An Efficient Approach to Adaptive-Implicit Compositional Simulation With an Equation of State

Abstract

A robust and efficient method for solving the equations corresponding to an equation-of-state compositional model is described. The method is developed for an adaptive-implicit simulator where only a small number of blocks need to be solved implicitly, while the remaining blocks are solved in an explicit manner. The salient features of the method is the decoupling of the solution of flow equations from the flash calculations. This allows an easy implementation of powerful flash- calcualtion schemes in the simulator. The adaptive-implicit approach is compared to a fully implicit approach and an explicit-transmissibility approach by using the Third SPE Comparative Solution Project. Results show that the adaptive-implicit approach is about twice as fast as the other two methods. Introduction Most EOS compositional models described in the literature use explicit transmissibilities. Because of the complexity of the equations and the use of large numbers of components (around 10), the explicit formulation has been the only feasible approach to field-scale simulation. This was demonstrated in the Third SPE Comparative Solution Project, where all the simulators used were explicit. The drawback of the explicit formulation is the timestep size limitation, which excludes its application in coning studies. Some attempts have been made to develop a fully implicit EOS compositional model. However, the application of such a model was restricted to very small problems (3 components and 80 blocks in Reference 2; and 3 components and 64 blocks in Reference 3), and no field-scale runs were reported. Recently, Bertiger and Kelsey described the use of the adaptive-implicit method in an EOS compositional model. The adaptive-implicit method introduced by Thomas and Thurnau is based on the idea that at a given time during a simulation, only some blocks need to be solved implicitly, while the remaining blocks are solved explicitly. Thus during a simulation, blocks will be switched automatically from explicit to implicit to allow the use of large timesteps. The alignment of equations and variables used by Bertiger and Kelsey 4 is similar to that of Coats. Their test examples were also restricted to relatively small systems (5 components and 56 blocks; 3 components and 150 blocks). This paper describes a robust and efficient formulation of an adaptive-implicit EOS compositional model. The salient features of the formulation are the selection of primary equations and variables, and the decoupling of the solution of the flow equations from the flash calculations. The flow equations are converged with Newton's method, while the phase-equilibrium equations can be solved by any technique (e.g., Newton's method and quasi-Newton successive substitutions). This is a departure from previous Newton's method approaches to compositional simulation where the flow and phase-equilibrium equations are converged simultaneously. These approaches are described by Fussell and Fussell and Young and Stephensons for explicit-transmissibility models and by Coats and Chien et al. for fully implicit models. The desirable features of the present approach are discussed later. The use of either analytical or numerical derivatives in the Jacobian construction has been a point of contenttion. Numerical differentiation is easy to program and does not require extensive revision of the code when property correlations are changed. However, it is computationally more time-consuming than analytical differentiation, especially in the evaluation of of equation-of-state fugacity derivatives. This paper shows a differentation method where both analytical and numerical differentationare used at different places to take advantage of both schemes. P. 395⁁

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