Summary
It is well known that the adjoint approach is the most efficient approach
for gradient calculation, and it can be used with gradient-based optimization
techniques to solve various optimization problems, such as the
production-optimization problem and the history-matching problem. The adjoint
equation to be solved in the approach is a linear equation formed with the
"transpose" of the Jacobian matrix from a fully implicit reservoir simulator.
For a large and/or complex reservoir model, generalized preconditioners often
prove impractical for solving the adjoint equation. Preconditioners specialized
for reservoir simulation, such as constrained pressure residual (CPR), exploit
properties of the Jacobian matrix to accelerate convergence, so they cannot be
applied directly to the adjoint equation. To overcome this challenge, we have
developed a new two-stage preconditioner for efficient solution of the adjoint
equation by adaptation of the CPR preconditioner (named CPRA: CPR
preconditioner for adjoint equation).
The CPRA preconditioner has been coupled with an algebraic multigrid (AMG)
linear solver and implemented in Chevron's extended applications reservoir
simulator (CHEARS(R)). The AMG solver is well known for its outstanding
capability to solve the pressure equation of complex reservoir models; solving
the linear system with the "transpose" of the pressure matrix is one of the two
stages of construction of the CPRA preconditioner.
Through test cases, we have confirmed that the CPRA/AMG solver with
generalized minimal residual (GMRES) acceleration solves the adjoint equation
very efficiently with a reasonable number of linear-solver iterations. Adjoint
simulations to calculate the gradients with the CPRA/AMG solver take
approximately the same amount of time (at most) as do the corresponding CPR/AMG
forward simulations. Accuracy of the solutions has also been confirmed by
verifying the gradients against solutions with a direct solver. A
production-optimization case study for a real field using the CPRA/AMG solver
has further validated its accuracy, efficiency, and the capability to perform
long-term optimization for large, complex reservoir models at low computational
cost.
© 2013. Society of Petroleum Engineers
View full textPDF
(
680 KB
)
History
- Original manuscript received:
7 January 2011
- Meeting paper published:
21 February 2011
- Revised manuscript received:
3 June 2012
- Manuscript approved:
11 June 2012
- Published online:
15 January 2013
- Version of record:
5 April 2013
View Cart
