Practical Applications of Optimal-Control Theory to History-Matching Multiphase Simulator Models

Abstract

This paper applies material presented by Chen et al. and by Chavent et al to practical reservoir problems. The pressure history-matching algorithm used is initially based on a discretized single-phase reservoir model. Multiphase effects are approximately treated in the single-phase model by multiplying the transmissibility and storage terms by saturation-dependent terms that are obtained from a multiphase simulator run. Thus, all the history matching is performed by a "pseduo" single-phase model. The multiplicative factors for transmissibility and storage are updated when necessary. The matching technique can change any model permeability thickness or porosity thickness value. Three field examples are given. Introduction History matching using optimal-control theory was introduced by two sets of authors. Their contributions were a major breakthrough in attacking the long-standing goal of automatic history matching. This paper extends the work presented by Chen et al. and Chavent et al. Specifically, we focus on three areas.We derive the optimal-control algorithm using a discrete formulation. Our reservoir simulator, which is a set of ordinary differential equations, is adjoined to the function to be minimized. The first variation is taken to yield equations for computing Lagrange multipliers. 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.We show detailed results of the application of the optimal-control algorithm to three field examples. DERIVATION OF ADJOINT EQUATIONS Most implicit-pressure/explicit-saturation-type, finite-difference reservoir simulators perform two calculation stages for each time step. The first stage involves solving an "expansivity equation" for pressure. 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. A typical expansivity equation is shown in Appendix B, Eq. B-1. 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. Eq. B-1 is a position become specific values. Eq. 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. 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. 1 and 2. SPEJ P. 347

Efficient real-time reservoir management using adjoint-based optimal control and model updating

“…The use of adjoints for history matching was pioneered by Chen et al [18] and Chavent et al [19], who applied it to single-phase problems. Since then, many other researchers have modified and improved the application of adjoint models for multiphase history matching including Wasserman et al [20], Watson et al [21], Wu et al [22], Li et al [23], Wu and Datta-Gupta [24], and Zhang et al [25]. Gavalas et al [26] introduced the use of an eigenfunction expansion for the efficient parameterization of reservoir properties, which was also used later by Oliver [27] and Reynolds et al [28].…”

Model-reduced gradient-based history matching

“…A particularly efficient class of optimization methods are those where a gradient of the objective function with respect to the model parameters is calculated by solving the adjoint, or co-state, problem as introduced by Courant and Hilbert [11]. In reservoir engineering, the adjoint method was used for the first time by Chen et al [9] and later applied by, among others, Chavent et al [8], Wasserman et al [42], Watson et al [43], Lee and Seinfeld [25], Yang et al [44], Zhang and Reynolds [47], Li et al [26], and Oliver et al [31]. In the adjoint approach, the history matching problem is treated as an optimal control problem where the control variables are the unknown model parameters and where the objective function is minimized subject to the constraint that the state variables obey the prescribed reservoir model.…”

Rapid History Matching Using a Generalized Travel-Time Inversion Method

“…15 For computing the sensitivity coefficients of the production data with respect to reservoir parameters, we resort to an optimal control method as discussed by various authors. 6,[16][17][18][19] A major limitation of the optimal control method has been the fact that the sensitivity computations require a solution of a number of linear equations (adjoint equations) equal to the number of observed data. 6 This imposes a severe computational burden, because in field situations we are typically faced with thousands of observed data.…”