Compositional Space Parameterization Approach for Reservoir Flow Simulation

Phase equilibrium calculations are the most challenging part of a compositional flow simulation. For every gridblock and at every time step, the number of phases and their compositions must be computed for the given overall composition, temperature, and pressure conditions. The conventional approach used in petroleum industry is based on performing a phase-stability test, and solving the fugacity constraints together with the coupled nonlinear flow equations when the gridblock has more than one phase. The multi-phase compositional space can be parameterized in terms of tie-simplexes. For example, a tie-triangle can be used such that its interior encloses the three-phase region, and the edges represent the boundary with specific two-phase regions. The tie-simplex parameterization can be performed for pressure, temperature, and overall composition. The challenge is that all of these parameters can change considerably during the course of a simulation. It is possible to prove that the tie-simplexes change continuously with respect to pressure, temperature, and overall composition. The continuity of the tie-simplex parameterization allows for interpolation using discrete representations of the tie-simplex space. For variations of composition, a projection to the nearest tie-simplex is used, and if the tie-simplex is within a predefined tolerance, it can be used directly to identify the phase-state of this composition. In general, our parameterization approach can be seen as the generalization of negative flash idea for systems with two or more phases. Theory of dispersion-free compositional displacements, as well as computational experience of general-purpose compositional flow simulation indicates that the displacement path in compositional space is determined by a limited number of tie-simplexes. Therefore, only few tie-simplex tables are required to parameterize the entire displacement. The small number of tie-simplexes needed in a course of a simulation motivates an adaptive tabulation procedure for the parameterization of the compositional space. Since a single tie-simplex 'supports' (i.e., identifies the phase-state of) a large number of compositions in its vicinity, the efficiency gains of adaptive construction are considerable. I will present the comparison of our adaptive tie-simplex parameterization method with conventional EoS (Equation-of-State) procedures for two- and three-phase displacements in homogeneous and heterogeneous reservoirs. The results indicate clearly that the new method is at least an order-of-magnitude more efficient than conventional EoS methods for two-phase flow. For multi-phase systems the approach helps to increase robustness of phase change identification that improves a stability of multi-phase flow simulation.