Summary
Streamline simulators have received increased attention in the petroleum
industry because of their ability to effectively handle multimillion-cell
detailed geologic models and large simulation models. The efficiency of
streamline simulation has relied primarily on the decoupling of the 3D
saturation equation into 1D equations along streamlines using the streamline
time of flight as the spatial coordinate. Until now, this decoupling has been
strictly valid for incompressible flow. Applications to compressible flow have
generally lacked strong theoretical foundations, and very often yielded mixed
or unsatisfactory results.
In this paper, for the first time we generalize streamline models to
compressible flow using a rigorous formulation while retaining much of its
favorable characteristics. Our new formulation is based on three major elements
and requires only minor modifications to existing streamline models. First, we
introduce an “effective density” for the total fluids along the streamlines.
This density captures the changes in the fluid volume with pressure and can be
conveniently and efficiently traced along streamlines. Thus, we simultaneously
compute time of flight and volume changes along streamlines. Second, we
incorporate a density-dependent source term in the streamline saturation
equation to account for compressibility effects. Third, the effective density,
fluid volumes, and the time-of-flight information are used to incorporate
cross-streamline effects through use of pressure updates and remapping of
saturations. Our proposed approach preserves the 1D nature of the saturation
calculations and all the associated advantages of the streamline approach. The
saturation calculations are fully decoupled from the underlying grid and can be
carried out using large timesteps without grid-based stability limits.
We demonstrate the validity and practical utility of our approach using
synthetic and field examples and comparison with a commercial finite-difference
simulator. A comparison of the number of pressure solutions and the CFL numbers
for the streamline and finite-difference simulation indicates that our proposed
compressible streamline approach is likely to offer substantial computational
advantage.
Introduction
Streamline simulators have become increasingly popular for high-resolution
reservoir simulation using multimillion-cell geologic models. For
incompressible or slightly compressible flow and under convection-dominated
conditions, streamline models are well known to outperform conventional
finite-difference simulation in terms of computational speed. Streamline models
can also result in improved accuracy because of subgrid resolution and reduced
numerical dispersion and grid-orientation effects (King and Datta-Gupta 1998;
Datta-Gupta 2000). To a large extent, the efficiency of the current streamline
formulation is a consequence of the incompressibility assumption that allows us
to easily and effectively decouple the pressure and saturation calculations
during flow simulation. This decoupling has been greatly facilitated by the
introduction of the streamline time of flight coordinate (Datta-Gupta and King
1995). Specifically, utilizing the time of flight as the spatial coordinate,
the multidimensional saturation calculations are reduced to a series of 1D
solutions along streamlines. These 1D solutions can be carried out
independently and using relatively large timesteps, as they are not impacted by
the underlying geologic grid-based stability limitations. This is the primary
advantage of streamline simulation. In addition, for heterogeneity-dominated
flow and adverse mobility ratio conditions, the streamlines need to be updated
infrequently, leading to further savings in computation time (King and
Datta-Gupta 1998; Datta-Gupta 2000).
© 2006. Society of Petroleum Engineers
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History
- Original manuscript received:
16 July 2005
- Revised manuscript received:
8 July 2006
- Manuscript approved:
16 August 2006
- Version of record:
20 December 2006
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