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Summary
This paper presents a new approach for constructing approximate analytical solutions for 1D, multicomponent gas displacement problems. The solution to mass conservation equations governing 1D dispersion-free flow in which components partition between two equilibrium phases is controlled by the geometry of key tie lines. It has previously been proven that for systems with an arbitrary number of components, the key tie lines can be approximated quite accurately by a sequence of intersecting tie lines. As a result, analytical solutions can be constructed efficiently for problems with constant initial and injection compositions (Riemann problems). For fully self-sharpening systems, in which all key tie lines are connected by shocks, the analytical solutions obtained are rigorously accurate, while for systems in which some key tie lines are connected by spreading waves, the analytical solutions are approximations, but accurate ones. Detailed comparison between analytical solutions with both coarse- and fine-grid compositional simulations indicates that even for systems with nontie-line rarefactions, approximate analytical solutions predict composition profiles far more accurately than coarse-grid numerical simulations. Because of the generality of the new approach, approximate analytical solutions can be obtained for any system having a phase behavior that can be modeled by an equation of state. The construction of approximate analytical solutions is shown to be orders of magnitude faster than the comparable finite difference compositional simulation. Therefore, the new approach is valuable in situations requiring fast compositional solutions to Riemann problems are required.
Introduction
Miscible gas-injection processes have become a widely used technique for enhanced oil recovery throughout the world. The understanding of the multiphase, multicomponent flow taking place in any miscible displacement process is essential for successful design of gas-injection projects. Due to complex reservoir geometry and reservoir fluid properties, numerical simulations of the flow processes are usually conducted to obtain such understanding. In principle, compositional simulation could be used to study such problems. In practice, however, conventional finite difference (FD) simulation is sufficiently slow that three-dimensional (3D) computations are feasible only for relatively coarse grids. As a consequence, such FD simulations can be strongly affected by numerical artifacts and may ultimately lead to misleading conclusions.
Recent progress in the application of streamline methods offers one way to overcome the limitations of 3D finite difference compositional simulations.1–4 In the streamline approach, a 1D solution is mapped onto streamlines that capture the effects of reservoir heterogeneity. Thiele et al.2 described 2D and 3D streamline compositional simulations in which analytical and finite difference approaches were used to solve the 1D flow problem. Thiele et al. used a numerical solution of the 1D problem to perform a compositional simulation for a heterogeneous 3D reservoir described with 518,000 gridblocks. At that time, analytical solutions for problems with an arbitrary number of components in the oil and injection gas were not available. In this paper, we describe an algorithm to obtain analytical solutions for that problem. Use of the analytical solutions in simulations like those of Thiele et al.2 could lead to substantial additional speed-ups in streamline calculations.
A substantial body of mathematical theory now exists for construction of analytical solutions to the dispersion-free 1D flow problem.5–10 Those investigations considered four-component systems primarily, but special case solutions for systems with more than four components were reported for fully self-sharpening displacements by pure injection gases. The first systematic attempts to describe multicomponent gas/oil systems were restricted to calculation of the minimum miscibility pressure (MMP).11–13 Those calculations were based on identifications of the key tie lines that control miscibility. Calculation of the full analytical solutions for multicomponent oils and gases was not required to determine the MMP and was not attempted. In this paper, we build on the approach for approximating key tie lines by adding tools from the analytical theory of gas displacement to obtain the full solution to the 1D, dispersion-free, two-phase flow problem for an arbitrary number of components in the oil or the injection gas.
Analytical Theory of 1D Miscible Displacements
The analytical theory of gas-injection processes describes the complex interactions between two-phase flow and phase equilibrium for 1D dispersion-free miscible displacements. Analytical solutions obtained in this paper are based on the following assumptions:The 1D porous medium has constant permeability and porosity.Instant thermodynamic equilibrium exists between phases present at any point.No gravity or capillary forces act on the fluid.Pressure and temperature are constant throughout the porous medium for the purpose of calculating phase equilibrium.Components do not change volume as they transfer between phases.
The assumption of no volume change is reasonable when pressures are high. For systems at lower pressures, in which solubility of light components in undisplaced oil is high but gas density is low, effects of volume change can be significant,10 and Dindoruk's formulation of the conservation equations should be used.

DOI: 10.2118/74700-PA

# Fast, Approximate Solutions For 1D Multicomponent Gas-Injection Problems

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