Standard facies models to incorporate all heterogeneity levels in a reservoir model

The construction of reservoir models is frustrated by the fact that core and well cover only a fraction of the reservoir volume and it is therefore difficult to determine features like facies shape, -size, and -distribution, inter- and intra-facies boundaries and lateral trends from them. These features are, however, critical to fluid flow and they should necessarily be incorporated in the reservoir model and we therefore propose to systematically describe geometry and distribution of facies. To this end we make use of “standard facies models” that a priori contain all elements and boundaries of facies for a number of typical depositional environments.     

Effective macroscopic transport parameters between parallel plates with constant concentration boundaries

A macroscopic transport model is developed, following the Taylor shear dispersion analysis procedure, for a 2D laminar shear flow between parallel plates possessing a constant specified concentration. This idealized geometry models flow with contaminant dissolution at pore-scale in a contaminant source zone and flow in a rock fracture with dissolving walls. We upscale a macroscopic transient transport model with effective transport coefficients of mean velocity, macroscopic dispersion, and first-order mass transfer rate. To validate the macroscopic model the mean concentration, covariance, and wall concentration gradient are compared to the results of numerical simulations of the advection–diffusion equation and the Graetz solution. Results indicate that in the presence of local-scale variations and constant concentration boundaries, the upscaled mean velocity and macrodispersion coefficient differ from those of the Taylor–Aris dispersion, and the mass transfer flux described by the first-order mass transfer model is larger than the diffusive mass flux from the constant wall. In addition, the upscaled first-order mass transfer coefficient in the macroscopic model depends only on the plate gap and diffusion coefficient. Therefore, the upscaled first-order mass transfer coefficient is independent of the mean velocity and travel distance, leading to a constant pore-scale Sherwood number of 12. By contrast, the effective Sherwood number determined by the diffusive mass flux is a function of the Peclet number for small Peclet number, and approaches a constant of 10.3 for large Peclet number.     

Coarsening of three-dimensional structured and unstructured grids for subsurface flow

We present a generic, semi-automated algorithm for generating non-uniform coarse grids for modeling subsurface flow. The method is applicable to arbitrary grids and does not impose smoothness constraints on the coarse grid. One therefore avoids conventional smoothing procedures that are commonly used to ensure that the grids obtained with standard coarsening procedures are not too rough. The coarsening algorithm is very simple and essentially involves only two parameters that specify the level of coarsening. Consequently the algorithm allows the user to specify the simulation grid dynamically to fit available computer resources, and, e.g., use the original geomodel as input for flow simulations. This is of great importance since coarse grid-generation is normally the most time-consuming part of an upscaling phase, and therefore the main obstacle that has prevented simulation workflows with user-defined resolution. We apply the coarsening algorithm to a series of two-phase flow problems on both structured (Cartesian) and unstructured grids. The numerical results demonstrate that one consistently obtains significantly more accurate results using the proposed non-uniform coarsening strategy than with corresponding uniform coarse grids with roughly the same number of cells.     

Multi-scale modeling of three-phase–three-component processes in heterogeneous porous media

Flow and transport processes in porous media occur on different spatial and temporal scales and may also be locally different. Additionally, the structure of the porous medium itself generally shows a high dependence on the spatial scale.     

Another look at the conceptual fundamentals of porous media upscaling

We suggest a critical look at the epistemic foundations of the porous media upscaling problem that focuses on conceptual processes at work and not merely on form manipulations. We explore the way in which critical aspects of scientific methodology make their appearance in the upscaling context, thus generating useful effective parameters in practice. The fons et origo of our approach is a conceptual blending of knowledge states that requires the revision of the traditional method of scientific argument underlying most upscaling techniques. By contrast to previous techniques, the scientific reasoning of the proposed upscaling approach is based on a stochastic model that involves teleologic solutions and stochastic logic integration principles. The syllogistic form of the approach has important advantages over the traditional reasoning scheme of porous media upscaling, such as: it allows the rigorous derivation of the joint probability distributions of hydraulic gradients and conductivities across space; it imposes no restriction on the functional form of the effective parameters or the shape of the probability laws governing the random media (non-Gaussian distributions, multiple-point statistics and non-linear models are automatically incorporated); it relies on sound methodological principles rather than being ad hoc; and it offers the rational means for integrating the multifarious core knowledge bases and uncertain site-specific information sources about the subsurface system. Previous upscaling results are derived as special cases of the proposed upscaling approach under limited conditions of porous media flow, a fact that further demonstrates the generalization power of the approach. Our hope is that looking at the upscaling problem in this novel way will direct further attention to the methodological exploration of the problem at the length and the detail that it deserves.I would like to thank Drs. A. Kolovos and D.T. Hristopulos for their valuable comments. The work was supported by grants from the Army Research Office (Grant no. DAAG55–98–1-0289) and the National Institute of Environmental Health Sciences (P42-ES05948 & P30-ES10126).     

Upscaling: Effective Medium Theory, Numerical Methods and the Fractal Dream

Upscaling is a major issue regarding mechanical and transport properties of rocks. This paper examines three issues relative to upscaling. The first one is a brief overview of Effective Medium Theory (EMT), which is a key tool to predict average rock properties at a macroscopic scale in the case of a statistically homogeneous medium. EMT is of particular interest in the calculation of elastic properties. As discussed in this paper, EMT can thus provide a possible way to perform upscaling, although it is by no means the only one, and in particular it is irrelevant if the medium does not adhere to statistical homogeneity. This last circumstance is examined in part two of the paper. We focus on the example of constructing a hydrocarbon reservoir model. Such a construction is a required step in the process of making reasonable predictions for oil production. Taking into account rock permeability, lithological units and various structural discontinuities at different scales is part of this construction. The result is that stochastic reservoir models are built that rely on various numerical upscaling methods. These methods are reviewed. They provide techniques which make it possible to deal with upscaling on a general basis. Finally, a last case in which upscaling is trivial is considered in the third part of the paper. This is the fractal case. Fractal models have become popular precisely because they are free of the assumption of statistical homogeneity and yet do not involve numerical methods. It is suggested that using a physical criterion as a means to discriminate whether fractality is a dream or reality would be more satisfactory than relying on a limited data set alone.     

Spatial variability of N2O concentrations and of denitrification-related factors in the surficial groundwater of a catchment in Northern Germany

N₂O concentrations and denitrification-related factors (NO₃, SO₄, dissolved organic carbon (DOC) and CO₂) were investigated in the surface groundwater of a catchment in northern Germany, the Fuhrberger Feld Aquifer (FFA). We sampled 79 plots that were selected according to the three criteria of land use, historical land use conversion (1954–1995) and groundwater level. We sampled three sites within each plot. The sampling depth was 0.5 m below the groundwater surface.We found no indication for the occurrence of autotrophic denitrification in the surface groundwater. Heterotrophic denitrification was identified as the main process for N₂O accumulation. The variability of N₂O concentrations on the plot-scale was extremely high and was poorly explained by the three sampling criteria. Other denitrification-related variables such as NO₃, SO₄ and DOC were less variable. The selection criteria land use and groundwater level clearly influenced the order of magnitude of N₂O concentrations in the surface groundwater. Under arable land, high NO₃ concentrations resulted in high N₂O concentrations. The surface groundwater under forest and pasture was almost NO₃-free and had also very small N₂O concentrations. Plots where the distance from the soil surface to the groundwater surface was large (>1 m up to 3.4 m) showed higher N₂O concentrations in the surface groundwater than plots where the distance was small (<1 m). A larger distance from the soil surface to the groundwater leads to a longer residence time and more decomposition of DOC in the soil. Consequently the less bioavailable DOC could inhibit the efficiency of the heterotrophic denitrification in the groundwater, yielding more N₂O. Elevated organic carbon levels in plots with historic land use conversion (pasture to arable) were very stable and did not influence N₂O concentrations. The high within plot variability showed that an upscaling of N₂O from the plot-scale to the catchment-scale is possible as long as the groundwater level regime and the land use do not change.