Pacific Northwest Numerical Analysis Seminar

Abstracts

The computational simulation of Methane Hydrates (MH), an ice-like substance abundant in permafrost regions and in subsea sediments, is useful for understanding of their impact on climate change as well as a possible energy source. We report here on the results from [N. Gibson, P. Medina, M. Peszynska, R. Showalter, "Evolution of phase transitions in methane hydrate", J. Math. Anal. Appl. Volume 409, Issue 2 (2014), 816-833], and consider a simplified model of MH evolution which is a scalar nonlinear parabolic PDE with two unknowns, solubility and saturation, bound by an inequality constraint. This constraint comes from thermodynamics and expresses maximum solubility of methane component in the liquid phase; when the amount of methane exceeds this solubility, methane hydrate forms. The problem can be seen as a free boundary problem somewhat similar to the Stefan model of ice-water phase transition. Mathematically, the solubility constraint is modelled by a nonlinear complementarity constraint and we extend theory of monotone operators to the present case of a spatially variable constraint. In our fully implicit finite element discretization we apply recently analyzed semismooth Newton method, and show that it converges superlinearly, also for other interesting test cases unrelated to MH but covered by the theory. As concerns error estimates and convergence order, we show that they are essentially of first or half-order, depending on the norm (L2 or L1), or the variable (the smooth solubility variable or the non-smooth saturation). These results are similar to those known for the temperature and enthalpy, respectively, for Stefan problem. Our current work is on extending the computational model and analysis to include more variables such as salinity, pressure, temperature, and gas phase saturation, as well as in considering realistic scenarios such as those that may occur in ocean observatories along Hydrate Ridge and Cascadia Margin.