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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.

This paper presents a detailed investigation on the reliability of the dynamic pseudo functions used to up-scale flow properties in reservoir simulation. A theoretical study and a real field application are used to evaluate Kyte and Berry (1975). Stone (1991), and a new flux weighted potential (FWP) method. A derivation of Stone's method is presented and shown to use 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 pseudo functions from the field application were not able to match the 2D simulations from which they were calculated, even when a different pseudo function was used for each coarse grid block. Improvements were obtained when directional pseudo functions 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 paper suggest that dynamic pseudo functions, 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 pseudo functions 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 flow properties (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. Several pseudo function 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. The dynamic pseudo functions of Jacks et al. (1973) and Kyte and Berry (1975) were designed with the purpose of reducing the dimensionality of a problem (e.g. including 3D effects within a 2D model), and in general, reducing numerical diffusion in coarse gridded models. Simple rules to average absolute permeability were also included in these methods. More recently, up-scaling due to heterogeneities has become the main issue. It was realized that heterogeneities not included in coarse grid simulation could have a major impact on oil recovery predictions. New methods were presented by Hewett and Berhens (1991), Stone (1991), Beier (1992) for dynamic pseudo functions, and by King (1989) and King et al. (1993) for single and two-phase renormalization. The main problem with renormalization is the systematic error that is introduced due to the artificial boundary conditions (e.g. no flow or constant pressure) imposed on each block at each renormalization step. Hewett and Berhens (1993) and Yamada and Hewett (1995) have explained the origin of this error as the forcing of incorrect flow paths or streamlines through the rescaled grid blocks. They have shown that the magnitude of the error can be large and yield incorrect up-scaling (also see Malick and Hewett (1995) for a quantification of the error). The errors in multiphase renormalization are larger than in single-phase since, in addition to incorrect boundary conditions in pressure, multiphase flow renormalization imposes incorrect boundary conditions for fractional flow. The method of Durlofsky et al. (1994) tries to remedy the problems with the renormalization technique. In this method, a non-uniform coarse grid is defined using the flow velocity from a single-phase flow solution (i.e., trying to honor the streamlines), and up-scaling using periodic instead of no flow boundary conditions. Significant errors are still introduced by the artificial boundary conditions (Yamada 1995), and only single-phase flow was up-scaled properly. P. 9

The measurement of geomechanical properties of reservoir rock and caprock for completion optimization, enhanced oil recovery (EOR), and disposal/storage of any kind is becoming an integral and key aspect of asset evaluation and appraisal. One of the most important of these characteristics is an in-situ evaluation of the magnitude and variation of the minimum in-situ stress, the measurement of which is critical for geomechanical modelling and thereby a range of applications such as well construction, caprock integrity, and completion optimization. A wireline formation testing (WFT) tool is a common approach for obtaining direct measurements of these stresses at a range of depths. This process is referred to as microfracturing and is most typically performed in an openhole environment. Typical toolstrings consist of a straddle packer arrangement, a pumping mechanism, gamma ray for accurate depth correlation, a motorized valve/manifold arrangement, and pressure/temperature gauges. To perform a stress test, a specific interval of the wellbore is isolated by inflating the straddle packers. The interval is then pressurized by incrementally pumping fluid until a tensile fracture has been initiated. In an open hole, the fracture will initiate and propagate normal to the minimum stress at the wellbore and multiple injection and falloff cycles are subsequently performed to ensure fracture growth beyond the influence of the hoop stress regime. The data are then analysed to determine fracture initiation, reopening, propagation and closure pressures. Additionally, it may be possible to approximate fracture orientation, if an image log is available. This paper describes the process of obtaining minimum in-situ stress measurements using a WFT and advanced integrated stress analysis (ISA) process, in an ultradeep reservoir at ultrahigh pressures. Lessons learned and best practices are highlighted along with their importance for efficient job execution. The integrated geomechanical analysis covers subsequent generation of a calibrated stress model with minimum horizontal stress measured during microfracturing. Factors include evaluation of the stress contrast in the target formations and evaluation of the overburden gradient and mechanics for microfracturing job design for future operations (breakdown pressure) and lessons learned such as station selection, backup packer availability, and influence of stress cage material on breakdown, to name but a few. Obtaining accurate knowledge of in-situ minimum stress values, based on actual measurements, is a key step on the road to effective execution, and the earlier that this is achieved, the more efficient the results of any development. This paper summarises the successful application of the WFT approach in delivering such data under extremely harsh depth and pressure conditions, but resulting in a measurement from which numerous subdisciplines can conduct their decision making and design.