Evaluation of Dynamic Pseudofunctions for Reservoir Simulation

Abstract

Summary We present a detailed investigation on the reliability of some of the dynamic pseudofunctions used to upscale flow properties in reservoir simulation. A one-dimensional example (1D) and a real field application are used to evaluate methods developed by Kyte and Berry and Stone, and a new flux weighted potential(FWP) method. A derivation of Stone's method (based on the description given by Stone above) is presented, which is found to give an inconsistent set of equations. Stone's analytical example was used to illustrate how pseudo relative permeabilities that exhibit non-physical behavior may still give acceptable results, but this success can disappear with changes in boundary conditions. The pseudofunctions from a field application were not able to match the 2Dsimulations from which they were calculated, even when a different pseudofunction was used for each coarse grid block. Improvements were obtained when directional pseudofunctions were used, but still the results were not satisfactory. Similar results were found when comparing fine and coarse grid 3D simulations for a quarter of a five-spot pattern in this field. The results presented in this article suggest that dynamic pseudofunctions, as applied here and as commonly used in industry, may not be an adequate approach to up-scaling. The possibility of large errors and the difficulty in predicting when they may occur make the use of pseudofunctions examined in this paper unreliable. Introduction Oil and gas reservoirs are very complex systems in which rock and flow properties vary at all scales (pore to reservoir scale). Rock properties(e.g., porosity and absolute permeability) and saturation functions (e.g., relative permeability and capillary pressure) show variations that can be significant to oil recovery at scales below the size of common simulation grid blocks. One of the most important problems in reservoir simulation is that of accurately accounting for such small scale variations. In addition to the low resolution, coarse grid solutions can be strongly affected by numerical diffusion. Many pseudofunction techniques have been proposed to reproduce fine grid results(including detailed descriptions of heterogeneity and with minimum numerical diffusion) using the typical coarse grids of field simulations. Baker and Dupouy1 have a useful review of some of the competing alternatives methods (including Kyte and Berry2 and Stone3 methods). For example, they discuss a variation of the Kyte and Berry method which requires averaging pressures over the volume of a coarse grid cell, rather than using the mobility weighted average pressure over a face. The volume weighting method is more consistent with the derivation of the simulator's finite difference equations and avoids the occurrence of directional pseudocapillary pressures. However, at the coarse grid sizes usually deployed, the capillary pressure would have negligible influence. Another upscaling method is introduced in this paper similar to the Kyte and Berry method. It is based on a flux weighting of potentials at a face, referred to as the flux weighted potential(FWP). Except for certain specific analytic upscaling methods (Li et al.4), there are three broad approaches to upscaling which are pursued in the petroleum industry with varying degrees of success. These may be summarized as follows.Solve a fine grid 3D problem for a large representation area of the reservoir to be modeled, and apply an upscaling algorithm to the proposed coarse grid in this area. It is then necessary to demonstrate that the resulting pseudofunctions used in the coarse grid solution adequately reproduce the fine grid results. Some ad hoc decisions are then required on how to allocate the derived family of pseudofunctions to the whole reservoir. The difficulties with this approach can be the large cost with the fine grid solution, failure to reproduce it adequately with the coarse grid, and that the chosen area is not representative of other parts of the reservoir. One of the principal advantages is that the geometry of some of the real wells can be properly included in the large model area.Choose a few moderately small representative elements of volume (REV), and obtain a fine grid solution for the REVs, and the corresponding pseudofunctions for the coarse grid of the intended reservoir model. 2D cross sections are frequently used for this purpose. They have the merit of being much less expensive than a large area model. However, cross sections do not give any representations of the changing viscous to gravity ratios associated with areal sweep effects. Sometimes this is alleviated using a typical stream-tube geometry as a varying width in the cross section (Hewett and Berhens5), but this cannot deal with the 3D problem caused by areal heterogeneities. Interactions between real wells are neglected, and the allocation of the generated families of pseudofunctions to the coarse grid reservoir model becomes more problematic.A procedure referred to as successive renormalization6–8 is used. In this approach the central idea is to use fast solutions of flow problems in small Cartesian blocks as a basis for upscaling. The small blocks have artificial boundary conditions applied (e.g., constant pressure on two opposing faces and no flows on the other faces). The block solutions can be very fast, being independent of each other, and no special ad hoc assumptions are made about representative volumes. Successive sweeps can then be made at increasing block sizes, until the desired coarse grid size is attained for the large reservoir simulation. The choice of block size (King6 used 2×2×2 blocks) and the number sweeps is arbitrary. For some problems, error canceling between successive sweeps can occur, but in others, such as with complicated shale distributions, the cancellation does not occur. For two-phase renormalization, the use of a water injection only boundary condition on a block can generate large errors, and the influences of gravity slumping are ignored.

Large-scale Invasion Percolation with Trapping for Upscaling Capillary-Controlled Darcy-scale Flow

“…Nevertheless, it is usually observed that at relatively small scales capillary forces dominate the displacement process, while viscous and gravitational forces may become more important at larger scales (Odsaeter et al 2015;Kumar and Jerauld 1996;Jonoud and Jackson 2008a). Numerous upscaling methods are available in the literature for two-phase flow (Guzman et al 1999;Jacks et al 1973;Kyte and Berry 1975;Stone 1991;Pickup and Sorbie 1996;Lohne et al 2006;Virnovsky et al 2004;King 1989;King et al 1993;Coll et al 2001;Wallstrom et al 2002;Durlofsky 1997) among which flow-based methods are considered the most reliable. Under this category, two fundamental approaches are employed: steady-state and dynamic pseudos.…”

Incorporating Multiphase Behaviours in Dynamic Simulation Models Using In-Situ-Generated Pseudofunctions

“…The same assumption followed some litirature is adopted here, namely that the high resolution model, in which all faults are included discretely, is assumed to give the most realistic results (e.g. Cao and Aziz, 1999;Guzmán et al, 1999;Pickup et al, 2000;Darman, 2001), since no observed data is available to provide a comparison. The following steps, see Fig.…”

The importance of incorporating the multi-phase flow properties of fault rocks into production simulation models

“…A common assumption of any upscaling study is that the high-resolution model gives the most realistic result (e.g. Cao and Aziz, 1999;Guzmán et al, 1999;Pickup et al, 2000;Darmen et al, 2002). Similarly, we will assume that the model with the local grid refinement gives the most realistic result.…”