This repository contains a set of models to diagnose slush formation in thermally-drilled glacier boreholes. This work was done to aid in design of a hot-point drill, the Ice Diver, at the Applied Physics Lab, University of Washington. Please cite the associated text (Hills et al., 2020) anywhere that this repository is used.

Mechanically-drilled glacier boreholes are easily stabilized with an antifreeze solution and can be held open for years. However, attempts to stabilize thermally-drilled holes have resulted in a plug of slush freezing in the hole which effectively freezes the hole shut. In essense, molecular diffusion (movement of the solute particles in the solution) is at least an order of magnitude slower than thermal diffusion in the solution, which creates an area of constitutional supercooling inside the hole (Worster, 2000). Therefore, refreezing happens in small particles within the solution rather than accretion on the borehole wall.

Prior work from Humphrey and Echelmeyer (1990) described the physics of slush formation in the hot-water case.

The obvious strategy to avoid slush formation is to add enough antifreeze to avoid freeze-back entirely. Unfortunately for the hot-water drilling case, the drilling process has accumulated enough energy as heat in the ice outside the borehole that even pure antifreeze would not prevent freezeback (Humphrey and Echelmeyer, 1990). One recommendation that they provided is drilling with an antifreeze solution; although, that creates its own engineering difficulties. While the hot-point drilling case is significantly slower, one possible benefit is the opportunity to inject antifreeze directly behind the downgoing drill. In this case, enough antifreeze could be added to stabilize the hole without freezeback. We use the models in this repository to explore that idea.

We document the properties for commonly used antifreeze agents (ethanol and methanol) within a few scripts here. The liquidus line from the Industrial Solvents Handbook (Flick, 1998) is saved as a function of percent by mass:

• ./data/methanol_freezingdepression_PBM.npy
• ./data/ethanol_freezingdepression_PBM.npy

Other constants that are used in the models:

• ./cylindricalstefan/lib/constants.py

Dimensional conversions used in the models:

• ./cylindricalstefan/lib/concentration_functions.py

The mathematical setup is a Stefan problem in cylindrical coordinates.

1. Analytical Solutions

There is an analytical solution for melting with a heat source that stays at the center of the hole (i.e. r=0) derived in Carslaw and Jaeger (1959; sec 11.6). We derive an analogous solution that better fits our case where the heat source follows the borehole wall as it moves out. While there is no true analytical solution for the freezing case, Crepeau and Siahpush derived an 'approximate' solution which we provide here as well. All three solutions are provided as functions in:

• ./cylindricalstefan/lib/analytical_pure_solution.py

The above functions are for the conventional Stefan problem with pure water in the borehole. The problem is significantly complicated by the addition of antifreeze into the borehole solution. Most of the theory on freezing into a binary solution was developed by Worster (2000). His theory is all in cartesian coordinates, but we provide some of the useful functions here as a reference:

• ./cylindricalstefan/lib/analytical_binary_solution.py In the above script, there are also functions for cylindrical coordinates, these are still in development and should only be used experimentally.

2. Instantaneous Mixing Model

This is a numerical model which uses the finite element software FEniCS. This model simulates borehole evolution through melting and refreezing, including with an antifreeze solution. It assumes that the borehole solution is always mixed instantaneously, so any time that the borehole wall moves, the solution concentration and the corresponding freezing temperature are updated immediately. This follows Humphrey and Echelmeyer (1990) exactly and is the predominant model explored by Hills et al. (2020).

• ./cylindricalstefan/lib/instantaneous_mixing_model.py

The instantaneous model is extended to two dimensions for testing any influence that advection of the downgoing drill may have on the ice temperature. As stated in Hills et al. (2020; Supplementary Material S.3), at short timescales (hours) or far distances behind the drill (~500 times the borehole radius in this case) the problem is exactly one-dimensional.

• ./cylindricalstefan/lib/thermal_model_2d.py

3. Double Diffusion Model

This was meant to be a more rigorous model which allows for thermal and molecular diffusion within the borehole (i.e. the mixing is no longer instantaneous). Unfortunately, we have had some numerical issues with this version of the model, so it is still in development.

• ./cylindricalstefan/lib/double_diffusion_model.py

Unit testing is done for all scripts.

• ./cylindricalstefan/tests/

Python 3 (other versions may work, but they are not tested). Also, numpy and scipy.

FEniCS. I recommend either the Anaconda or the Docker install.

• Carslaw, H. S., & Jaeger, J. C. (1959). Conduction of Heat in Solids (Second). London: Oxford University Press.
• Crepeau, J., & Siahpush, A. (2008). Analytical solutions to the Stefan problem with internal heat generation. Heat and Mass Transfer, 44, 787–794.
• Flick, E. W. (1998). Industrial Solvents Handbook (5th ed.). Westwood, NJ: Noyes Data Corporation.
• Hills, B. H., Winebrenner, D. P., Elam, W. T., Kintner P. (2020). Avoiding slush for hot-point drilling of glacier boreholes. Annals of Glaciology.
• Humphrey, N., & Echelmeyer, K. (1990). Hot-water drilling and bore-hole closure in cold ice. Journal of Glaciology, 36(124), 287–298.
• Worster, M. G. (2000). Solidification of Fluids. In G. K. Batchelor, H. K. Moffat, & M. G. Worster (Eds.), Perspectives in Fluid Dynamics. Cambridge University Press.