Multiphase flow modeling with DAE/MOL methods

Modeling multiphase flow through porous media is a prerequisite for modeling common subsurface phenomena. Standard modeling approaches result in a complex class of nonlinear partial differential equation formulations, most of which must be solved with specialized numerical methods software. Software development, model formulation, and numerical solution are thus significant stages of the environmental modeling process.; Object-oriented techniques have been advocated for developing complex software such as environmental simulators. Efficiency is a fundamental concern, however, and we first confirm that the application of these techniques to numerical modeling software will not incur a significant performance penalty. We implement adaptive integration methods for differential-algebraic equation in the object-oriented programming language C++ and compare to procedural Fortran software. Quantitative comparisons demonstrate the efficiency of the object-oriented C++ approach and yield recommendations for further software development.; A method of lines approach has been advocated for the simplified Richards' equation multiphase model. We derive full two-phase model formulations suitable for this approach and propose a formulation of the Richards' equation model that addresses some of the shortcomings of earlier work. We analyze the formulations in the theoretical framework of differential-algebraic equations and from the perspective of mass conservation for discrete numerical models.; We apply the multiphase simulator to several one-dimensional problems to study higher order temporal discretizations. A set of numerical experiments demonstrates that higher order temporal discretizations are more efficient than standard first-order methods for a variety of practical grids and integration tolerances and that mass balance can be maintained efficiently for variable-order integration using appropriately formulated models.; Two- and three-dimensional simulations require effective preconditioners for the linear systems which arise in the numerical solution approach. We studied multilevel Schwarz domain decomposition preconditioners based on aggregation concepts because of their effectiveness on related model problems. We implemented the methods and performed a set of numerical experiments to evaluate the behavior of the methods when applied to multiphase flow models. The results demonstrate consistent reductions in iteration counts and, thus, that further study of the scalability of these methods is warranted.