On the development of component- or phase-based Eulerian-Lagrangian formulation for compositional flow and transport in porous media

Compositional models describe the simultaneous flow and transport processes of multiple components flowing in coexisting phases in porous media [1, 2]. Because each component can transfer between different phases, the mass of each phase or a component within a particular phase is no longer conserved. Instead, the total mass of each component among all the phases must be conserved, leading to strongly coupled systems of transient nonlinear partial differential equations of convection-diffusion type. These equations are closely coupled to a set of constraining equations, which are strongly nonlinear, implicit functions of phase pressure, temperature, and composition. These equations need to be solved in all spatial cells within the two-phase region at each iterative step of each time step via thermodynamic flash calculation. In industrial applications, upwind methods have commonly been used to stabilize the numerical approximations [1, 2]. However, these methods often generate excessive numerical dispersion and serious spurious effects due to grid orientation. Eulerian-Lagrangian methods combine the convection and capacity terms in the mass transport equations to carry out the temporal discretization in a Lagrangian coordinate, and discretize the diffusion-dispersion term on a fixed mesh. Eulerian-Lagrangian methods symmetrize the mass transport equations and stabilize their numerical approximations. They generate accurate numerical solutions and significantly reduce the numerical diffusion and grid-orientation effect present in upwind methods, even if large time steps are used. Eulerian-Lagrangian methods have been successfully applied in single-phase flow [4, 5, 6, 7] and in immiscible two-phase flow [8, 9]. In this paper we are concerned with the development of an Eulerian-Lagrangian formulation for two-phase multicomponent flow and transport processes in porous media. Such a formulation retains the numerical advantages of earlier Eulerian-Lagrangian methods for single or two-phase flow and transport. The accurate solution of the mass transport equations provides an accurate initial guess for flash calculation and, thus, speeds up the flash. Conversely, accurate flash calculation improves the solutions to the mass transport equations in the next iterative or time step. Finally, larger time steps can be used, leading to further reduction of computational storage and cost. Because of the complexities and strong nonlinearity and coupling of these processes, different Eulerian-Lagrangian formulations could be proposed based on different considerations of the physical and mathematical properties of compositional flow and transport. In this paper, we explore both component-based approach and phase-based approach. Preliminary numerical experiments of two-phase multicomponent compositional flow and transport in a two-dimensional reservoir reveals the following observations: (1) on the same spatial partition, using a timestep > 100 times larger than that of the upwind method, the Eulerian-Lagrangian method generates more accurate solutions with steeper fronts than the upwind method; (2) both the Eulerian-Lagrangian method and upwind methods use comparable CPU time per time step.