Summary
Growing interest in understanding, predicting, and controlling advanced
oil-recovery methods emphasizes the importance of numerical methods that
exploit the nature of the underlying physics. The fully implicit method offers
unconditional stability of the discrete approximations. This stability comes at
the expense of transferring the inherent physical stiffness onto the coupled
nonlinear residual equations that are solved at each timestep. Current
reservoir simulators apply safeguarded variants of Newton’s method that can
neither guarantee convergence nor provide estimates of the relation between
convergence rate and timestep size. In practice, timestep chops become
necessary and are guided heuristically. With growing complexity, such as in
thermally reactive compositional flows, convergence difficulties can lead to
substantial losses in computational effort and prohibitively small timesteps.
We establish an alternative class of nonlinear iteration that converges and
associates a timestep to each iteration. Moreover, the linear solution process
within each iteration is performed locally.
By casting the nonlinear residual equations for a given timestep as an
initial-value problem, we formulate a continuation-based solution process that
associates a timestep size with each iteration. Subsequently, no iterations are
wasted and a solution is always attainable. Moreover, we show that the rate of
progression is as rapid as that for a convergent standard Newton method.
Moreover, by exploiting the local nature of nonlinear wave propagation typical
to multiphase-flow problems, we establish a linear solution process that
performs computation only where necessary. That is, given a linear convergence
tolerance, we identify a minimal subset of solution components that will change
by more than the specified tolerance. Using this a priori criterion, each
linear step solves a reduced system of equations. Several challenging examples
are presented, and the results demonstrate the robustness and computational
efficiency of the proposed method.
© 2009. Society of Petroleum Engineers
View full textPDF
(
1,128 KB
)
History
- Original manuscript received:
14 November 2008
- Meeting paper published:
2 February 2009
- Revised manuscript received:
28 May 2009
- Manuscript approved:
1 June 2009
- Published online:
29 December 2009
- Version of record:
17 June 2010
View Cart
