Summary This paper presents the results of the 10th SPE Comparative Solution Project on Upscaling. Two problems were chosen. The first problem was a small 2D gas-injection problem, chosen so that the fine grid could be computed easily and both upscaling and pseudoization methods could be used. The second problem was a waterflood of a large geostatistical model, chosen so that it was hard (though not impossible) to compute the true fine-grid solution. Nine participants provided results for one or both problems. Introduction The SPE Comparative Solution Projects provide a vehicle for independent comparison of methods and a recognized suite of test data sets for specific problems. The previous nine comparative solution projects1–9 have focused on black-oil, compositional, dual-porosity, thermal, or miscible simulations, as well as horizontal wells and gridding techniques. The aim of the 10th Comparative Solution Project was to compare upgridding and upscaling approaches for two problems. Full details of the project, and data files available for downloading, can be found on the project's Web site.10 The first problem was a simple, 2,000-cell 2D vertical cross section. The specified tasks were to apply upscaling or pseudoization methods and to obtain solutions for a specified coarse grid and a coarse grid selected by the participant. The second problem was a 3D waterflood of a 1.1-million-cell geostatistical model. This model was chosen to be sufficiently detailed so that it would be hard, though not impossible, to run the fine-grid solution and use classical pseudoization methods. We will not review the large number of upscaling approaches here. For a detailed description of these methods, see any of the reviews of upscaling and pseudoization techniques, such as Refs. 11 through 14. Description of Problems Model 1. The model is a two-phase (oil and gas) model that has a simple 2D vertical cross-sectional geometry with no dipping or faults. The dimensions of the model are 2,500 ft long×25 ft wide×50 ft thick. The fine-scale grid is 100×1×20, with uniform size for each of the gridblocks. The top of the model is at 0.0 ft, with initial pressure at this point of 100 psia. Initially, the model is fully saturated with oil (no connate water). Full details are provided in Appendix A. The permeability distribution is a correlated, geostatistically generated field, shown in Fig. 1. The fluids are assumed to be incompressible and immiscible. The fine-grid relative permeabilities are shown in Fig. 2. Residual oil saturation was 0.2, and critical gas saturation was 0. Capillary pressure was assumed to be negligible in this case. Gas was injected from an injector located at the left of the model, and dead oil was produced from a well to the right of the model. Both wells have a well internal diameter of 1.0 ft and are completed vertically throughout the model. The injection rate was set to give a frontal velocity of 1 ft/D (about 0.3 m/d or 6.97 m3/d), and the producer is set to produce at a constant bottomhole pressure limit of 95 psia. The reference depth for the bottomhole pressure is at 0.0 ft (top of the model). The specified tasks were to apply an upscaling or pseudoization method in the following scenarios.2D: 2D uniform 5×1×5 coarse-grid model.2D: 2D nonuniform coarsening, maximum 100 cells. Directional pseudorelative permeabilities were allowed if necessary. Model 2. This model has a sufficiently fine grid to make the use of any method that relies on having the full fine-grid solution almost impossible. The model has a simple geometry, with no top structure or faults. The reason for this choice is to provide maximum flexibility in the selection of upscaled grids. At the fine geological model scale, the model is described on a regular Cartesian grid. The model dimensions are 1,200×2,200×170 ft. The top 70 ft (35 layers) represent the Tarbert formation, and the bottom 100 ft (50 layers) represent Upper Ness. The fine-scale cell size is 20×10×2 ft. The fine-scale model has 60×220×85 cells (1.122×106 cells). The porosity distribution is shown in Fig 3. The model consists of part of a Brent sequence. The model was originally generated for use in the PUNQ project.15 The vertical permeability of the model was altered from the original; originally, the model had a uniform kV/kH across the whole domain. The model used here has a kV/kH of 0.3 in the channels and a kV/kH of 10–3 in the background. The top part of the model is a Tarbert formation and is a representation of a prograding near-shore environment. The lower part (Upper Ness) is fluvial. Full details are provided in Appendix B. Participants and Methods Chevron. Results were submitted for Model 2 using CHEARS, Chevron's in-house reservoir simulator. They used the parallel version and the serial version for the fine-grid model and the serial version for the scaled-up model. Coats Engineering Inc. Runs were submitted for both Model 1 and Model 2. The simulation results were generated with SENSOR. GeoQuest. A solution was submitted for Model 2 only, with coarse-grid runs performed using ECLIPSE 100. The full fine-grid model was run using FRONTSIM, a streamline simulator,16 to check the accuracy of the upscaling. The coarse-grid models were constructed with FloGrid, a gridding and upscaling application. Landmark. Landmark submitted entries for both Model 1 and Model 2 using the VIP simulator. The fine grid for Model 2 was run with parallel VIP. Phillips Petroleum. Solutions were submitted for both Model 1 and Model 2. The simulator used was SENSOR. Roxar. Entries were submitted for both Model 1 and Model 2. The simulation results presented were generated with the black-oil implicit simulator Nextwell. The upscaled grid properties were generated using RMS, specifically the RMSsimgrid option. Streamsim. Streamsim submitted an entry for Model 2 only. Simulations were run with 3DSL, a streamline-based simulator.17 TotalFinaElf. TotalFinaElf submitted a solution for Model 2 only. The simulator used for the results presented was ECLIPSE; results were checked with the streamline code 3DSL. U. of New South Wales. The U. of New South Wales submitted results for Model 1 only, using CMG's IMEX simulator.

To evaluate whether the location of bacterial endophyte communities contributes to disease resistance in potato tubers (Solanum tuberosum), the population density, biodiversity and antibiotic activity of endophytic bacteria was examined from the tuber peel (periderm plus top 3 mm of tissue) of four cultivars (Russet Burbank, Kennebec, Butte and Shepody). There were no significant differences for population density of bacteria among the layers of peel examined and no cultivar × peel layer interaction. Endophytic bacteria from several layers of peel were challenged in in vitro bioassays to the soil-borne plant pathogens Fusarium sambucinum, Fusarium avenaceum, Fusarium oxysporum and Phytophthora infestans (mating types A1 and A2). In general, antibiosis of bacterial endophytes against these pathogens was significantly higher (P ¼ 0·01) in isolates recovered from the outermost layer of tuber peel and decreased progressively toward the centre of the tuber. Antibiosis against P. infestans was variable, with a progressive decrease in antibiotic activity from outer to inner layers of peel occurring in cvs Russet Burbank and Kennebec only. For antibiosis there were significant cultivar × peel, and cultivar × pathogen interactions (P ¼ 0·01). In all cases the inhibitory activity of endophytic bacteria was significantly greater (P ¼ 0·01) against the A1 than the A2 mating type of P. infestans. In four of seven cases, where the same species of bacteria were recovered from all three peel layers, antibiosis to pathogens decreased significantly (P ¼ 0·01) with depth of recovery (from the periderm to inside the tuber), indicating that in certain communities of endophytic bacteria, defence against pathogens may be related to bacterial adaptation to location within a host and may be tissue-type and tissue-site specific.

Summary. This paper describes applications of linear and nonlinear simulations to unstable miscible flooding. The first section describes a method of calculating linear growth of unstable modes by use of finite differences in the direction of flow and Fourier decomposition perpendicular to flow. This work extends the previous long- and short-wavelength analytic results to cover the whole wave-number range. Results obtained are used to help validate a two-dimensional (2D) code to study nonlinear evolution of viscous fingers and to identify likely fingering regimes in the computed solution by identifying the range of wave numbers that dominates the linear growth behavior. The second section describes the numerical scheme for calculations on a fine grid of the nonlinear development of an instability. Results are presented on calculations of nonlinear growth at several mobility ratios and levels of diffusion. Comparisons with Blackwell's experimental data are presented. Good agreement is obtained. suggesting that the physical processes governing fingering are being correctly modeled. Introduction Miscible displacement at an unfavorable mobility ratio is known to be unstable to viscous fingering of the solvent into the oil. There have been many attempts to characterize this phenomenon through laboratory experiments, empirical methods. and direct simulation. Blackwell et al. studied experimentally the factors determining fingering behavior in miscible displacement at adverse mobility ratios. Their studies investigated the effects of viscosity ratio, system size, and heterogeneities on the production characteristics of horizontal and vertical linear floods. Habermann carried out experiments with five-spot geometries to examine the effects of miscible slug size on recovery from a heterogeneous system in the presence of fingering. Further linear experiments by Handy were reported by Dougherty. Empirical methods have been developed to fit experimental data. Kovat's K-factor method allows for viscosity ratio and heterogeneity effects by calculating an effective mixture viscosity that is then used in a simple fractional-flow formula. The parameters in the effective viscosity equation were determined by calibrating the model against experiments. Todd and Longstaff developed an approach for modeling fingering effects in a three-phase miscible simulator that uses a mixing parameter for viscosities and densities. A more recent approach by Fayers developed the concept of a "fingering function" to predict the flow behavior. Fayers was able to achieve good fits to the experimental data reported by Blackwell by using a consistent set of parameters in his model. To match Handy's data, however, it was necessary to adjust one of the parameters. This change was attributed to the length-to-width ratio of Handy's experiment. Detailed simulation has also been used to study viscous fingering, but the calculations generally have been on coarse meshes. An early study of fingering in a miscible system was undertaken by Peaceman and Rachford. They used a 40 × 20 grid and initiated fingers with a 2.5 or 5 % permeability fluctuation. More recently, Farinerg used moving-point techniques to study fingering in linear geometry and compared his results with an analytic solution for the single-finger case obtained by Jacquard and Seguier Another recent study was reported by Glimm et al., who used an interface-tracking technique to investigate the effects of converging and diverging geometry (near injection and production wells). Two techniques were used to initiate fingers-random initial conditions or permeability variations-and similar results were obtain from each. This paper addresses the following two questions.In an unstable displacement, what is the range of wavelengths that will dominate the linear growth?What is the nonlinear evolution of these instabilities'? The first section extends previous work on linear stability theory, applicable to the long and short wavelength limits, to cover the whole range. This provides a toot to calculate growth rate at any value of wave number. Linear theory is limited to small-amplitude behavior of fingering; therefore, other techniques must be used to investigate the large-amplitude regime. The second section describes the use of direct simulation to investigate the nonlinear regime of viscous fingering. The major difference between work described in the second section and previous work is the use of simulation on grids fine enough to resolve effects of both diffusion and fingering. This allows investigation of the nonlinear de-velopment of viscous fingering and effects caused by boundarie(e.g., length/width in a core). Calculation of Growth Rates by Use of Linear Stability Theory Linear stability theory has been used in several studies to predict growth rates for both miscible and immiscible displacement. In particular, two recent papers described stability analyses for miscible displacement in the presence of gravity and diffusion. The purpose of this section is first to discuss these analyses and to show that the results predicted are valid only for the short-wavelength region and second to describe a method of calculating linear growth rates that is valid for all wavelengths. Analytic Calculations of Growth Rates. To derive the results in this section. we assume 2D horizontal miscible flow. The governing, equations can then be written ..........................................(1) and ..........................................(2) If we perform a linear stability analysis for a Fourier mode that varies as ei (wt+ z) (where u) is the growth rate, a is the wave number. and z is a general coordinate perpendicular to the unperturbed flow), we can obtain analytic expressions for the growth rate as a function of wave number in both long- and short-wave length limits. ..........................................(3) SPERE P. 514^

This paper reviews the formulation and parameters for three principal empirical viscous-fingering models: the Koval, Todd and Longstaff, and Fayers methods. All three methods give similar levels of accuracy when compared with linear homogeneous experiments, but they differ in performance in 2D applications. This arises from differences in the formulation of the total mobility terms. The superiority of the Todd and Longstaff and Fayers methods is demonstrated for 2D and gravity-influenced flows by comparison with experiments and high-resolution simulation. The use of high-resolution simulation to calibrate empirical models in a systematic manner is described. Results from detailed simulation demonstrate the sensitivity of empirical model parameters to viscous/gravity ratio, recovery process [secondary, tertiary, or water-alternating-gas (WAG)], and geological heterogeneity. For large amplitude heterogeneities with short correlation lengths, the accuracy of the empirical models is shown to be less satisfactory, but improved by the addition of a diffusive term.

Sum.m~ry. A the?ry of compositional viscous fingering with no adjustable parameters that reduces to the Todd and Longstaff model for mIscIble floo?s IS prese~ted. The t~~ory gi~es excellent predictive agreement with simulation results for a wide range of recovery ?rocesses. The hlgh-r~s~lutlOn compOSItIonal SImulations used to validate the theory are the first to resolve viscous fingering adequately III flow other than mIscIble flow.