A New Method for Thermodynamic Equilibrium Computation of Systems With an Arbitrary Number of Phases

Reservoir recovery processes involve complex mass and heat transfer between the injected fluid and the resident rock-fluid system. Thermal-compositional reservoir simulators can be used to plan such displacement processes, in which the phase behavior is computed with an Equation of State (EoS). These thermodynamic-equilibrium computations include phasestability tests and flash calculations, and can consume a significant fraction of the total simulation time, especially for highly detailed reservoir models and a large number of components. Here, we propose a general Compositional Space Parameterization (CSP) method for complex mixtures, especially those where more than two fluid phases can coexist in parts of the parameter space. For a given pressure (P), temperature (T) and overall composition, a unique tie-simplex (tie-line for two phases, tie-triangle for three phases, etc.) can be defined. For a particular composition at P and T, the tiesimplex provides the necessary phase equilibrium information (i.e., phase state and phase compositions). For compositional flow simulation, a set of tie-simplexes can be calculated in a preprocessing step, or adaptively constructed during the simulation. The tie-simplex representation can be used to replace standard phase-equilibrium calculations completely, or it can be used as an initial guess for standard EoS calculations. Challenging examples with two and three phases are presented to validate this tie-simplex CSP approach. Standard EoS methods, which are widely used in industrial compositional simulators, are compared with CSP-based simulations for problems with large numbers of components and complex twoand three-phase behaviors spanning wide ranges of pressure and temperature. The numerical experiments indicate that our multi-dimensional tie-simplex representation combined with linear pressure and temperature interpolation in tie-simplex space, which is implemented as an adaptive tabulation strategy, leads to highly robust and efficient computations of the phase behavior associated with compositional flow simulation. Introduction Enhanced oil recovery (EOR) of heavy hydrocarbons mixtures requiring thermal processes are commonly described using a compositional model Aziz and Settari (1979). The equations that describe the flow, transport and thermodynamic behavior include mass and energy conservation and thermodynamic-equilibrium relations. The accurate description of the phase behavior of the multi-component multiphase mixtures as a function of pressure and temperature is a major challenge in compositional flow simulation. A thermodynamic approach using an Equations of State (EoS) is usually employed to describe the phase behavior. It is possible to determine the state of a mixture (phases) based only on the overall component compositions, pressure, and temperature. The use of an EoS based approach makes the simulation process time consuming. The thermodynamic-equilibrium equations are usually solved iteratively for each computational gridblock at every Newton iteration during a time step. Accurate and efficient integration of thermodynamic computations in compositional simulation has been the subject of several studies ( Firoozabadi, 1999, Michelsen and Mollerup, 2004). One of the widely used simplifications is the assumption of constant K-values (Aziz and Wong, 1988). In this approach, the component equilibrium ratios between the vapor and molar fraction are taken to be a function of pressure and temperature only, and this reduces the complexity of dealing with the nonlinear behaviors due to compositional dependence of the 2 MULTIPHASE NEGATIVE FLASH SPE 118996 phase equilibrium equations. Extension for this K-value approach has been proposed by Nghiem and Li (1988), where the equilibrium ratios are assumed to depend on the overall fraction of one of the components. However, this approximation is accurate only for low temperatures and can lead to difficulties in representing the phase behavior in the near-critical region. The determination of the phase state of the mixture is necessary to solve the system of conservation laws that describes multi-component, multi-phase transport in porous media. Once the phase state of a given gridblock is determined for the current nonlinear (Newton) iteration, the solution for the next iteration can be computed. The phase-state definition is an important aspect of compositional simulation. One needs to track the appearance and disappearance of the various phases. The disappearance of a phase is detected based on the saturation. When a phase saturation becomes negative, the phase is assumed to have disappeared. Phase appearance is usually treated using a phase stability analysis (Michelsen, 1982a). In this case, we need to determine the equilibrium state by computing the tangent plane distance (TPD) of the Gibbs energy function. This procedure must be performed for every single-phase cell at every nonlinear iteration of the flow computation process. It can be time consuming, especially when the number of phases and components is large. Rasmussen et al. (2006) presented a method to speed up the phase behavior computations for compositional mixtures in pipe flow. They proposed a criteria for avoiding the phase stability test for single-phase fluids. This approach is very efficient for transient simulation where the compositions change slowly. Since most nonlinear solvers for reservoir simulation include limits on the compositional changes during an iteration, this method can be applied to gas injection processes as well. However, it is difficult to extend this method for systems with more than two phases, or for more sophisticated nonlinear solvers, in which significant changes in the mixture composition of a gridblock are allowed. While two-phase EoS computations associated with compositional flow simulation are becoming quite robust and efficient, reliable methods for handling multi-component fluids with more than two phases are generally lacking. The purpose of this work is to provide an alternative method for phase-behavior computations in flow simulation for system with an arbitrary number of phases. Specifically, we propose a negative-flash based strategy within a compositional space parametrization (CSP) framework. Comparisons with standard EoS based approaches are used to demonstrate the effectiveness of our method. Compositional space parametrization for systems with arbitrary numbers of phases We summarize the compositional space parameterization (CSP) framework, which serves as the basis of our approach. The details of the CSP framework are given by Voskov and Tchelepi (2008c). General Thermodynamic Parameterization The overall mole fraction of a component, zi, is usually used in thermodynamic equilibrium calculations. The independent mole fractions form a subset of <c , where nc is the number of components, and can be expressed as follows: ∆nc−1 = {(z1 . . . znc) ∈ <c | ∑ zi = 1, zi ≥ 0, ∀i}. (1) This relation defines an (nc − 1)-dimensional simplex in <c . Geometrically, this means we can represent the compositional space using a line for two components, a triangle for three component (∆), a tetrahedron for four components (∆), and so on. In a multi-component multi-phase systems, the state of thermodynamic equilibrium is defined by the volume fractions of the phases, νj , and can be represented using an (np − 1)-dimensional tie-simplex defined as ∆np−1 = {(ν1 . . . νnp) ∈ <p | ∑ νj = 1, νj ≥ 0 ∀j}. (2) where np is the number of phases. Thus, we have a tie-line for two phases, a tie-triangle for three-phase, etc. The tie-simplex can be represented by the np vertices that correspond to the equilibrium fractions, namely, V1 = xi,1, V2 = xi,2, . . . Vnp = xi,np , (3) where xi,j is the mole fraction of component i in phase j. The relationship between the overall mole fraction, zi, and the phase equilibrium fractions is given by