Production Optimization With Adjoint Models Under Nonlinear Control-State Path Inequality Constraints

Abstract

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.

Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

“…This includes partial and sometimes heuristic approaches, valid for particular types of constraints [5,38,39,43], and more systematic approaches, valid for a broader range of constraint equations [9,14,22,33,37]. An important feature in simulations involving highly compressible fluids, which we have in the systems considered here since we inject gas, is the occurrence of transient peaks in the rate in response to changes in well bottom-hole pressure (bottom-hole pressure, or BHP, is the wellbore pressure at a specified depth within the reservoir).…”

“…A more efficient way of approximating these gradients is to 'lump' the constraints over the full simulation time frame [33]. This lumping can be performed on a well-by-well basis, in which case O(N w N) linear systems must be solved for the evaluation of the gradient, or over the entire model, in which case only O(N) linear systems must be solved.…”

“…In subsequent work, the focus was on gradient-based optimization (and in some cases on the optimization of 'smart wells') for water flooding [1,5,34,36,39]. Recent studies have addressed the implementation of adjoint-based procedures into general purpose simulators, the treatment of general constraints, and regularization and other numerical issues [15,20,24,33]. Refer to [21] for a more complete overview of adjoint-based optimization methods.…”

A review of closed-loop reservoir management

“…One is the first-order methods, which only require the derivative information. For example, the steepest ascent algorithm and the conjugate gradient method have been widely used by many researchers such as Brouwer and Jansen (2004), Sarma et al (2008a), and Wang et al (2009). The other category is the second-order methods, which not only require the derivative information but also require the Hessian matrix.…”

Model-reduced gradient-based history matching

“…In that case, the objective function is ultimate recovery or net present value, and the control variables are the well rates, well pressures, or well valve settings. Initially, this was done for the optimization of tertiary recovery processes, see Ramirez [32], later followed by water flooding optimization, see, e.g., [3,5,34,36,45] and [15]. In both the history matching and the recovery optimization problems, the necessary conditions for optimality lead to the gradients, which are now the total derivatives with respect to the controls of the objective function, modified to include the reservoir model constraint.…”