Fully coupled generalised hybrid-mixed finite element approximation of two-phase two-component flow in porous media. Part II: numerical scheme and numerical results

Abstract

We consider the modeling and simulation of compositional two-phase flow in a porous medium, where one phase is allowed to vanish or appear. The modeling of Marchand et al. (in review) leads to a nonlinear system of two conservation equations. Each conservation equation contains several nonlinear diffusion terms, which in general cannot be written as a function of the gradients of the two principal unknowns. Also the diffusion coefficients are not necessarily explicit local functions of them. For the generalised mixed finite elements approximation, Lagrange multipliers associated to each principal unknown are introduced, the sum of the diffusive fluxes of each component is explicitly eliminated and the static condensation leads to a "global" nonlinear system of equations only in the Lagrange multipliers also including complementarity conditions to cope with vanishing or appearing phases. After time discretisation, this system can be solved at each time step using a semi-smooth Newton method. The static condensation involves "local" nonlinear systems of equations associated to each element, solved also by a semismooth Newton method. The algorithm is successfully applied to 1D and 2D examples of waterhydrogen flow involving gas phase appearance and disappearance.

Compositional two-phase flow in saturated–unsaturated porous media: benchmarks for phase appearance/disappearance

“…Two of the participating teams :INRIA-Rocquencourt, France (Jérôme Jaffré, Ibtihel Ben Gharbia) and Friedrich-Alexander Universitat FAU, Erlangen-Nurnberg, Germany (Peter Knabner, Estelle Marchand, Torsten Muller)) ; are using the total hydrogen concentration and the pressure for "persistent" variables ; but they formulate the solubility conditions as complementary conditions, which complement the conservation law equations (see [23] and [32]). Namely : they add the "complementary solubility constraints" to the nonlinear algebraic equations, coming from the discretization of the conservation laws and the constitutive equations.…”

Extending the persistent primary variable algorithm to simulate non-isothermal two-phase two-component flow with phase change phenomena

“…The second benchmark simulates the classical heat-pipe problem, where a thermal convection process gradually develops itself and eventually reaches equilibrium ('Benchmark II: heat pipe problem' section). The numerical results produced by OpenGeoSys were verified against analytical solution and also against results from other numerical codes (Marchand et al 2012). Furthermore, details of numerical techniques regarding how to solve the non-linear EOS system were discussed ('Numerical solution of EOS' section).…”

Energy stable numerical methods for porous media flow type problems

“…In [29], the authors combine a Picard-type fixed point strategy with Newton's method (i.e., they perform a few fixed points iterations before running Newton's algorithm). An alternative approach would consist in keeping both s and p as unknowns together with the additional equation p 2 pðsÞ that is often rephrased as a complementary constraint and then solving the problem with a non-smooth Newton method (see for instance [30][31][32]). Another classical solution consists in partitioning X at each time t into a part X s ðtÞ where s is chosen as a primary variable and a part X p ðtÞ where p is chosen as a primary variable.…”