# Optimal Waterflood Design Using the Adjoint Method

- Authors
- J.F.B.M. Kraaijevanger (Shell Intl. E&P BV) | P.J.P. Egberts (TNO Built Environment and Geosciences) | J.R. Valstar (TNO Built Environment and Geosciences) | H. W. Buurman (LogicaCMG)
- DOI
- https://doi.org/10.2118/105764-MS
- Document ID
- SPE-105764-MS
- Publisher
- Society of Petroleum Engineers
- Source
- SPE Reservoir Simulation Symposium, 26-28 February, Houston, Texas, U.S.A.
- Publication Date
- 2007

- Document Type
- Conference Paper
- Language
- English
- ISBN
- 978-1-55563-197-0
- Copyright
- 2007. Society of Petroleum Engineers
- Disciplines
- 5.4.1 Waterflooding, 1.7.5 Well Control, 4.1.2 Separation and Treating, 5.1.2 Faults and Fracture Characterisation, 2.3 Completion Monitoring Systems/Intelligent Wells, 5.5 Reservoir Simulation, 4.1.5 Processing Equipment, 2.2.2 Perforating, 5.7.2 Recovery Factors, 4.3.4 Scale, 5.5.8 History Matching, 6.5.2 Water use, produced water discharge and disposal
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**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.

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