BAIL 2006 Book of Abstracts - Institut für Numerische und ...

H. WANG: A Component-Based Eulerian-Lagrangian Formulation for Compositional
Flow in Porous Media
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A Component-Based Eulerian-Lagrangian Formulation
for Compositional Flow in Porous Media
Hong Wang ∗
Introduction Compositional models describe the simultaneous transport of multiple components
flowing in coexisting phases in porous media [1, 4]. 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 advection-diffusion equations.
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 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, 4]. However, these methods often generate excessive
numerical dispersion and serious spurious effects due to grid orientation.
Eulerian-Lagrangian methods symmetrize the transport equations and generate accurate
numerical solutions even if large time steps and coarse spatial grids are used. They have demonstrated
excellent performance in the numerical simulations of single-phase flow [3, 5] and immiscible
two phase flow [2]. However, there exist serious mathematical and numerical difficulties in
the development of Eulerian-Lagrangian methods for multiphase multicomponent compositional
flow. We present a component-based Eulerian-Lagrangian formulation for compositional flow,
which can be used by many Eulerian-Lagrangian methods.
Numerical experiments We simulate the transport of Methane, Propane, and n-Hexane,
flowing in coexisting liquid and vapor phases in a horizontal porous medium reservoir Ω =
(0, 1000) × (0, 1000) ft 2 with a thickness of 1 ft over a time period of 20 years. An injection well
is located at the upper-right corner of Ω with a volumetric injection rate of Q =15ft 3 /day. A
production well is located at the lower-left corner with a production rate of Q = −15 ft 3 /day.
The porosity φ = 0.1. The permeability is 60 md. The effect of capillary pressure is neglected.
The relative permeability k r,l =(s l ) 2 and k r,v =(1−s l ) 2 . The initial reservoir pressure is 2100
psia and the reservoir temperature is 350 o K. The composition of the resident fluid is cMethane
= 0.5, cP ropane =0.2, and cn−Hexane =0.3, which is in liquid phase at the given temperature
and pressure. The composition of the injected fluid is ¯cMethane = 0.8, ¯cP ropane =0.15, and
¯cn−Hexane =0.05, which is in vapor phase.
We use a uniform coarse spatial grid of ∆x =∆y= 25 ft. We use a time step of ∆tel =1
year for the Eulerian-Lagrangian method, and of ∆tup = 2 days for the upwind method that is
the largest possible. In Figure 1(a) we present the plots of the overall mole fraction cMethane
generated by the Eulerian-Lagrangian method at t = 5 and 20 years. In Figure 1(b) we present
the plots of the normalized molar amount n v Methane = ρv s v c v Methane /(ρ cMethane), where ρ v and
sv are density and saturation of the vapor phase and ρ is bulk molar density of the fluid mixture.
Note that nv i represents the fraction of ci in the vapor phase. Thus, nv i = 1 or 0 indicates that
component i stays in the vapor phase or liquid phase completely. 0