A Novel Eulerian-Lagrangian Formulation for Compositional Flow in the Subsurface

Time:	December 14, 2004, 4:00 p.m. – 5:30 p.m.
Lecturer:	Hong Wang, Department of Mathematics, University of South Carolina, U.S.A.
Venue:	Pfaffenwaldring 61, Raum U1.003 (MML), Universität Stuttgart
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Compositional models describe the simultaneous transport of multiplecomponents flowing in coexisting phases through porous media. Becauseeach component can transfer between different phases, the mass of eachphase or a component within a particular phase is no longer conserved.Instead, the total mass of each component among all the phases must beconserved, leading to very large, strongly coupled systems of transientnonlinear advection-diffusion equations. These equations are closelycoupled to a very large set of constraining equations, which are stronglynonlinear, implicit functions of phase pressure, phase temperature, andmole fractions and need to be solved on all computational cells at eachtime step in thermodynamic flash calculations. Additional difficultiesinclude the strong influence of singular sources and sinks, heterogeneitiesof the porous media, high compressibilities of the fluids, large adversemobility ratio, anisotropic dispersion in tensor form, and enormous sizeof field-scale applications. Consequently, these models present severemathematical and numerical difficulties. Classical second-order methodstend to yield numerical solutions with nonphysical oscillations. Inindustrial applications, upwind methods with fully coupled and fullyimplicit temporal discretization have commonly been used to stabilizethe numerical approximations. However, these methods often generateexcessive numerical dispersion and serious spurious effects due to gridorientation.

Eulerian-Lagrangian methods symmetrize the transport equations andgenerate accurate numerical solutions even if very large time steps andcoarse spatial grids are used. They have shown great performance in thenumerical simulations of single-phase flow and immiscible two phase flow.However, there exist serious mathematical and numerical difficultiesthat hinder the development of such methods for multiphase multicomponentcompositional flows in multiple space dimensions: (1) Eulerian-Lagrangianmethods require the governing equations to have a well-defined transportvelocity in terms of their primary unknowns. The molar mass balanceequations in compositional flows are expressed as a weighted sum of molefractions and phase velocities in different phases. (2) Although theexcessive numerical diffusion present in upwind methods severely smearsthe moving steep fronts and introduces grid orientation effect, itfirmly subdues and hides various numerical difficulties. Eulerian-Lagrangianmethods minimize numerical diffusion in upwind methods, leading tosignificantly improved accuracy. However, the numerical difficultiessubdued by the excessive numerical diffusion in upwind methods reoccur.Moreover, the use of Lagrangian coordinates in Eulerian-Lagrangian methodsintroduces extra difficulties. All these numerical difficulties are inaddition to the mathematical and numerical difficulties of compositionalmodeling.

In this talk, we present a novel Eulerian-Lagrangian formulation formultiphase and multicomponent compositional flow. Our preliminarynumerical experiments show that the resulting numerical scheme generatesstable and physically reasonable numerical solutions even if extremelylarge time steps (of more than 0.1 pore volume injected) is used. Thisshows that the strong potential of the proposed mathematical formulation.