Summary A fully implicit, parallel, compositional reservoir simulator has been developed that includes both a cubic equation of state model for the hydrocarbon phase behavior and Hand's rule for the surfactant/oil/brine phase behavior. The aqueous species in the chemical model include surfactant, polymer, and salt. The physical property models include surfactant/oil/brine phase behavior, interfacial tension, viscosity, adsorption, and relative permeability as a function of trapping number. The fully implicit simulation results were validated by comparison with results from our IMPEC chemical flooding simulator (UTCHEM). The results indicate that the simulator scales well using clusters of workstations. Also, simulation results from parallel runs are identical to those using a single processor. Field-scale surfactant/polymer flood simulations were successfully performed with over 1,000,000 gridblocks using multiple processors. Introduction Chemical flooding is a method to improve oil recovery that involves the injection of a solution of surfactant and polymer followed by a polymer solution. The surfactant causes the mobilization of oil by decreasing interfacial tension, whereas the polymer increases the sweep efficiency by lowering the mobility ratio. Chemical flooding has the potential to recover a very high fraction of the remaining oil in a reservoir, but the process needs to be designed to be both cost effective and robust, which requires careful optimization. Several reservoir simulators with chemical flooding features have been developed as a tool for optimizing the design (Delshad et al. 1996; Schlumberger 2004; Computer Modeling 2004). The University of Texas chemical flooding simulator, UTCHEM (Delshad et al. 1996) is an example of a simulator that has been used for this purpose. However, because UTCHEM is an Implicit Pressure and Explicit Concentration (IMPEC) formulation and in its current form cannot run on parallel computers, realistic surfactant/polymer flooding simulations are limited to around 100,000 gridblocks because of small timestep restrictions and insufficient memory. Recently, the appropriate chemical module was added to the fully implicit, parallel, EOS compositional simulator called GPAS (General Purpose Adaptive Simulator) based on a hybrid approach (John et al. 2005). GPAS uses a cubic equation of state model for the hydrocarbon phase behavior and the parallel and object-based Fortran 95 framework for managing memory, input/output, and the necessary communication between processors (Wang et al. 1999; Parashar et al. 1997). In the hybrid approach implemented in GPAS, the material balance equations for hydrocarbon and water components are solved implicitly first. Then, the material balance equations for the aqueous components such as surfactant, polymer, and electrolytes are solved explicitly using the updated phase fluxes, saturations, and densities.

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It is well known that the adjoint approach is the most efficient approach for gradient calculation, and it can be used with gradient-based optimization techniques to solve various optimization problems, such as the production-optimization problem and the history-matching problem. The adjoint equation to be solved in the approach is a linear equation formed with the "transpose" of the Jacobian matrix from a fully implicit reservoir simulator. For a large and/or complex reservoir model, generalized preconditioners often prove impractical for solving the adjoint equation. Preconditioners specialized for reservoir simulation, such as constrained pressure residual (CPR), exploit properties of the Jacobian matrix to accelerate convergence, so they cannot be applied directly to the adjoint equation. To overcome this challenge, we have developed a new two-stage preconditioner for efficient solution of the adjoint equation by adaptation of the CPR preconditioner (named CPRA: CPR preconditioner for adjoint equation).The CPRA preconditioner has been coupled with an algebraic multigrid (AMG) linear solver and implemented in Chevron's extended applications reservoir simulator (CHEARS V R ). The AMG solver is well known for its outstanding capability to solve the pressure equation of complex reservoir models; solving the linear system with the "transpose" of the pressure matrix is one of the two stages of construction of the CPRA preconditioner.Through test cases, we have confirmed that the CPRA/AMG solver with generalized minimal residual (GMRES) acceleration solves the adjoint equation very efficiently with a reasonable number of linear-solver iterations. Adjoint simulations to calculate the gradients with the CPRA/AMG solver take approximately the same amount of time (at most) as do the corresponding CPR/ AMG forward simulations. Accuracy of the solutions has also been confirmed by verifying the gradients against solutions with a direct solver. A production-optimization case study for a real field using the CPRA/AMG solver has further validated its accuracy, efficiency, and the capability to perform long-term optimization for large, complex reservoir models at low computational cost. Mathematical Derivation of the CPRA PreconditionerDefinition of the Optimization Problem. The optimization problem discussed in the preceding requires a sequence of control vectors u n be found for n ¼ 0, 1, …, N -1, where n is the control-step index and N is the total number of control steps, to maximize (or minimize) a performance measurement J(u 0 , …, u N-1 ) (Sarma et al. 2005). The

Equation-of-state (EOS) compositional and surfactant models are coupled in a fully implicit parallel reservoir simulator using the equivalent alkane carbon number (EACN) of the oleic phase. The EACN of the oleic phase is computed using a mole-fractionweighted carbon number for each component present in the oleic phase. Important microemulsion properties such as optimum salinity and optimum solubilization parameter as a function of the EACN of the oleic phase are implemented on the basis of known correlations. Type II(-) surfactant phase behavior is considered in this study. The simulator developed is validated using our implicit-pressure/explicit-concentration (IMPEC) chemical-flooding simulator. Case studies, including a large-scale simulation, emphasize that surfactant floods should be modeled carefully, taking the EACN of crude oil into consideration for more realistic and accurate oil-recovery predictions.