IMPES Stability: Selection of Stable Timesteps
- K.H. Coats (Coats Engineering, Inc.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2003
- Document Type
- Journal Paper
- 181 - 187
- 2003. Society of Petroleum Engineers
- 5.4.2 Gas Injection Methods, 4.1.2 Separation and Treating, 5.5 Reservoir Simulation, 4.1.5 Processing Equipment, 4.6 Natural Gas, 7.1.8 Asset Integrity, 5.2.1 Phase Behavior and PVT Measurements
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An IMPES stability criterion is derived for multidimensional three-phase flow for black-oil and compositional models. The grid may be structured or unstructured. Tensor considerations are neglected. The criterion can be used to set the time steps in an IMPES formulation or as a switching criterion in an adaptive implicit model.
The criterion extends previous work by accounting for three-phase flow, including capillary, gravity, and viscous forces, with all the possible cocurrent and countercurrent flow configurations in a general grid. The criterion derivation uses stability theory to the limits of its applicability, augmented by numerical experimentation, including extensive 1D tests and numerous field study datasets.
A reservoir simulation model consists of N*N nonlinear difference equations which express conservation of mass of Nc components in each of N gridblocks comprising the reservoir. The form of each of these equations, for a given block, is
where MI,n=the mass of component I in the block at time level n. The first sum is over all block neighbors; the second sum is over all wells completed in the block. Well terms are assumed to be treated fully implicitly and are dropped from consideration.
The IMPES formulation1,2 treats the interblock flow rates implicitly in pressure, but explicitly in saturations and compositions. This explicit treatment gives rise to a conditional stability for IMPES,
where ?t=maximum stable timestep and Fi=some function of rates and reservoir and fluid properties. This paper derives the function Fi for compositional and black oil models, accounting for viscous, gravity, and capillary pressure forces in cocurrent and countercurrent three-phase flow. The grid may be unstructured, or structured (e.g., Cartesian) with or without nonneighbor connections. The flow may be 1D, 2D, or 3D. Tensor considerations are neglected. Condition 2 gives a different stable step value for each block. In an IMPES model, the time step used is the minimum of all blocks' stable step values. In an AIM (adaptive implicit) model,3 each block's stable step size can be used to determine if the block needs implicit treatment. The effects on stable step size of (a) interphase mass transfer and (b) the pressure and composition dependence of fluid properties are assumed small and neglected.
Two functions Fi are derived for use in Condition 2. The first relates to effects of explicit treatment of the saturation-dependent terms (relative permeability and capillary pressure) in the interblock flow rates. The second relates to the explicit treatment of compositions in the interblock flow rates.
Derivations of the function Fi are lengthy and, at various points, tedious. This tends to obscure the simplicity and low cpu expense associated with the final results. Therefore, a Summary of Final Results section gives the final results, followed by sections describing the derivations.
Summary of Final Results
For the unstructured grid case, the subscript i denotes a gridblock, and the subscript j denotes one of its neighbors. Derivations using a Cartesian grid use subscripts i, j, k as the gridblock indices in the x, y, and z directions, respectively. In all equations throughout the paper, each phase mobility and its derivatives are evaluated at the upstream block for the phase.
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