Summary
Streamline models are routinely used for waterflood optimization and
management and are being extended to more complex processes (e.g.,
compositional simulation). Despite these new developments, no systematic study
has examined the underlying numerical spatial and temporal discretization
errors in streamline simulation and their convergence. Such studies are a
prerequisite to determining the optimal density of streamlines during
simulation and ensuring the resulting accuracy of the solution.
In this paper, we first examine transverse spatial errors (e.g., errors
resulting from the number or placement of streamlines). We provide an analytic
proof and a numeric demonstration of the order of spatial convergence of the
mass-balance discretization error. Both global and local calculations are
performed, and they demonstrate the impact of stagnation regions on the order
of convergence. A second transverse error arises for faulted grids, where lack
of flux continuity at cell faces can lead to incorrect trajectories. These
trajectory errors are of zeroth order and can be resolved only by introducing
additional degrees of freedom into the streamline velocity model. Longitudinal
spatial errors also arise and are associated with the inaccurate calculation of
time of flight across cells.
We show that the commonly used algorithm for corner-point cells leads to
inaccurate time-of-flight calculations for stratigraphic grids, depending upon
aspect ratio. We provide a simple and exact means of calculating the time of
flight for arbitrary corner-point cells, or unstructured grids, in two or three
dimensions, for either compressible or incompressible flow. Finally, using this
new time-of-flight formulation, we analyze a series of cross-sectional
finite-difference simulations to identify grid-orientation errors in the
numerical calculation of flux and spatial error.
Introduction
Streamline simulation has developed rapidly within the oil industry over the
last 10 years (Datta-Gupta and King 1995; King and Datta-Gupta 1998; Osako et
al. 2003; Mallison et al. 2004; Matringe and Gerritsen 2004; Bratvedt et al.
1993, 1996; Prévost et al. 2001; Blunt et al. 1996). Unlike the earlier
streamtube calculations, which date back to the 1930s, streamline simulators
have dispensed with the explicit construction of volume elements (the tubes)
and replaced them with calculations along lines. Each line may be thought of as
tracing out the center of a streamtube, with the velocity obtained from a
numerical finite-difference calculation. In contrast, within a streamtube,
fluid velocity is obtained from the volumetric flux per unit area, where the
area must be calculated explicitly as part of the streamtube construction. With
streamlines, the geometry is implicit, making it simple to perform calculations
in three dimensions. To leading order, streamline simulation appears as a sum
of 1D simulations, and so calculations in one, two, or three dimensions are
essentially equivalent. It is this ease of formulation that has transformed the
class of problems that we can study with streamline simulation. Where
streamtube calculations emphasized 2D sweep and pattern floods, streamline
simulation has been applied to the full range of multiphase and multicomponent
physical and chemical processes in three dimensions (Lolomari et al. 2000;
Crane et al. 2000).
Although streamline simulators have received widespread attention over the
past decade, no systematic study has been performed to understand the sources
of spatial error and their convergence properties. The spatial discretization
in streamline simulation generally takes two forms: first, a transverse
discretization of the domain in terms of streamlines (the number of streamlines
used during simulation determines the degree of transverse resolution), and
second, a longitudinal discretization along streamlines for numerical solution
of the transport equations along streamlines. In fact, much of the
computational advantage of streamline models comes from the decomposition of
the 3D saturation calculations into these 1D calculations along streamlines.
There is an additional zeroth-order truncation error associated with faulted
grids, which also will be discussed.
© 2007. Society of Petroleum Engineers
View full textPDF
(
1,813 KB
)
History
- Original manuscript received:
7 December 2004
- Revised manuscript received:
30 May 2006
- Manuscript approved:
8 March 2007
- Version of record:
20 June 2007
View Cart
