## Abstract

The adjoint gradient method is well recognized for its efficiency in large-scale production optimization. When implemented in a nonlinear programming (NLP) algorithm, adjoint gradients of the objective function and nonlinear constraints enable the construction of a convex approximation of the original optimization problem using just one forward and one backward simulation. Here we focus on the performance of the adjoint gradients with respect to the time step strategy applied in the underlying forward and backward simulations. First, we demonstrate that the NPV objective is sensitive to the details of mass transfer in the forward reservoir simulation. Using simple examples with uniform time steps, we show that the adjoint gradients and optimal solutions for bottom-hole pressure (BHP) controls are less consistent with respect to time step refinement compared to the gradients and optimal solutions using rate controls. Next, we investigate an adaptive time step strategy within the simulator which generates a time step refinement immediately after the control update. We consider adjoint gradients of NPV with respect to injection BHP controls. We observe that time step refinement after control update improves the quality of adjoint gradients. Although the increase in the number of time steps does increase the number of objective function evaluations required to achieve a prescribed Karush-Kuhn-Tucker condition tolerance, the resulting optimal NPV is higher. The instantaneous update of the BHP controls results in a sharp increase in the injection rates. The presence of high instantaneous rates influences the outcome of reservoir simulation when well rate constraints are present. We consider a strategy where constraints are applied directly during the forward simulations against the strategy with nonlinear constraints applied in the optimization process. We demonstrate on practical examples the influence of time step size on both strategies. In the case of optimization with constrained simulation, we observe that the response to the time step refinement is similar to an unconstrained production optimization. However, in the case of constrained optimization, the advantage of small time step refinement may be counterbalanced by excessive constraint violations after control updates.

Original language | English |
---|---|

Title of host publication | 14th European Conference on the Mathematics of Oil Recovery 2014, ECMOR 2014 |

Publisher | EAGE |

ISBN (Electronic) | 9781634391689 |

Publication status | Published - 2014 |

Externally published | Yes |

Event | 14th European Conference on the Mathematics of Oil Recovery 2014 - Catania, Italy Duration: 8 Sep 2014 → 11 Sep 2014 Conference number: 14 |

### Conference

Conference | 14th European Conference on the Mathematics of Oil Recovery 2014 |
---|---|

Abbreviated title | ECMOR 2014 |

Country | Italy |

City | Catania |

Period | 8/09/14 → 11/09/14 |