Summary
A particularly efficient reservoir simulator can be obtained by combining a
recent multiscale mixed finite-element flow solver with a streamline method for
computing fluid transport. This multiscale-streamline method has shown to be a
promising approach for fast flow simulations on high-resolution geologic models
with multimillion grid cells. The multiscale method solves the pressure
equation on a coarse grid while preserving important fine-scale details in the
velocity field. Fine-scale heterogeneity is accounted for through a set of
generalized, heterogeneous basis functions that are computed numerically by
solving local flow problems. When included in the coarse-grid equations, the
basis functions ensure that the global equations are consistent with the local
properties of the underlying differential operators. The multiscale method
offers a substantial gain in computation speed, without significant loss of
accuracy, when basis functions are updated infrequently throughout a dynamic
simulation.
In this paper, we propose to combine the multiscale-streamline method with a
recent “generalized travel-time inversion” method to derive a fast and robust
method for history matching high-resolution geocellular models. A key point in
the new method is the use of sensitivities that are calculated analytically
along streamlines with little computational overhead. The sensitivities are
used in the travel-time inversion formulation to give a robust quasilinear
method that typically converges in a few iterations and generally avoids much
of the time-consuming trial-and-error seen in manual history matching.
Moreover, the sensitivities are used to enforce basis functions to be
adaptively updated only in areas with relatively large sensitivity to the
production response. The sensitivity-based adaptive approach allows us to
selectively update only a fraction of the total number of basis functions,
which gives substantial savings in computation time for the forward flow
simulations.
We demonstrate the power and utility of our approach using a simple 2D model
and a highly detailed 3D geomodel. The 3D simulation model consists of more
than 1,000,000 cells with 69 producing wells. Using our proposed approach,
history matching over a period of 7 years is accomplished in less than 20
minutes on an ordinary workstation PC.
Introduction
It is well known that geomodels derived from static data only—such as
geological, seismic, well-log, and core data—often fail to reproduce the
production history. Reconciling geomodels to the dynamic response of the
reservoir is critical for building reliable reservoir models. In the past few
years, there have been significant developments in the area of dynamic data
integration through the use of inverse modeling. Streamline methods have shown
great promise in this regard (Vasco et al. 1999; Wang and Kovscek 2000;
Milliken et al. 2001; He et al. 2002; Al-Harbi et al. 2005; Cheng et al. 2006).
Streamline-based methods have the advantages that they are highly efficient
“forward” simulators and allow production-response sensitivities to be computed
analytically using a single flow simulation (Vasco et al. 1999; He et al. 2002;
Al-Harbi et al. 2005; Cheng et al. 2006). Sensitivities describe the change in
production responses caused by small perturbations in reservoir properties such
as porosity and permeability and are a vital part of many methods for
integrating dynamic data.
Even though streamline simulators provide fast forward simulation compared
with a full finite-difference simulation in 3D, the forward simulation is still
the most time-consuming part of the history-matching process. A streamline
simulation consists of two steps that are repeated: (i) solution of a 3D
pressure equation to compute flow velocities; and (ii) solution of 1D transport
equations for evolving fluid compositions along representative sets of
streamlines, followed by a mapping back to the underlying pressure grid. The
first step is referred to as the “pressure step” and is often the most
time-consuming. Consequently, history matching and flow simulation are usually
performed on upscaled simulation models, which imposes the need for a
subsequent downscaling if the dynamic data are to be integrated in the
geomodel. Upscaling and downscaling may result in loss of important fine-scale
information.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
5 December 2006
- Meeting paper published:
26 February 2007
- Revised manuscript received:
2 July 2007
- Manuscript approved:
12 July 2007
- Version of record:
20 March 2008
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