Optimal Waterflood Design Using the Adjoint Method

Abstract

We address the problem how to operate the injectors and producers of an oil field so as to maximize the value of the field. Instead of agressively producing and injecting fluids at maximum rate aiming at large short term profits, we are after optimizing the total value (e.g. discounted oil volume) over the whole lifecycle of the field. An essential tool in tackling this optimization problem is the adjoint method from optimal control theory. Starting from a base case reservoir simulation run, this extremely efficient method makes it possible to compute the sensitivities of the total (lifecycle) value with respect to all (time-dependent) well control variables in one go, at a cost less than that of an extra reservoir simulation run. These sensitivities can be used in an optimization loop to iteratively improve well controls. We implemented the adjoint method and an associated optimization algorithm in our in-house reservoir simulator. In addition to conventional well control options based on the well's pressure or total rate, we have also implemented smart well control options which allow the separate control of individual inflow intervals. Special adaptations of the optimization algorithm were required to allow the inclusion of inequality constraints on well control (pressure and rate constraints). We applied the optimization algorithm to a number of cases, and found interesting, non-trivial solutions to some optimal waterflood design problems, that would not easily have been found otherwise. In this paper, we also present a self-contained elementary derivation of the adjoint method, which is different from, but equivalent to the well-known derivation based on the Lagrange formalism. Introduction We focus on the problem of designing an optimal waterflood for an oil field. We limit ourselves to the situation where the well configuration and well types are given, so the only degree of freedom left is the way the injector and producer wells are operated. The waterflood design we are looking for should be optimal in the lifecycle sense, i.e., it should maximize the lifecycle integral Equation (1) where the integrand is a weighted sum of field rates, Equation. Here the weights are denoted by the letter and the field rates by the letter . The subscripts and refer to "oil" and "water", while the superscripts "prod" and "inj" refer to "production" and "injection", respectively. We note that the weights, which are given functions of time, can have arbitrary sign. This makes it possible to combine oil revenues and water costs in the lifecycle integral. Well control is modeled with one or more time-dependent well control variables per well. For conventional wells, there is just one control variable, which can be tubinghead pressure (THP), bottomhole pressure (BHP) or a rate. For smart wells there are generally more control variables corresponding to the setting of downhole inflow or outflow devices which can be controlled independently. Well control is not completely free as it should take into account certain operational limits such as rate and pressure constraints. These operational limits correspond to inequality constraints, either directly on the control variables or indirectly, in terms of certain state variables of the well/reservoir system. Apart from constraints on individual wells, there can also be global constraints dealing with several wells. Examples can be (equality or inequality) constraints imposed by surface facilities, or constraints imposed by reservoir management considerations (e.g., voidage balance constraints). The mathematical optimization problem to be solved is to maximize the lifecycle integral by choosing the optimal well control while satisfying all 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).…”

Optimization of Geothermal-Well-Doublet Placement

“…Therefore, automatic optimization algorithms pose a solution. Multiple of such optimization methods have been developed by the oil and gas industry to determine optimal positions for producers and injectors in heterogeneous reservoirs (e.g., Güyagüler and Horne 2004;Kraaijevanger et al 2007;Zandvliet et. al 2008;Kalla and White 2007;Wang et al 2012).…”

“…Various methods have been developed to calculate this gradient. Best known and most used among these methods are various perturbation methods, the forward sensitivity equations method and the (backwards) adjoint method (Biegler 2000;Kraaijevanger et al 2007). In this thesis a perturbation method is used and elaborated on in the next section.…”

“…The perturbation method as described by e.g., Biegler 2000 andKraaijevanger et al 2007., is used in this thesis. This perturbation method is, as opposed to other gradient calculation methods, quite easy to implement.…”

Effect of Time Stepping Strategy on Adjoint-based Production Optimization

“…This method serves to eliminate PDE constraints and calculates the total variation of the objective function and remaining non-PDE constraints, with respect to the optimization variables. It is accomplished by calculating sensitivities of the objective function with respect to the PDE constraints, called adjoint variables [16,18]. For a given state of the reservoir, the adjoint variables can be accurately computed using one backward in time simulation with a linearized reservoir system.…”