An Analytical Model for One-Dimensional, Three-Component Condensing and Vaporizing Gas Drives

An analytical model based on the method of characteristics is presented for the calculation of one- dimensional (1D), three-component condensing and vaporizing gas dives. The model describes (1) mass transfer between oil and gas, (2) swelling and shrinkage, (3) viscosity and density changes, (4) gravity stabilization, and (5) rock/fluid interaction. The main assumptions of the model are local thermodynamic equilibrium and the absence of dispersion, diffusion, and capillarity. Example calculations are presented that bring out the main features of both condensing and vaporizing gas drives and also indicate the importance of mass transfer between the phases. In the special case of "developed miscibility," the model predicts a piston-like displacement having a complete recovery at gas breakthrough. The main applications of the model are in (1) conceptual studies of gas drives in which mass transfer plays an important role and (2) the calibration and checking of numerical reservoir simulators for multicomponent, multiphase flow. Introduction Gas injection is increasingly being applied as a secondary or tertiary recovery technique. In many applications injection gas is not directly miscible and is not in thermodynamic equilibrium with reservoir oil. As a consequence, component transfer takes place between gas and oil, which has a direct bearing on the displacement efficiency of the gas-injection process. Depending on the component transfer, two different processes are commonly distinguished: condensing and vaporizing gas drives. In condensing gas drives, the composition of the gas phase becomes progressively leaner on contact with the reservoir oil; the heavier components in the injection gas "condense" in the oil phase. Condensing gas drives occur when relatively rich gas is injected and are therefore called "rich" or "enriched" gas drives. In vaporizing gas drives, the reverse process occurs: the gas phase becomes progressively richer owing to vaporization of the middle components of the reservoir oil. Vaporizing gas drives occur when relatively lean gas is injected and are therefore called "lean" gas drives. A mechanistic understanding of oil displacement by immiscible, nonequilibrium gases is no simple matter. In these processes the flow of the two phases--gas and oil--is strongly influenced by the phase behavior of the multicomponent gas/oil mixture. This is compounded by the nonconstant physical properties of gas and oil resulting from compositional changes during the displacement. To investigate multicomponent gas drives theoretically, two approaches can be taken. First, the numerical approach: the basic differential equations are directly cast in a difference form and subsequently solved. In principle, this approach can handle many components and three dimensions. The drawback of the numerical approach is that possible sharp fronts are smeared out by numerical dispersion, which may obscure the results and make interpretation rather difficult. The second approach is the analytical one: the basic differential equations are simplified such that they become amenable to analytical mathematical analysis, notably the method of characteristics. This approach is less versatile in that it generally will be restricted to one dimension and a small number of components. Analytical models, however, are very helpful in obtaining a mechanistic understanding of the process. In addition, these models can accommodate sharp fronts and can therefore be used to calibrate and check numerical models. The first successful attempt to describe the coupling of two-phase flow and phase behavior in gas drives analytically was made by Welge et al. They investigated a 1D, three-component condensing gas drive and developed a calculation method essentially based on the method of characteristics. The problem of coupled multiphase flow and phase behavior also occurs in alcohol and surfactant flooding. Here the problems also can be formulated such that they can be solved by the method of characteristics. Wachmann presented a theory for alcohol flooding along these lines. Larson and Hirasaki and Larson applied the theory of characteristics to surfactant flooding. Recently Helfferich presented a general theory on 1D multiphase, multicomponent fluid flow in porous media. based on concepts developed in the area of theoretical multicomponent chromatography. Hirasaki applied these concepts to surfactant flooding. SPEJ P. 169^