ice Technology 371 CONCLUSIONS Natural features, especially those which behave slowly in terms of human observation periods, are difficult to investigate. This is especially the case where direct drilling or digging is a problem through cost or remoteness. Yet it is important to know how such features respond (or have responded) to climate change. Further, in some areas, permafrost may provide the only possible foundation for structures such as cable car stations or mine buildings. The long term observational problems, whether geotechnical or geomorphological, can be remedied by linking observations of features (e.g. Fig.l) with short term (present day) flow data. The FEM allows recognition of temperature variations, which can dramatically alter flow rates as well as the material properties.the approach used here is a start in making valid interpretations of geomorphological features and engineering behaviour of ice and frozen soils. ACKNOWLEDGEMENTS. Special thanks are due to various people with whom we have discussed the content of this paper. BIBLIOGRAPHY [ 1 ] Azizi, F. 1992. Assessment of the mathematical modelling of the creep of ice and frozen soils. Proceeding of the third International Conference on ice Technology. M.I.T, Boston. USA. pp.3-14 [2] Hodge, S.M. Two dimensional time dependent modelling of an arbitrarily shaped ice mass with the finite element technique. Journal ofglaciology vo!31, 109.pp 350-359. [3] Hooke, R.B, Raymond, C.F, Hotchkiss, R.L, and Gustafson, R.J. 1979. Calculations of velocity and temperature in a polar glacier using the finite element method. Journal ofglaciology, Vol.24, 90.pp 131-146 [4] Paterson, W.S.B. 1981. The Physics of Glaciers. Pergamon press. [5] Sollid, J.L and S0rbel, L. 1992. Rock glaciers in Svalbard and Norway. Permafrost and Periglacial Processes. 3, pp.215-220 [6] Stolle, D.F.E and Killeavy, M.S. 1986. Determination of particle paths using the finite element method. Journal ofglaciology. vol 32, lll.pp 219-223 [7] Washburn, A.L. 1979. Geocryology. Arnold, London.. [8] Whalley, W.B and Azizi, F. 1994. Rheological models of active rock glaciers : evaluation, critique and a possible test. Permafrost and Periglacial Processes. 5, pp.37-51. [9] Whalley, W.B and Martin, H.E. 1992. Rock glaciers : II models and mechanisms. Progress in Physical Geography. 16, pp. 127-186.