(580d) Compositional Space Parameterization for Compositional Reservoir Flow Simulation

Phase equilibrium calculations can be the most expensive kernel in 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 is based on (1) performing a phase-stability test for single-phase gridblocks, and (2) solving the fugacity constraints together with the coupled nonlinear flow equations when the gridblock has more than one phase.

One can parameterize the multi-phase compositional space 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. For a given composition, parameterization in pressure and temperature can be obtained using a multi-phase negative flash procedure. In order to correctly identify the phase state for a composition that lies outside a higher dimensional tie-simplex, parameterization can be performed for all lower dimensional phase states (e.g., tie-lines around a tie-triangle) at a fixed pressure and temperature.

Continuity of the tie-simplexes in pressure, temperature, and overall composition is a necessary condition for the validity of the parameterization. The challenge is that all of these parameters can change considerably during the course of a simulation. Here, we prove that the tie-simplexes change continuously with respect to pressure, temperature, and overall composition. We also show that by changing any of these three parameters, the computed tie-simplex through a given composition can degenerate to a lower dimensional tie-simplex, which also varies continuously in terms of the three parameters. The continuity of the tie-simplex parameterization allows for interpolation using discrete representations of the tie-simplex space as a function of pressure, temperature and composition. Here, we interpolate in tie-simplex space as a function of pressure and temperature using tables. For variations of composition, we use projection to the nearest tie-simplex, and if the tie-simplex is within a predefined tolerance, it can be used directly to identify the phase-state of this composition.

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.

We compare 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 flow simulation.