Extensions to Dietz Theory and Behavior of Gravity Tongues in Slightly Tilted Reservoirs
- F. John Fayers (BP Research Centre) | Ann H. Muggeridge (BP Research Centre)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- November 1990
- Document Type
- Journal Paper
- 487 - 494
- 1990. Society of Petroleum Engineers
- 5.1.1 Exploration, Development, Structural Geology, 4.1.2 Separation and Treating, 5.5 Reservoir Simulation, 5.3.2 Multiphase Flow, 5.2.1 Phase Behavior and PVT Measurements, 4.1.5 Processing Equipment
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This paper investigates accuracy implications in the modeling of unstable gravity tongues in slightly tilted, anisotropic reservoirs. An extended form of Dietz's equation is examined and its range of validity established by reference to high-resolution simulation. The theory is also used to investigate the onset of viscous fingering with a new viscous-to-gravity number.
When oil is displaced by either miscible or immiscible gas in a reservoir, the displacement rate is usually sufficiently slow that the resulting vertical sweep is dominated by the effects of gravitational segregation. In this type-of flow, the transition zone between oil and gas is often relatively thin compared with the reservoir thickness. The problem of simulating the displacement (whether miscible or immiscible) can then be reduced to that of tracking the motion of an interface that divides a zone containing only mobile oil from a zone above containing only the invading gas in the presence of immobile residual oil. Approximate solutions to this type of problem can be obtained with the well-known Dietz equation; problem can be obtained with the well-known Dietz equation; however, extensions to this theory that give improved accuracy are provided by solving the less familiar equation derived by Sheldon provided by solving the less familiar equation derived by Sheldon and Fayers. This theory, which applies to the motion of a slightly tilted interface in a thin, slightly tilted reservoir, introduces terms that are dependent on the reservoir tilt and the curvature of the gas/oil interface. The analysis is derived for the gas/oil problem, but equations also apply to segregated water/oil displacement.
The simulation of segregated flow is still a significant problem in the present era of black-oil simulation codes because the thickness of the transition zone between the oil and gas is generally thin compared with the dimensions of the gridblocks typically used in finite-difference schemes. Numerical diffusion effects can then sig-nificantly influence the finite-difference solution. It is there form-portant to have a benchmark theory against which to test the capabilities of these black-oil models.
As a first step, a very accurate procedure for solving the general, 2D, two-phase flow and miscible displacement equations that incorporates flux-corrected transport techniques is used to investigate the importance of physical and numerical dispersion effects by simulating the vertical-section miscible experiments reported by Crane et al. We show that, with care, this high-resolution modecan be used to infer a nondispersive solution that is in excellent agreement with the extended Dietz theory. We then investigate the effects of reduced vertical/horizontal-permeability ratios, kV/kH, on segregated flow. We also show that the extended Dietz method still applies when the vertical anisotropy is not too small, but progressively fails as the ratio is reduced below about 0.3. progressively fails as the ratio is reduced below about 0.3. Finally, we illustrate some aspects of flow stability and the breakdown of segregated flow for miscible displacements in a tilted, thin bed.
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