A General Formulation for Simulating Physical Dispersion and a New Nine-Point Scheme
- Gautam S. Shiralkar (Amoco Production Co.) | Robert E. Stephenson (Amoco Production Co.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- February 1991
- Document Type
- Journal Paper
- 115 - 120
- 1991. Society of Petroleum Engineers
- 5.3.2 Multiphase Flow, 4.1.2 Separation and Treating, 1.10.1 Drill string components and drilling tools (tubulars, jars, subs, stabilisers, reamers, etc), 5.3.1 Flow in Porous Media, 4.3.4 Scale, 5.7.2 Recovery Factors, 5.2.1 Phase Behavior and PVT Measurements, 2.4.3 Sand/Solids Control, 5.4.9 Miscible Methods, 1.2.3 Rock properties, 5.4 Enhanced Recovery, 5.5 Reservoir Simulation, 5.4.1 Waterflooding
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Summary. This paper presents a general numerical formulation for simulating physical dispersion that avoids the convergence and grid-orientation problems associated with numerical dispersion. The formulation is applicable to a wide range of problems, including enhanced gasdrives with phase discontinuities. A new nine-point scheme is also presented, along with general rules for any nine-point scheme to obey Darcy's law. Converged solutions are obtained for displacements at adverse mobility ratios of 5,000 with no grid-orientation effects.
Physical dispersion plays a critical role in determining phase behavior and recovery for many reservoir displacement processes. In miscible gasdrives, dispersion influences both sweep and displacement efficiency. Only recently, however, have attempts been made to include dispersion effects in reservoir simulators, partly because of numerical difficulties.
The straightforward inclusion of physical dispersion in conventional reservoir simulators is complicated by upstream weighting of mobilities. Upstream weighting introduces a truncation error that has a form similar to that of physical dispersion. This "numerical dispersion" cannot be related one-to-one to physical dispersion; moreover, the magnitude of this numerical dispersion depends not only on grid size, but also on grid orientation, resulting in the well-known grid-orientation effect. The nine-point scheme of Yanosik and McCrackens and other similar schemes reduce grid-orientation effects, but often do not converge as the grid is refined when upstream weighting is used.
Young's approach to modeling physical dispersion used centered differences for evaluating convective and dispersive fluxes. The method lacked generality, however, because it was not applicable to problems involving (a) gravity or capillary forces and (b) spatial phase discontinuities. The latter class of problems includes the important cases of enhanced gasdrives. This paper presents a general formulation for simulating physical dispersion that is not subject to these limitations.
Grid-orientation effects can be important in miscible floods with adverse mobility ratios. These require more powerful numerical schemes than traditional five-point differencing. A new nine-point scheme is presented that always gives positive transmissibilities.
In the absence of dispersion, the reservoir flow of Component i is
For convenience, source/sink terms associated with wells have been omitted. In immiscible floods (e.g., waterfloods), neglecting physical dispersion is appropriate. In miscible floods, however, physical dispersion can be important, enhancing sweep efficiency and reducing displacement efficiency. To include physical dispersion, we can write (2)
Bear gives the following form for the dispersion tensor:
where i =x, y, z and j =x, y, z. Usually, molecular diffusion is negligible compared with mechanical mixing. Dispersivities cent and , are not purely rock properties, but may depend on the flow, viz., the extent of viscous fingering, saturations, and other conditions. 6 Young's approach, 4 which we follow, is to account for the primary dependence on phase velocity. For isotropic dispersion, = ,= , Eq. 3 simplifies to
This approach may be considered a first, but important, step in including the effects of physical dispersion on a field scale. At the same time, the present work has allowed more detailed studies of laboratory-scale corefloods with a view to a better understanding and quantification of the dispersivity dependence on flow variables. Inserting Eq. 4 into Eq. 2 gives
Numerical solutions to Eq. 1 that do not introduce sufficient dispersion of some kind exhibit physically unrealistic spatial oscillations or wiggles in Cim. These numerical wiggles represent the exact solution to the finite-difference equations. The physical dispersion term in Eq. 5 eliminates the wiggles, provided that the grid size is smaller than a certain value dictated by the "maximum principle."
Flows, in general, have a miscible and an immiscible aspect. We associate the convection and dispersion of concentrations with the miscible aspect. The convection term is proportional to cim, while the dispersion term is proportional to 2Cim. Beyond this, any truncation error that involves 2Cim introduces, through a numerical scheme such as upstream weighting, an undesirable dispersion of concentrations that makes the numerical solution grid-size and grid-orientation dependent.
We associate the convection and dispersion of phase saturations with the immiscible aspect. Steep saturation gradients tend to be smeared by capillary pressure forces-e.g., in the "stabilized zone" of a waterflood front. For most field-scale problems, capillary pressure effects are small enough to be unimportant. An artificial dispersion term much larger than the actual capillary pressure may then be used to avoid spatial numerical oscillations in saturation. For laboratory-scale corefloods where capillary pressure can be important, choosing a small enough grid makes the artificial dispersion negligible. The "immiscible" or "artificial" dispersion is related to 2Sn or 2[function(Sm)].
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