Abstract
In the past miscible displacement calculations have been approximated with two-phase reservoir simulators. Such calculations have neglected diffusional mixing between miscible components. In fact, no analog bas been proposed for rigorously treating miscible simulations with two-phase programs. This paper describes requirements that programs. This paper describes requirements that permit such a rigorous simulation. permit such a rigorous simulation. The sets of partial differential equations describing each of the displacement processes are shown to be exactly analogous if relative permeability and capillary pressure functions are permeability and capillary pressure functions are adjusted in a special manner. Application of the "miscible" analogy in a two-phase simulator, however, has several limitations, the most severe of which is the truncation error (numerical diffusion) typical of an immiscible formulation. Since this error is time-step and/or block-size dependent, numerical smearing can, in principle, be made as small as necessary. But this feature limits the practical applicability of the "miscible" analogy practical applicability of the "miscible" analogy to cases with rather large physical diffusion. The range of applicability and other limitations are outlined in the paper. Also, illustrative sample calculations are presented for linear, radial and layer-cake systems. Component densities and viscosities are varied in the linear model.
Introduction
In recent years, use of two- and three-phase reservoir simulators to calculate the performance of immiscible fluid displacement has become widespread. Reservoir simulators capable of calculating miscible displacement problems, however, have been limited to special use programs. The primary reason for this limitation has been the significant truncation error (numerical diffusion) typical of ordinary finite difference approximations to the miscible equations. The method of characteristics provided a means of making miscible displacement calculations without significant truncation error. A recently proposed second calculation technique, based on variational methods, also reduces numerical diffusion. Both of these calculational techniques can be used for immiscible calculations. Still, general miscible displacement applications such as gas cycling, enriched-gas injection, or tracer injection have historically required use of immiscible reservoir simulators for performance predictions. Larson et al. have reported an example of such use of a two-phase computer program.
Displacement involving two components flowing within a single phase would appear to be analogous to a two-phase displacement. Yet, past miscible calculations using immiscible simulators made the two-phase saturation profile as near piston-like as possible and neglected component mixing due to possible and neglected component mixing due to diffusional processes. The capillary pressure function was chosen to minimize capillary flow. Also, in these miscible approximations, no provision had been made for viscosity variations provision had been made for viscosity variations with component concentration. Though mixing due to diffusional processes had been neglected, countercurrent diffusion due to component concentration differences in a miscible process should be essentially analogous to countercurrent capillary flow due to saturation differences in a two-phase system. This paper describes a method by which two- and three-phase reservoir simulators can be made to calculate miscible displacement rigorously. The only requirement of the method is that relative permeability and capillary pressure be special permeability and capillary pressure be special functions of saturation. With these properly chosen functions, the set of partial differential equations describing immiscible displacement becomes completely analogous to the partial differential equations describing miscible displacement.
SPEJ
P. 192
A Regularization Homotopy Strategy for the Constrained Parameter Inversion of Partial Differential Equations