Summary
In this work, we present a numerical procedure that combines the mixed
finite-element (MFE) and the discontinuous Galerkin (DG) methods. This
numerical scheme is used to solve the highly nonlinear coupled equations that
describe the flow processes in homogeneous and heterogeneous media with mass
transfer between the phases. The MFE method is used to approximate the phase
velocity based on the pressure (more precisely average pressure) at the
interface between the nodes. This approach conserves the mass locally at the
element level and guarantees the continuity of the total flux across the
interfaces. The DG method is used to solve the mass-balance equations, which
are generally convection-dominated. The DG method associated with suitable
slope limiters can capture sharp gradients in the solution without creating
spurious oscillations. We present several numerical examples in homogeneous and
heterogeneous media that demonstrate the superiority of our method to the
finite-difference (FD) approach. Our proposed MFE-DG method becomes orders of
magnitude faster than the FD method for a desired accuracy in 2D.
Introduction
There has been gradual progress in the development of algorithms for the
compositional simulation of hydrocarbon reservoirs in the last 15 years. Before
that, there were several major advances in the numerical solution of the
combined flow equations and the thermodynamic equilibrium with the equations of
state. Despite the advances of the last 25 to 30 years and the enormous
progress in the speed of computers in the same period, we cannot yet perform
field-scale compositional modeling satisfactorily in heterogeneous reservoirs.
The main problem is the continued use of the FD discretization scheme and its
inherent limitations. Most of the current compositional simulators use the
upstream weighted FD method to approximate the flow equations. Because of the
fact that the flow processes are usually convection-dominated, FD methods may
produce significant numerical diffusion (Coats 1980). The excessive numerical
diffusion requires unrealistic gridding, especially with heterogeneities.
Recently, the DG methods have been successfully implemented to approximate
various physical problems, notably hyperbolic systems of conservative laws. One
property of these methods is that they conserve mass at the element level in a
finite-element framework. Consequently, they enhance the flexibility of finite
elements in describing flow in complicated geometries. Furthermore, the choice
of the spatial approximation without the continuity across inter-element
boundaries allows a simple treatment of combined finite-element cells with
different geometries as well as different degrees of approximating polynomials.
These methods associated with suitable slope limiters can capture
discontinuities or sharp gradients in the solution. The DG method was first
implemented for nonlinear scalar conservative laws by Chavent and Salzano
(1982). However, these authors noted that a very restrictive timestep should be
used to keep stability of the scheme.
© 2006. Society of Petroleum Engineers
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History
- Original manuscript received:
26 October 2004
- Revised manuscript received:
14 September 2005
- Manuscript approved:
21 September 2005
- Version of record:
20 March 2006
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