Summary
The major issues for parallel solvers in a modern reservoir simulator are
robustness, scalability, efficiency, and flexibility. There is significant
interest in running fast field-scale simulations for complex giant Middle
Eastern reservoirs, which will require tens of millions to hundreds of millions
of grid cells to give reasonable resolution. At the same time, significant
geologic complexity will require the treatment of dual-permeability regions,
faulting and fractures, and high variations of reservoir and fluid properties.
Of course, the methods should also work well for extracted-sector simulation
with local grid refinements in both the structured and unstructured
discretization. The preconditioning methods considered in this work include
both the single-stage and multistage frameworks. In the single-stage framework,
a novel method is considered in addition to the well-known variants of
incomplete lower-upper (ILU) factorizations [ILU0, ILU(k), and ILUT]. The new
method is a highly parallel method, which, in this paper, will be referred to
as the unstructured line-solve power-series (LSPS) method. The method will be
discussed and contrasted in light of key issues for parallel linear solvers.
The unstructured LSPS has certain interesting properties in the parallel
construct, which make it a highly effective component.
The multistage method researched in this work is of the constraint pressure
residual (CPR) framework. The method uses approximate pressure solve as the
first-stage preconditioning to the full-system preconditioning. A number of
original adaptations based on this concept were researched. Here, the use of
the parallel algebraic multigrid (PAMG) method and other single-level methods
mentioned previously in combinations within the multistage CPR framework were
explored. Certain methods constructed in this way are found to be highly
efficient, scalable, and robust. The methods developed are discussed, and
several test problems are included, in this paper. The largest simulation model
tested to date using these solver methods is a 172-million-cell full-field
model of a supergiant carbonate complex with more than 3,000 wells and 60 years
of history simulation.
Introduction
Parallel reservoir simulation involving millions of grid cells is now common
practice and is an essential component for the management of many giant
carbonate complexes in the Middle East. The recent advances are aided in part
by the computational power offered by inexpensive PC clusters. Many of today’s
parallel machines are built with mass-produced commodity-based components. At
the same time, research and development on parallel highly scalable methods in
the modern reservoir simulator have made routine field-scale simulation an
effective and useful part of resource planning and analysis.
Field-scale analyses are often desired over sector simulation for a
comprehensive understanding of overall reservoir-behavior and
recovery-processes performance. Special study involving an area of interest
frequently arises in a full-field project. For example, evaluation of
alternative designs for expensive maximum-reservoir-contact wells with
intelligent downhole controls and production equipment requires near-wellbore
reservoir simulation and optimization workflow. Thus, the demand is high for
simulation capabilities with mixed structured and unstructured grids for fast
field-scale megacell modeling. The capability to refine and coarsen at ease
regionally and perform simulation and analyses at multiple scales within a
single project is a primary near-term goal.
This paper addresses one critical component of the tool set required to
accomplish this mission--the linear solver. The primary solver methods in the
old generation of reservoir simulators typically use nested factorization or
variants of ILU-factorization method for preconditioning. While extension to
small-scale parallel processing was achieved in the late 1990s, these methods
have limitations in terms of scalability or robustness for the very-large-scale
simulations where parallel processing with hundreds or even thousands of
processors is required for speed and performance.
Previously, within the structured-grid framework, a solver method known as
the z-line Neumann series, which is more scalable for parallel
field-scale simulation of structured grid, was documented by Dogru et al.
(2002). Later, a parallel structured multigrid method was introduced by Fung
and Dogru (2000) for treating the local-grid-refinement problems. The
additional solver method for the dual-porosity dual-permeability system was
later described by Fung and Al-Shaalan (2005).
In this work, new ideas in the fully unstructured setting are being
researched and developed. These ideas involve both the single-stage method and
the multistage method. In the single-stage method, a novel idea of building an
approximate inverse preconditioner through matrix substructuring of the
Jacobian matrix was investigated. This substructuring method, which we refer to
as LSPS, is a powerful generalization of the z-line Neumann series
method. The method is fully unstructured. It increases robustness by tracing
the maximum-transmissibility direction of the 3D unstructured graph. The
strategy is particularly beneficial for reservoirs with fracture corridors and
superpermeability (super-K) regions that cause difficulties for other solver
methods. Furthermore, parallel efficiency is maintained, which is crucial for
large-scale multiprocessor applications of the method.
In the multistage method, the two-stage CPR method was investigated. The CPR
method was first introduced into the petroleum literature by Wallis (1983) and
Wallis et al. (1985). It was recently applied by Gratien et al. (2004) and Cao
et al. (2005) in a new simulator development in which they have used the PAMG
method as the pressure preconditioner. The research documented here explores
the quasi-implicit-pressure-explicit-saturation (quasi-IMPES) reduction methods
and the use of various approaches to solve the pressure approximately as a
first-stage preconditioning to the full-system matrix. Solver results for
several sample problems are included for comparison of the various methods.
These include the public-domain data sets for the SPE1 (Odeh 1981) and SPE10
(Christie and Blunt 2001) comparative-solution projects and several
megacell-simulation models. To add some challenge for the solver methods, the
SPE1 grid system has been refined uniformly to 300,000 cells.
To put all the methods into proper prospective, the three variants of the
ILU factorizations [ILU0, ILU(k), and ILUT] are used as baseline comparison for
some problems. The ILU preconditioners are well-known and are described in Saad
(2003), thus descriptions of them are not included here. Interested readers can
refer to Saad (2003) or the many other reference papers concerned with
them.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
4 December 2007
- Meeting paper published:
26 February 2007
- Revised manuscript received:
30 April 2008
- Manuscript approved:
2 May 2008
- Version of record:
15 December 2008
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