We present a general framework for thermodynamic equilibrium calculations of multi-phase, multi-component mixtures. We use the fact that the compositional space can be represented as a high-dimensional simplex. For given values of pressure and temperature, the phase behavior of a particular system can be described using a low-dimensional tie-simplex, for example, tie-lines and tie-triangles for two and three phase systems, respectively. In general, a high-dimensional compositional space can be parameterized using a lower dimensional tie-simplex sub-space. Tie-simplex computation and interpolation procedures complement the parameterization to complete the mathematical framework. The robustness and efficiency of the method is demonstrated using several multi-phase equilibrium problems of practical interest. One type of problem is the equilibrium flash calculation of systems with a large number of phases. The complexity and strong nonlinear behaviors associated with such systems pose serious difficulties for standard techniques. The tie-simplex representation of the equilibrium data, which may be obtained using a particular equation-of-state for example, can be pre-processed. The parameterized space can be used to obtain the phase compositions (i.e., flash results) or used as an initial guess to accelerate convergence of standard Equation of State (EoS) based procedures. The second type of application is multi-phase multi-component displacement problems. In the standard compositional simulation approach, an EoS is used to describe the phase behavior. For each gridblock, given the temperature, pressure and overall compositions, the EoS is used to detect the phase state (e.g., one, two, or more phases), and to calculate the phase compositions, if multiple phases are present. These EoS computations can dominate the overall simulation cost. Adaptive computation of the tie-simplex space can be used to speed up the EoS computations of large-scale problems of practical interest by an order of magnitude, or more.


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