These Lagrange multipliers are then used for computing a gradient vector.

The discrete formulation keeps the adjoint equations consistent with the reservoir simulator. We include the effects of saturation change in history-matching pressures. We do this in a fashion that circumvents the need for developing a full multiphase optimal-control code.

## Optimal Control of Partial Differential Equations: Theory, Methods and Applications

We show detailed results of the application of the optimal-control algorithm to three field examples. The first stage involves solving an "expansivity equation" for pressure.

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The expansivity equation is obtained by summing the material-balance equations for oil, gas, and water flow. Once the pressures are implicitly obtained from the expansivity equation, the phase saturations can be updated using their respective balance equations.

## Control theory - Wikipedia

A typical expansivity equation is shown in Appendix B, Eq. When we write the reservoir simulation equations as partial differential equations, we assume that the parameters to be estimated are continuous functions of position. The partial-differential-equation formulation is partial-differential-equation formulation is generally termed a distributed-parameter system.

However, upon solving these partial differential equations, the model is discretized so that the partial differential equations are replaced by partial differential equations are replaced by sets of ordinary differential equations, and the parameters that were continuous functions of parameters that were continuous functions of position become specific values.

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B-1 is a position become specific values. B-1 is a set of ordinary differential equations that reflects lumping of parameters. Each cell has three associated parameters: a right-side permeability thickness, a bottom permeability thickness, and a pore volume.

Once the discretized model is written and we have one or more ordinary differential equations per cell, we can then adjoin these differential equations to the integral to be minimized by using one Lagrange multiplier per differential equation. The ordinary differential equations for the Lagrange multipliers are now derived as part of the necessary conditions for stationariness of the augmented objective function. These ordinary differential equations are termed the adjoint system of equations. A detailed example of the procedure discussed in this paragraph is given in Appendix A.

The ordinary-differential-equation formulation of the optimal-control algorithm is more appropriate for use with reservoir simulators than the partial-differential-equation derivation found in partial-differential-equation derivation found in Refs. Log in to your subscription Username.

Peer reviewed only Published between: Published from year: and Published to year: Advanced search Show search help. Rock Mechanics Symposium and 5th U. Authors M. As a result, the usefulness of the present method was demonstrated.

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Transactions of the Society of Mechanical Engineers. Transactions of the Japan Society of Mechanical Engineers.

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