Interplay of Phase Behavior and Numerical Dispersion in Finite-Difference Compositional Simulation
- Kristian Jessen (Stanford U.) | Erling H. Stenby (IVC-SEP, Technical U. of Denmark) | Franklin M. Orr Jr. (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2004
- Document Type
- Journal Paper
- 193 - 201
- 2004. Society of Petroleum Engineers
- 5.2 Reservoir Fluid Dynamics, 5.4.2 Gas Injection Methods, 5.6.9 Production Forecasting, 4.6 Natural Gas, 4.1.2 Separation and Treating, 5.2.1 Phase Behavior and PVT Measurements, 5.4.9 Miscible Methods, 4.1.4 Gas Processing, 4.3.4 Scale, 4.1.5 Processing Equipment
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Compositional simulation of multicontact miscible (or near-miscible) gas injection problems is typically performed by use of numerical finite-difference (FD) schemes. Such simulations are intrinsically affected by numerical dispersion.* In this paper, we propose a method to assess the sensitivity of a particular displacement calculation to the effects of numerical dispersion for systems with realistic multicomponent fluid descriptions. We use two simple ternary systems to illustrate how dispersion and convection interact to determine displacement composition paths and to define a "dispersive distance." We show how the dispersive distance relates to displacement performance with a detailed analysis of grid size effects on the recovery predictions for 1D displacements of a CH4-C4-C10 system by pure N2 and mixtures of N2 and CH4 . We then show that the sensitivity of multicomponent compositional simulations also correlates well with the dispersive distance. We demonstrate that quantitative difference between oil recovery predicted by coarse-grid numerical simulations, and values obtained from analytical solutions to the conservation equations, can be estimated well from calculating the distance between the dilution line and the composition path predicted by the method of characteristics. Finally, we apply the assessment of sensitivity to 2D displacements in a heterogeneous porous medium to demonstrate the interplay of phase behavior, adverse mobility, heterogeneity, and numerical dispersion.
The effects of numerical dispersion in conventional FD compositional simulations have been studied extensively ever since computers were first applied to prediction of reservoir performance and generation of production forecasts. The papers of Stalkup,1 Stalkup et al.,2 Lim et al.,3 Walsh and Orr,4 Haajizadeh et al.,5 and Wang and Peck6 are examples of such studies. This paper focuses on the apparent system-specific sensitivity to numerical dispersion observed for the numerical simulations reported by Jessen.7 Walsh and Orr4 demonstrated, on the basis of 1D ternary displacement problems, that the sensitivity to numerical dispersion for a given system is related to the phase behavior of the system in terms of the size and the shape of the two-phase region. In this paper, we show how to evaluate the quantitative impact of numerical dispersion on calculated displacement performance.
This paper considers the simple but still often applied one-point upstream weighting formulation of the mass conservation equations. Two types of FD simulators were used:
1. A simplified slimtube model described by Dindoruk,8 in which the pressure drop along the displacement is considered negligible for the purpose of calculating phase equilibrium.
2. The commercial simulator Eclipse 300.
Numerical dispersion in this type of simulation emerges partly from truncation errors introduced by the FD representation of the convective term,9 and partly from the fact that FD simulations of this kind basically correspond to a sequence of interconnected mixing cells. Aris and Amundson10 demonstrated the asymptotic equivalence of mixing cells in series and the convection-diffusion equation. In mixing cell terminology, the characteristic of numerical dispersion is that material entering one cell can be allowed to enter the next cell faster than normal flow would allow. The magnitude of the artificial dispersion is of the order ?z/2, which for reservoir-scale modeling often exceeds what is physically realistic.1
To illustrate the effects of numerical dispersion, and to demonstrate the method we propose for predicting when displacement systems are sensitive to numerical dispersion, we consider displacements for a very simple ternary system with equilibrium K-values that are constants, independent of composition. We also fix the viscosity ratio and use simple quadratic Corey curves for the relative permeability functions. A closed-form analytical solution for that system was obtained by Wang.11
We begin by considering a displacement that is relatively sensitive to the effects of numerical dispersion. Results of FD simulations are compared with the analytical solution in Figs. 1 and 2 for a system with KCH4=2.5, KCO2=1.5, KC10=0.05, and with viscosity ratio (oil to gas) of 5. The numerical solutions were calculated with 20, 100, and 1,000 gridblocks. The analytical solution, shown as a solid line in the ternary diagram and profiles in Fig. 1, includes a leading shock along the tie line that extends through the initial oil composition, a short continuous variation along that tie line, a long continuous variation along a curved path that connects the initial tie line to the injection gas tie line, and a trailing shock.
With 20 gridblocks, the FD solution differs significantly from the analytical composition path. As the number of gridblocks is increased to 100 and then to 1,000, the FD path approaches the analytical solution, and correspondingly, the FD saturation and composition profiles also approach the analytical values. The calculated path for 10,000 gridblocks (not shown) is indistinguishable from the analytical solution on the scale of the ternary diagram in Fig. 1. Clearly, the FD solutions converge to the analytical solution, though accurate solution requires a substantial number of gridblocks.
The saturation and composition profiles are plotted in Fig. 1 against the propagation velocity (?/t). To obtain the profile at a particular time, the propagation velocity is simply multiplied by the time in pore volumes injected (PVI). With 20 gridblocks, for example, the region of two-phase flow is extended substantially at either end of the transition zone, and all the composition profiles are smeared significantly. As the number of gridblocks increases, however, the features of the solution profiles are resolved more accurately. For example, the CO2 bank that follows the leading shock is barely resolved at all with 20 gridblocks, but it is better delineated with 100 and 1,000 gridblocks. While more than 20 gridblocks would be used for most 1D compositional simulations, 2D and 3D simulations might very well be limited to 20 or fewer gridblocks in one direction by limits on computation time. Fig. 1 shows that such coarse grids alter compositional behavior substantially.
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