Analytical solutions for dispersion-free flow of two-phase, four-component mixtures confirm the existence of condensing/vaporizing gas drives and reveal how they behave.
Summary Analytical solutions are presented that confirm the existence of a combined condensing and vaporizing displacement mechanism in enriched gas drives. The solutions are derived for dispersion-free, one-dimensional displacements in four-component hydrocarbon systems. A simple geometric construction is used to find a key tie line in the solution, the "crossover" tie line. This tie line is shown to control the development of miscibility in condensing/vaporizing systems, and it connects the condensing and vaporizing portions of the displacement.
Introduction Pseudoternary diagrams have traditionally been used to explain the behavior of multicontact miscible (MCM) gas drive processes. According to three-component theory, only two types of displacements are possible, vaporizing or condensing gas drives. Both qualitative mixing cell arguments and more rigorous mathematical approaches show that a ternary displacement can be MCM only if either the oil composition (vaporizing gas drive) or the injection gas composition (condensing gas drive) lies outside the region of tie-line extensions on a ternary phase diagram. Thus, a ternary displacement can be either a condensing or a vaporizing drive but not both simultaneously.
Zick, however, used experimental and simulation results to show that real oil displacements by gas have characteristics of both vaporizing and condensing gas drives. Stalkup also observed that classical displacement mechanisms cannot explain the behavior of real oil displacements by enriched gases. Novosad et al. agreed with this view, but concluded that at high temperatures a pure vaporizing mechanism is responsible for high recovery in enriched gas displacements. Tiffin et al ., however, concluded that enriched gas displacements were primarily condensing, and any vaporizing mechanism that might contribute would be inconsequential to the oil recovery. Lee et al. attempted to explain these contradictory results by showing that real field displacements can be either vaporizing, condensing, or both.
To investigate whether both condensing and vaporizing mechanisms contribute to development of miscibility we consider displacements in four-component systems. We report the first definitive analytical proof that Zick's description of a combined condensing and vaporizing mechanism is correct for some enriched gas drives. We also extend significantly the analytical solution technique described by Orr et al. and Monroe et al. for pure gas injection to a two-component gas mixture. Two types of solutions are obtained for condensing/vaporizing (C/V) drives. We illustrate these types for two enriched gas displacement systems. In the first, mixtures of methane (CH4) and propane (C3) displace an oil containing CH4, n-hexane (C6), and n-hexadecane (C1). In the second, the injection gas is a mixture of CH4 and ethane (C2), while the oil remains unchanged. In both solutions, flow behavior is strongly influenced by a key tie line, the "crossover" tie line. We demonstrate how to find the crossover tie line in each case, and show that it is the volatility (equilibrium K-values) of the C2 or C3 that determines the type of solution. Finally, we examine how development of miscibility occurs for C/V systems.
Conservation Equations Dispersion-free flow in one-dimension is modeled using hyperbolic conservation laws under the assumptions stated by Helfferich. To simplify the analysis presented here, we assume that the total volume of both phases is invariant with component mixing. Extension of the theory to include the effects of volume change on mixing, where flow velocity varies, is described elsewhere. The dimensionless conservation equations areEquation 1
Where Ci and Fi are defined by Helfferich asEquation 2
The gas phase fractional flow, fg, used in the calculations isEquation 3
With the residual oil saturation Sor=0.20. Phase viscosities were determined with the Lohrenz-Bray-Clark correlation using phase densities calculated with the Peng-Robinson equation of state (PREOS). The equlibrium mole fractions, xij, were calculated using the PREOS with critical properties, acentric factors, and binary interaction parameters given in Table 1. Component densities used to convert mole fractions to volume fractions are also given in Table 1.
The analytical solution is constructed using the method of characteristics (MOC) to calculate the velocity at which a given composition propagates through the porous medium. An eigen-value problem results from manipulation of Eq. 1, in which the eigenvalues are the characteristic velocities and the eigenvectors are the characteristic directions in composition space. The solution route can then be constructed by following a sequence of eigenvector paths.