We present a mathematical model for the breakdown of porous rock by the growth of ice within cracks. The model is founded upon well-established principles of fracture mechanics and recent advances in soil physics, iJong with the assumption that progressive crack growth results from water migrating to ice bodies in cracks, much as water migrates to ice lenses in freezing soil. Our model predicts crack-growth rates compatible with empirical data. Calculations for a granite and a marble indicate that sustained freezing is most effective in producing crack growth when temperatures range from —4 °C to -15 °C. At higher temperatures, thermodynamic limitations prevent ice pressure from building up sufficiently to produce significant crack growth; at lower temperatures, the migration of water necessary for sustaining crack growth is strongly inhibited. In hydraulically "open" systems, in which pore-water pressure remains near atmospheric pressure during the freezing process, crackgrowth rates during continuous cooling will generally be greatest at low rates of cooling, less than -0.1-0.5 °C/h. At higher rates of cooling, the influx of water to growing cracks is significantly inhibited. The model delineates clearly the role of material parameters (elastic moduli, fracture-mechanical properties, grain size and shape, and crack size), environmental conditions (temperature, temperature gradient, water pressure), and time in frost damage to rocks. Our calculations, along with recent experimental work on water migration in freezing rocks (Fukuda, 1983), lead us to question the widely accepted importance of two phenomena—freezing of water in sealed cracks and freeze-thaw cycling—in the fracture of rock exposed to natural freezing conditions.

A theoretical model of the fracture of rock during freezing

In this paper, we present a phase field model for a fluid-driven fracture in a poroelastic medium. In our previous work, the pressure was assumed given. Here, we consider a fully coupled system where the pressure field is determined simultaneously with the displacement and the phase field. To the best of our knowledge, such a model is new in the literature. The mathematical model consists of a linear elasticity system with fading elastic moduli as the crack grows, which is coupled with an elliptic variational inequality for the phase field variable and with the pressure equation containing the phase field variable in its coefficients. The convex constraint of the variational inequality assures the irreversibility and entropy compatibility of the crack formation. The phase field variational inequality contains quadratic pressure and strain terms, with coefficients depending on the phase field unknown. We establish existence of a solution to the incremental problem through convergence of a finite dimensional approximation. Furthermore, we construct the corresponding Lyapunov functional that is linked to the free energy. Computational results are provided that demonstrate the effectiveness of this approach in treating fluid-driven fracture propagation.

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Phase-field modeling of a fluid-driven fracture in a poroelastic medium

In this paper we present a quasi-static formulation of a phase-field model for a pressurized crack in a poroelastic medium. The mathematical model represents a linear elasticity system with a fading Gassman tensor as the crack grows, that is coupled with a variational inequality for the phase-field variable containing an entropy inequality. We introduce a novel incremental approximation that decouples displacement and phase-field problems. We establish convergence to a solution of the quasi-static problem, including Rice's condition, when the time discretization step goes to zero. Numerical experiments confirm the robustness and efficiency of this approach for multidimensional test cases.

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A quasi-static phase-field approach to pressurized fractures

Elastic compliance for rock-like materials with random cracks

Iterative coupling of flow, geomechanics and adaptive phase-field fracture including level-set crack width approaches

I. N. Sneddon/M. Lowengrub, Crack Problems in the Classical Theory of Elasticity. (The SIAM Series in Applied Mathematics). VIII + 221 S. m. 57 Fig. New York/London/Sydney/Toronto 1969. John Wiley & Sons, Inc. Preis geb. 140 s. net

W. Pompe

Papers

I. N. Sneddon/M. Lowengrub, Crack Problems in the Classical Theory of Elasticity. (The SIAM Series in Applied Mathematics). VIII + 221 S. m. 57 Fig. New York/London/Sydney/Toronto 1969. John Wiley & Sons, Inc. Preis geb. 140 s. net