Active-Set Reduced-Space Methods with Nonlinear Elimination for Two-Phase Flow Problems in Porous Media

Abstract

Abstract. Fully implicit methods are drawing more attention in scientific and engineering applications due to the allowance of large time steps in extreme-scale simulations. When using a fully implicit method to solve two-phase flow problems in porous media, one major challenge is the solution of the resultant nonlinear system at each time step. To solve such nonlinear systems, traditional nonlinear iterative methods, such as the class of the Newton methods, often fail to achieve the desired convergent rate due to the high nonlinearity of the system and/or the violation of the boundedness requirement of the saturation. In the paper, we reformulate the two-phase model as a variational inequality that naturally ensures the physical feasibility of the saturation variable. The variational inequality is then solved by an active-set reduced-space method with a nonlinear elimination preconditioner to remove the high nonlinear components that often causes the failure of the nonlinear iteration for convergence. To validate the effectiveness of the proposed method, we compare it with the classical implicit pressure-explicit saturation method for two-phase flow problems with strong heterogeneity. The numerical results show that our nonlinear solver overcomes the often severe limits on the time step associated with existing methods, results in superior convergence performance, and achieves reduction in the total computing time by more than one order of magnitude.

A Lattice Boltzmann Model for Two-Phase Flow in Porous Media

“…Remark I : We would like to point that although Eq. (15) or (18) (18). We note that Ginzburg has proposed two LB models (E and L models) for general CDEs [32], and these two models also seem to be suitable for Eq.…”

“…However, due to the nonlinearity and coupling of these macroscopic continuum models, it is difficult or even impossible to obtain their analytical solutions [1,2,9]. Fortunately, with the development of computational technology, some numerical approaches, including finite-difference method [10], finite-volume method [11,12], finite-element method [2,9,13,14], operatorsplitting method [15,16], implicit pressure-explicit saturation method [2,17] and active-set reduced-space method [18], have been proposed to solve the macroscopic continuum models, and also gained a great success in the study of two-phase flows in porous media [2,19]. In this work, we will present an alternative, i.e., lattice Boltzmann (LB) method, for two-phase flows in porous media.…”

Fully implicit hybrid two-level domain decomposition algorithms for two-phase flows in porous media on 3D unstructured grids

“…However, their stability may degrade on unstructured grids that are necessary for complex geometries [20]. Other widely used discretization methods for subsurface systems include finite volume methods [28,41,46] and mixed finite element methods [11,24,60,62]. In the past decade, discontinuous Galerkin (DG) finite element methods have been shown to be competitive with respect to other standard methods for transport problems, such as single phase flows [47], miscible displacement [48] and reactive transport [53,54].…”

“…Efforts have been made in employing Newton method and its variants to solve the two-phase flow problems [16,42,39,60,61,62,63,64]. Good candidates for solving the linear Jacobian system are preconditioned Krylov subspace methods [50].…”

A fully mass-conservative iterative IMPEC method for multicomponent compressible flow in porous media

“…Modeling and simulation of compositional flow in subsurface media are prevalent and of interest in hydrogeology and hydrocarbon reservoirs (cf. [8,28,29,20,37,36,18,26,27,40,31,32,33,34] and references therein). Mathematical model of multicomponent compressible flow in porous media includes the Darcy's law, the conservation of mass and the equation of state, which is a coupled nonlinear system.…”