Two-Stage Upscaling of Two-Phase Flow: From Core to Simulation Scale

SummaryIn the coarse-scale simulation of heterogeneous reservoirs, effective or upscaled flow functions (e.g., oil and water relative permeability and capillary pressure) can be used to represent heterogeneities at subgrid scales. The effective relative permeability is typically upscaled along with absolute permeability from a geocellular model. However, if no subgeocellular-scale information is included, the potentially important effects of smaller-scale heterogeneities (on the centimeter to meter scale) in both capillarity and absolute permeability will not be captured by this approach.In this paper, we present a two-stage upscaling procedure for two-phase flow. In the first stage, we upscale from the core (fine) scale to the geocellular (intermediate) scale, while in the second stage we upscale from the geocellular scale to the simulation (coarse) scale. The computational procedure includes numerical solution of the finite-difference equations describing steady-state flow over the local region to be upscaled, using either constant pressure or periodic boundary conditions. In contrast to most of the earlier investigations in this area, we first apply an iterative rate-dependent upscaling (iteration ensures that the properties are computed at the appropriate pressure gradient) rather than assume viscous or capillary dominance and, second, assess the accuracy of the two-stage upscaling procedure through comparison of flow results for the coarsened models against those of the finest-scale model.The two-stage method is applied to synthetic 2D reservoir models with strong variation in capillarity on the fine scale. Accurate reproduction of the fine-grid solutions (simulated on 500×500 grids) is achieved on coarse grids of 10×10 for different flow scenarios. It is shown that, although capillary forces are important on the fine scale, the assumption of capillary dominance in the first stage of upscaling is not always appropriate, and that the computation of rate-dependent effective properties in the upscaling can significantly improve the accuracy of the coarse-scale model. The assumption of viscous dominance in the second upscaling stage is found to be appropriate in all of the cases considered.

An efficient space-angle subgrid scale discretisation of the neutron transport equation

Integration of dynamic data typically requires the solution of an inverse problem that can be computationally intensive and practically infeasible for fine scale reservoir models. In this paper we present a new methodology to directly update fine scale geostatistically-based reservoir models by combining gradual deformation parameterization for the fine scale geostatistical model and an upscaling technique for the coarse scale flow simulation model. The proposed methodology includes: Perturbation of the fine scale geostatistical model using the gradual deformation parameterization. Gradual deformation ensures the preservation of the overall geostatistical properties of the fine model. Generation of the coarse scale flow simulation model by upscaling the fine scale geostatistical model. Sensitivity computation of the flow simulation results with respect to the fine scale parameterization. This sensitivity computation is analytical and takes into account the upscaling process. Direct updating of the fine scale geostatistical model using classical optimization process. Direct updating ensures consistency between the fine and coarse scale models. The accuracy of the proposed methodology was improved by calibrating the flow simulation model. The objective of this calibration is to reduce the error introduced by the upscaling step during the flow simulation. We applied successfully our methodology for fine scale reservoir description by integrating permanent down-hole gauge measurements directly into a three-dimensional geostatistical model containing about two million grid blocks. This test is designed to highlight several key issues of the proposed methodology: Efficiency of the upscaling step coupled with gradient-based optimization to speed up the history matching process. Usefulness of the calibration step for a correct integration of upscaling techniques in history matching. Capability of the methodology for maintaining consistency and coherency between fine scale and coarse scale models. Improvement of the reservoir characterization by integrating dynamic data at the fine geostatistical scale. We conclude that the proposed methodology can be used effectively and efficiently for reservoir characterization purposes.

The association between invasive and native species varies across spatial scales and is affected by phylogenetic relatedness, but these issues have rarely been addressed in aquatic ecosystems. In this study, we used a non-native, highly invasive species of Poaceae (tropical signalgrass) to test the hypotheses that (i) tropical signalgrass success correlates negatively with success of most native species of macrophytes at fine spatial scales, but its success correlates positively or at random with natives at coarse spatial scales, and that (ii) tropical signalgrass is less associated with native species belonging to the family Poaceae than with species belonging to other families (Darwin’s naturalization hypothesis). We used a dataset obtained at fine (0.25 m2) and coarse (ca. 1,000 m2) scales. The presence/absence of all species was recorded at both scales, and their biomass was also measured at the fine scale. We tested the association between tropical signalgrass biomass and individual native species with logistic regressions at the fine scale, and using the T-score index between tropical signalgrass and each native species at both scales. The likelihood of the occurrence of six species (submersed and free-floating) was negatively affected by tropical signalgrass biomass at the fine scale. T-scores showed that three species were less associated with tropical signalgrass than expected by chance, but 22 species co-occurred more than expected by chance at the coarse scale. Associations between species of Poaceae and tropical signalgrass were null at the fine scale, but were positive or null at the coarse scale. In addition to showing that spatial scale affects the patterns of association among the non-native and individual native species, our results indicate that phylogeny did not explain associations between the invasive and native macrophytes, at both scales.

SUMMARYThis paper presents residual‐based turbulence models for problems with moving boundaries and interfaces. The method is developed via a hierarchical application of variational multiscale ideas and the models are cast in an arbitrary Lagrangian–Eulerian (ALE) frame to accommodate the deformation of domain boundaries. An overlapping additive decomposition of velocity and pressure fields into coarse and fine scale components leads to coarse and fine scale mixed‐field problems. The problem governing fine scales is subjected to a further decomposition of the fine scale velocity into overlapping components termed as fine scales level I and level II. In turn, in the bottom‐up integration of scales, the model for level II fine scales serves to stabilize the problem governing level I fine scales, and model for level I fields yields the turbulence models. From the computational perspective, the coarse scales are represented in terms of the standard Lagrange shape functions, whereas level I and level II scales are represented via quadratic and fourth order polynomial bubbles, respectively. Because of the bubble functions approach employed in the consistently derived fine scale models, the resulting method is free of any embedded or tunable parameters. The proposed turbulence models share a common feature with the LES models in that the largest scales in the flow are numerically resolved, whereas the subgrid scales are modeled. The method is applied to flow around a plunging airfoil at Re = 40,000, and results are compared with experimental and numerical data published in the literature. Also presented are the results for the plunging airfoil at Re = 60,000 to show the robustness and range of applicability of the method. Copyright © 2013 John Wiley & Sons, Ltd.

In order to run reservoir simulation efficiently, a coarse scale (CS) dynamic model is created by upscaling of a fine scale (FS) static model. All history match (HM) changes usually done in the CS dynamic model need to be downscaled to FS for geological justifications and consistency maintenance between the FS static and CS dynamic models. This paper proposes a robust downscaling method for integration of FS static and CS dynamic models. The proposed method downscales a HMDM (dynamic model) to HMSM (static) in multiple steps. Scale-up the ISM (initial) to CS to create an IDM. Identify the cell changes between HMDM and IDM, and transfer the changes to FS to create a MSM (modified). Scale-up the MSM to CS to create to a MDM and calculate the ratios between HMDM and MDM for all cell properties. Transfer the ratios to FS to create a HMSM. Scale-up the HMSM to CS to confirm its identity to the HMDM. Selection of sampling and zone mapping methods is critical in all steps. The proposed method has been successfully applied in a giant carbonate oil field in the Caspian Sea that consists of a matrix dominated platform and a fracture/karst dominated rim. Due to the field's complex geology and high H2S content (15%), a dual porosity, dual permeability compositional model has been created to model compositional sour crude flow within/between matrix and fracture/karst. The FS static model contains a 236m × 236m horizontal grid with 593 layers while the CS dynamic model has the horizontal cell sizes in a range of 236m to 944m with 73 layers. Rock regions, permeability, and reservoir connectivity in the CS dynamic model were calibrated using the field historical production data (e.g., static pressure, PLT, interference test, and GOR/water-cut data) to create a HMDM. Since the HM process was performed only in the CS dynamic model, the FS static model and HMDM became inconsistent. Appling the proposed downscaling method has helped the HM team to resolve this issue and resulted in a seamless link between the FS static and CS dynamic models for current and future HM and model updates.

Summary Geoscientists and engineers commonly build geologic or geostatis-tical reservoir models that contain more than 106 grid cells. For flow simulation, the number of grid cells must be reduced by a factor of 10 or more. Such scaling up necessarily involves a loss of information that must be restored through the use of effective or pseudorelative permeabilities. This paper describes an approach to generate such functions that, when combined with global absolute permeability scaleup,1 offers a significant improvement over existing "dynamic pseudo" methods that require extensive fine-grid simulation and that are very sensitive to flow conditions.1 The method builds on previous work to scale up absolute permeability through a global technique that minimizes the loss of permeability variance and the spatial correlation in an unequally sized grid system. This paper shows that, when the residual permeability following global absolute permeability scaleup is spatially uncorrected, the velocity of a fluid-displacement shock front correlates with a well-defined universal heterogeneity number that is related to the permeability distribution. The paper presents an analytic computation of the pseudo as a superposition of the shock velocities for the residual field, found from a table look-up, and the fine-grid field when the fine-grid relative permeabilities (usually based on measured relative permeabilities) have the same normalized form. The method is then generalized to multiple functional forms of fine-grid-cell relative permeabilities with new relative permeability and capillary pressure averaging methods. The procedure eliminates the need for fine-scale simulation and is equally applicable to geologic deterministic or stochastic models. The paper describes the technique in detail and demonstrates the procedure with one two-dimensional (2D) and one three-dimensional (3D) reservoir model waterflood simulation. In the latter case, the number of cells is reduced from 23,600 (fine scale) to 4,000 (coarse scale) with very little change in the water breakthrough and water-cut behavior.

Where land surface air temperature data are not available, satellite land surface temperature are used. However, the coarse spatial resolution of satellite‐derived products may yield errors at the local scale. This work shows the differences between the MODIS Land Surface Temperature and Emissivity (MOD11A1) product and ground measurements at two different scales. We used data from 21 SNOTEL stations across the northern Front Range of Colorado to represent the coarse scale and 17 iButton temperature sensors across the Colorado State University Mountain Campus to represent the fine scale. We found significant differences in the temperature and its changes with elevation for the two spatial scales. At the fine scale, cold air drainage can induce an inversion of the temperature gradient with elevation. A higher correlation was found during the nighttime at the fine scale, while, at the coarse scale, higher correlations were observed during the daytime. On windy nights, temperatures do not cool as much as on calmer nights, and the coarse scale near‐surface temperature gradient with elevation persists, though the fine scale inversions do not develop. The near‐surface temperature gradients with elevation based on the MODIS pixels are similar to the ground‐based data at the coarse scale but not at the fine scale. Thus, one must be cautious in selecting the near‐surface temperature gradients with elevation for mountainous terrain when different scales are considered, and a proper validation of satellite products is necessary prior to their use to avoid the propagation of uncertainties.

<p><b>Recent ecological studies have started integrate to spatial variation of ecological patterns into the study design rather than treating it as a statistical nuisance. In particular, the influence of the spatial scale at which ecological patterns are measured has gained much attention over the last two decades. Since, for example, sensory abilities as well as the ability to disperse vary among species, different species-specific responses to heterogeneous environments may be expected.</b></p> <p>Plant-insect interactions in heterogeneous landscapes, in particular, have gained much attention as experiments can be conducted on a more accessible scale and may yield new applications for crop and horticulture. Two hypotheses that describe insect herbivore aggregations in the landscape are: a) the resource concentration hypothesis which predicts higher numbers of specialist insect herbivores per unit biomass in dense and pure stands of their host plant, and b) the resource dilution hypothesis which predicts that insect herbivore numbers will decline with increasing plant density. I investigated resource dilution and resource concentration patterns in egg distributions of Pieris rapae and Tyria jacobaeae in relation to host plant density, which I defined differently by applying varying spatial scales of measurement. I also tested for effects of host plant density and the scale of measurement on flight patterns of P. rapae females.</p> <p>In a natural population of Lepidium oleraceum I investigated effects of scale of measurement of plant density, as well as white rust and hymenopteran parasitoids on P. rapae egg and larvae distributions. In a separate experiment I tested for any potential effects of arthropod predators on P. rapae egg distributions at different spatial scales. The number of P. rapae eggs per plant conformed to predictions made by the resource dilution hypothesis. However, such a pattern was only found for fine scale plant density but not for medium or coarse scale plant density. In contrast, the number of T. jacobaeae egg clutches per plant showed a resource concentration pattern for medium scale plant density but not for fine or coarse scale plant density. However, this result occurred only in one of two experiments with T. jacobaeae. A resource dilution pattern was also found for the number of visits per plant by P. rapae females at both coarse and fine scale measurement. Female flight paths were less directional when plants were present in the study area during fine scale observations and butterflies were attracted to areas containing host plants. Flight observations at coarse scale did not show any change in turning behaviour and butterflies moved at random across the study area. No effect of parasitism, or predation through arthropods was found on the distribution of P. rapae eggs. However, infection by white rust lead to a decreased number of eggs per plant in the natural L. oleraceum population. The results of my thesis underline the importance of spatial scale in ecological studies. Careful thought should be given to the scale of measurement and method of abstraction when describing real world patterns.</p>

In routine core analysis, porosity and permeability, both relative and absolute, are measured on rock samples which are not under net overburden (confining) pressure. Using these data to predict reservoir performance or estimate reserves can lead to serious errors since all reservoirs are under net overburden pressure. Data collected from constant rate, dynamic displacement experiments were utilized to study the effect of net overburden pressure on porosity and absolute and relative permeabilities. These experiments were conducted on small, consolidated rock samples under net overburden pressures up to 41.37 MPa (6000 psi) and room temperature. The pore pressure was maintained atmospheric. Examination of the data shows a decrease in porosity and permeability with increase in overburden pressure. A correlation between porosity and overburden pressure and also between permeability and overburden pressure has been developed using linear regression analysis. Both correlations are found to be logarithmic. The irreducible water saturation and residual oil saturation increase with increased overburden pressure levels. While the relative permeability to oil decreases with increasing overburden pressure, a corresponding negligible decrease in water relative permeability is observed.

The objective of this paper is to provide reservoir engineers fresh insight and a new approach to the description of heavy oil and water relative permeability functions in reservoir simulation. This paper proposes a plausible mechanism of heavy oil flow in porous media influenced by the effect of water-oil emulsification to address the inability of simulation models to replicate primary oil production rates during cold producing operations. A number of heavy oil reservoirs that naturally flow without stimulation show higher field oil production rates than often can be predicted by numerical simulation. To match the field production, reservoir simulation engineers often exhaust all valid parameter changes in the numerical model and resort to questionable changes on parameter values that usually enjoy high confidence level by physical measurements. To overcome these shortcomings, we first identify in situ water-oil emulsification, a plausible mechanism of heavy oil flow in water-oil systems, and then offer a practical approach to modeling water oil relative permeability for heavy oil-water systems. Beyond water-oil relative permeability curve adjustments, the parameters that are changed by simulation engineers during reasonable history matching may consist of an unrealistic increase in oil saturation and/or a significant decrease in oil viscosity. Both parameters are usually measured in labs and should not be altered significantly. In addition, sometimes during simulation model calibration, the increase in absolute permeability goes so high that it cannot be supported by any logic or physical measurement. The new relative permeability modeling approach for heavy oil and water systems considers the fact that naturally occurring heavy oil contains a large quantity of emulsifying agent and the density contrast between heavy oil and water is so low that heavy oil in porous media may flow by a continuous oil and stable emulsion phase. A part of the total oil flow occurs via oil phase that is similar to light oil and water system and the other part flow via a quasi-stable emulsion/water phase. This practical approach for heavy oil water relative permeability modeling removes the difficulties that a simulation engineer often faces during heavy oil water history matching and calibration. The new modeling approach presented in this paper is successfully applied in modeling cold heavy oil production performance for a heavy oil reservoir in California. One field example is provided in this paper and some results are presented to estimate heavy oil water relative permeability from standard lab data.

A variational multiscale approach to strain localization – formulation for multidimensional problems

A novel technique has been developed to implicitly estimate relative permeability by history matching the production data with the ensemble Kalman filter (EnKF) method. Water and oil relative permeability curves were approximated using the B-spline model, which has been modified to better represent the relative permeability curves. Compared to the existing implicit approaches, the newly developed technique did not require the use of the gradient of the objectives function and thus, was easy to implement. The newly developed technique has been validated by accurately evaluating relative permeability in a two-dimension synthetic heterogeneous reservoir. The case study showed that the estimated relative permeability was improved gradually as observation data were assimilated and that a good estimation of relative permeability curves can be effectively obtained. In this study, three estimation scenarios assimilating various types of observation data were examined and compared. It was also found that the oil relative permeability was sensitive to the oil production rate (OPR) data, as OPR was directly affected by oil relative permeability. On the other hand, water relative permeability was more sensitive to the bottomhole pressure data of the producers. Additionally, the production data, obtained prior to water breakthrough, contributed more to the estimation of the two-phase relative permeability curves.

Subsurface flows are affected by geological variability over a range of length scales. The modeling of well singularity in heterogeneous formations is important for simulating flow in aquifers and petroleum reservoirs. In this paper, two approaches in calculating the upscaled well index to capture the effects of fine scale heterogeneity in near-well regions are presented and applied. We first develop a flow-based near-well upscaling procedure for geometrically flexible grids. This approach entails solving local well-driven flows and requires the treatment of geometric effects due to the nonalignment between fine and coarse scale grids. An approximate coarse scale well model based on a well singularity analysis is also proposed. This model, referred to as near-well arithmetic averaging, uses only the fine scale permeabilities at well locations to compute the coarse scale well index; it does not require solving any flow problems. These two methods are systematically tested on three-dimensional models with a variety of permeability distributions. It is shown that both approaches provide considerable improvement over a simple (arithmetic) averaging approach to compute the coarse scale well index. The flow-based approach shows close agreement to the fine scale reference model, and the near-well arithmetic averaging also offers accuracy for an appropriate range of parameters. The interaction between global flow and near-well upscaling is also investigated through the use of global fine scale solutions in near-well scale-up calculations.

Functional relationships for unsaturated flow in soils, including those between capillary pressure, saturation, and relative permeabilities, are often described using analytical models based on the bundle‐of‐tubes concept. These models are often limited by, for example, inherent difficulties in prediction of absolute permeabilities, and in incorporation of a discontinuous nonwetting phase. To overcome these difficulties, an alternative approach may be formulated using pore‐scale network models. In this approach, the pore space of the network model is adjusted to match retention data, and absolute and relative permeabilities are then calculated. A new approach that allows more general assignments of pore sizes within the network model provides for greater flexibility to match measured data. This additional flexibility is especially important for simultaneous modeling of main imbibition and drainage branches. Through comparisons between the network model results, analytical model results, and measured data for a variety of both undisturbed and repacked soils, the network model is seen to match capillary pressure–saturation data nearly as well as the analytical model, to predict water phase relative permeabilities equally well, and to predict gas phase relative permeabilities significantly better than the analytical model. The network model also provides very good estimates for intrinsic permeability and thus for absolute permeabilities. Both the network model and the analytical model lost accuracy in predicting relative water permeabilities for soils characterized by a van Genuchten exponent n≲3. Overall, the computational results indicate that reliable predictions of both relative and absolute permeabilities are obtained with the network model when the model matches the capillary pressure–saturation data well. The results also indicate that measured imbibition data are crucial to good predictions of the complete hysteresis loop.