Summary
The general petroleum-production optimization problem falls into the
category of optimal control problems with nonlinear control-state path
inequality constraints (i.e., constraints that must be satisfied at every time
step), and it is acknowledged that such path constraints involving state
variables can be difficult to handle. Currently, one category of methods
implicitly incorporates the constraints into the forward and adjoint equations
to address this issue. However, these methods either are impractical for the
production optimization problem or require complicated modifications to the
forward-model equations (the simulator). Therefore, the usual approach is to
formulate this problem as a constrained nonlinear-programming (NLP) problem in
which the constraints are calculated explicitly after the dynamic system is
solved. The most popular of this category of methods for optimal control
problems has been the penalty-function method and its variants, which are,
however, extremely inefficient. All other constrained NLP algorithms require a
gradient for each constraint, which is impractical for an optimal control
problem with path constraints because one adjoint must be solved for each
constraint at each time step in every iteration.
The authors propose an approximate feasible-direction NLP algorithm based on
the objective-function gradient and a combined gradient for the active
constraints. This approximate feasible direction is then converted into a true
feasible direction by projecting it onto the active constraints and solving the
constraints during the forward-model evaluation itself. The approach has
various advantages. First, only two adjoint evaluations are required in each
iteration. Second, the solutions obtained are feasible (within a specified
tolerance) because feasibility is maintained by the forward model itself,
implying that any solution can be considered a useful solution. Third, large
step sizes are possible during the line search, which may lead to significant
reductions in the number of forward- and adjoint-model evaluations and large
reductions in the magnitude of the objective function. Through two examples,
the authors demonstrate that this algorithm provides a practical and efficient
strategy for production optimization with nonlinear path constraints.
Introduction
One of the primary goals of the reservoir modeling and management process is
to enable decisions that maximize the production potential of the reservoir.
Among the various existing approaches to accomplish this, real-time model-based
reservoir management, also known as the “closed-loop” approach, has recently
generated significant interest. This methodology entails model-based
optimization of reservoir performance under geological uncertainty while also
incorporating dynamic information in real time, which acts to reduce model
uncertainty. For such schemes to be practically applicable, a number of
algorithmic advances are required. Some earlier papers by the authors (Sarma et
al. 2006b; Sarma et al. 2005b) and also papers by other authors such as Brouwer
et al. (2004) have discussed efficient algorithms for such closed-loop
production optimization problems.
This paper, however, focuses only on the optimization component of the
closed-loop process, which is essentially a large-scale optimal control
problem. A large variety of methods for solving discrete-time optimal control
problems now exist in the control-theory literature, including dynamic
programming, neighboring extremal methods, and gradient-based
nonlinear-programming (NLP) methods. These are discussed in detail in Stengel
(1985) and Bryson and Ho (1975). Of these approaches, the NLP method combined
with the Maximum Principle (Bryson and Ho 1975) (adjoint models)
generates a class of NLP methods in which only the control variables are the
decision variables and the state variables are obtained from the dynamic
equations. These algorithms are generally considered more efficient compared to
the other methods. Furthermore, within this class of NLP methods, there are
many existing techniques available for handling nonlinear control-state path
inequality constraints (Bryson and Ho 1975; Mehra and Davis 1972; Feehery 1998;
Fisher and Jennings 1992). However, as will be discussed later, these
techniques are either impractical for the production-optimization problem or
difficult to implement with existing reservoir simulator codes.
In the petroleum-engineering literature, papers by various authors such as
Asheim (1988), Vironovsky (1991), Brouwer and Jansen (2004) have discussed in
significant detail the application of adjoint models and gradient techniques to
the production-optimization problem. However, an important element that is
missing from most of these papers is an effective treatment of nonlinear
control-state path inequality constraints (for example, a maximum
water-injection-rate constraint). Such constraints are always present in
practical production-optimization problems, and therefore appropriate
treatments are essential for such algorithms to be useful. In an earlier paper
by the authors (Sarma et al. 2005a), two methods to handle such constraints
were discussed; however, they either do not satisfy the constraints exactly or
are applicable only for small problems. Zakirov et al. (1996) also discussed an
approach to implementing path constraints; there are, however, certain issues
with this approach, as discussed in a later section.
It should be noted that adjoints and gradient methods have also been applied
to the history-matching problem. Such approaches were pioneered by Chen et al.
(1974) and Chavent et al. (1975) and have more recently been applied by Wu et
al. (1999), Li et al. (2001), and Zhang et al. (2005), among others. However,
the problem of nonlinear path constraints usually does not appear in the
history-matching problem.
In this paper, an approximate feasible-direction optimization algorithm is
proposed, suitable for large-scale optimal control problems, that is able to
handle nonlinear inequality path constraints effectively while maintaining
feasibility within a specified tolerance. Other advantages of this approach are
that only two adjoint simulations are required for each iteration and that
large step sizes are possible during the line search in each iteration,
potentially leading to large reductions in the magnitude of the objective
function. This method belongs to the class of NLP methods combined with the
Maximum Principle (adjoint models) discussed previously. Although the
algorithmic components implemented here have been applied previously in various
contexts, to the authors’ knowledge this is the first integration of a
feasible-direction algorithm, constraint lumping (with the particular lumping
function used), and a feasible-line search algorithm. Thus the resulting
feasible-direction optimization algorithm can be considered to be a new
treatment for an important problem.
This paper proceeds with a brief description of the mathematical formulation
of the problem and the application of adjoint models for efficient calculation
of objective-function gradients with respect to the controls. This is followed
by a discussion of existing methods for handling nonlinear path constraints for
optimal control problems. The next section discusses the traditional
feasible-direction optimization algorithm, which is the basis of the proposed
algorithm. This is followed by detailed discussions of the proposed approximate
feasible-direction and feasible-line search algorithms. The validity and
effectiveness of the approach for handling nonlinear path inequality
constraints is demonstrated through two examples, one with a maximum
water-injection constraint, and the other with a maximum liquid-production
constraint (both of these are nonlinear with respect to the BHP controls).
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
20 January 2006
- Meeting paper published:
11 April 2006
- Revised manuscript received:
9 June 2007
- Manuscript approved:
5 January 2008
- Version of record:
25 April 2008
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