We propose an analytical method to calculate effective block permeabilities as a full tensor based on the geometry, block size, full local tensorial permeabilities, and the geology within the block. The method was validated with a finite-element numerical simulator and was applied to a simulation of flow through an outcrop of the eolian Page sandstone. Results show that the relative positions of the main geologic features and the ratio between the grain-flow and wind-ripple permeabilities are more important than bounding surfaces and dispersion in determining flow behavior. IntroductionAll natural permeable media are heterogeneous because they have different pore sizes, different spatial structures, or both. Pore-level heterogeneities are smaller than the core plugs from which many ofthe reservoir data are obtained. On the other hand, spatial heterogeneities require a deeper understanding of the averaging-volume concept in flow through permeable media. Haldorsen and Lake 1 described four scales of averaging volume: microscopic, the size of several pores; macroscropic, the size of core plugs; megascopic, the size of the simulation blocks; and gigascopic, the size of an entire field. Core-plug measurements usually cannot be used for megascopic simulation blocks.Although the development of more powerful computers and simulation techniques allows us to use very fine grids and to include rather detailed features in numerical simulations, the size of the permeability variations in naturally occurring media is still smaller than the smallest grids. This situation requires calculation of an effective permeability for simulation blocks that contain spatial heterogeneities. An effective permeability preserves the fluid-flux/ potential-drop quotient between a heterogeneous block and an equivalent homogeneous block (same size, geometry, and fluid viscosity).How to calculate an effective permeability for a gridblock in a numerical simulation of a heterogeneous reservoir remains a difficult reservoir characterization problem. Although several numerical and statistical procedures have appeared in the literature, none can be inserted easily into a simulation because they either were developed for a very specific problem or require prior simulation runs.In this study, we propose an analytical method to calculate effective block permeabilities as a full tensor using the geometry, block size, tensoriallocal permeabilities, and the heterogeneity (geology) within the block. The method uses flow through parallel and serial crossbeds, which subsequently is rotated from parallel to serial crossbeds to get tensorial permeabilities from the bed orientations. The method was validated with a finite-element numerical simulator that can model detailed permeability anisotropy and heterogeneity explicitly. The analytical method can be inserted into numerical simulators to calculate effective tensorial permeabilities of heterogeneous gridblocks.

Summary The formation-rate-analysis (FRASM) technique is introduced. The technique is based on the calculated formation rate by correcting the piston rate with fluid compressibility. A geometric factor is used to account for irregular flow geometry caused by probe drawdown. The technique focuses on the flow from formation, is applicable to both drawdown and buildup data simultaneously, does not require long buildup periods, and can be implemented with a multilinear regression, from which near-wellbore permeability, p * and formation fluid compressibility are readily determined. The field data applications indicate that FRA is much less amenable to data quality because it utilizes the entire data set. Introduction A wireline formation test (WFT) is initiated when a probe from the tool is set against the formation. A measured volume of fluid is then withdrawn from the formation through the probe. The test continues with a buildup period until pressure in the tool reaches formation pressure. WFTs provide formation fluid samples and produce high-precision vertical pressure profiles, which, in turn, can be used to identify formation fluid types and locate fluid contacts. Wireline formation testing is much faster compared with the regular pressure transient testing. Total drawdown time for a formation test is just a few seconds and buildup times vary from less than a second (for permeability of hundreds of millidarcy) to half a minute (for permeability of less than 0.1 md), depending on system volume, drawdown rate, and formation permeability. Because WFT tested volume can be small (a few cubic centimeters), the details of reservoir heterogeneity on a fine scale are given with better spatial resolution than is possible with conventional pressure transient tests. Furthermore, WFTs may be preferable to laboratory core permeability measurements since WFTs are conducted at in-situ reservoir stress and temperature. Various conventional analysis techniques are used in the industry. Spherical-flow analysis utilizes early-time buildup data and usually gives permeability that is within an order of magnitude of the true permeability. For p* determination, cylindrical-flow analysis is preferred because it focuses on late-time buildup data. However, both the cylindrical- and spherical-flow analyses have their drawbacks. Early-time data in spherical-flow analysis results in erroneous p* estimation. Late-time data are obtained after long testing times, especially in low-permeability formations; however, long testing periods are not desirable because of potential tool "sticking" problems. Even after extended testing times, the cylindrical-flow period may not occur or may not be detectable on WFTs. When it does occur, permeability estimates derived from the cylindrical-flow period may be incorrect and their validity is difficult to judge. New concepts and analysis techniques, combined with 3-D numerical studies, have recently been reported in the literature.1–7 Three-dimensional numerical simulation studies1–6 have contributed to the diagnosis of WFT-related problems and the improved analysis of WFT data. The experimental studies7 showed that the geometric factor concept is valid for unsteady state probe pressure tests. This study presents the FRA technique8 that can be applied to the entire WFT where a plot for both drawdown and buildup periods renders straight lines with identical slopes. Numerical simulation studies were used to generate data to test both the conventional and the FRA techniques. The numerical simulation data are ideally suited for such studies because the correct answer is known (e.g., the input data). The new technique and the conventional analysis techniques are also applied to the field data and the results are compared. We first review the theory of conventional analysis techniques, then present the FRA technique for combined drawdown and buildup data. A discussion of the numerical results and the field data applications are followed by the conclusions. Analysis Techniques It has been industry practice to use three conventional techniques, i.e., pseudo-steady-state drawdown (PSSDD), spherical and cylindrical-flow analyses, to calculate permeability and p* Conventional Techniques Pseudo-Steady-State Drawdown (PSSDD). When drawdown data are analyzed, it is assumed that late in the drawdown period the pressure drop stabilizes and the system approaches to a pseudo-steady state when the formation flow rate is equal to the drawdown rate. PSSDD permeability is calculated from Darcy's equation with the stabilized (maximum) pressure drop and the flowrate resulting from the piston withdrawal:9–11 $$k {d}=1754.5\left({q\mu \over r {i}\Delta p {{\rm max}}}\right),\eqno ({\rm 1})$$where kd=PSSDD permeability, md. The other parameters are given in Nomenclature.

Summary Reservoir simulations are limited to large scale grid blocks due to prohibitive computational costs of fine grid simulations. Rock properties, such as permeability, are measured on smaller scale than coarse scale simulation grid blocks. Therefore, the properties defined on a smaller scale are upscaled to a coarser scale. Few prior studies on permeability upscaling paid special attention to the problem of the radial flow in the vicinity of a wellbore. This study presents an analytical method to calculate effective permeability for a coarse-grid well-block from fine-grid permeabilities. The method utilizes serial and parallel averaging procedures modified for radial flow. The method is validated with numerical simulations of primary and secondary recovery processes involving 2- and 3-D systems. The results of coarse grid simulations with the permeabilities upscaled through the new well-block approach agree well with the results of the fine grid simulations with initial permeability distributions. Introduction Reservoir heterogeneity can be described on a fine scale by using current stochastic models. In spite of the recent developments in computational speed, it is still not feasible to flow simulate reservoirs with stochastic grid blocks. Coarsened grid-block scheme, therefore, is required to ease the computational burden. Rock properties for coarse scale grid blocks are obtained by utilizing a proper upscaling procedure. Permeability, by far, is the most important property that affects flow performance. Therefore, unlike porosity, permeability upscaling requires a robust upscaling procedure. Several authors have discussed methods to upscale permeability, but few have addressed the problem of upscaling in the vicinity of wellbore although the well performance is more affected by the permeability of this region than by any permeability away from it. The large portion of pressure drop in a reservoir is realized in the near-wellbore region. White and Horne proposed a method to calculate well-block transmissibility. Their method was based on running simulations with fine scale grid blocks for different boundary conditions. The pressures and fluxes from fine scale simulations were averaged and summed to obtain pressures and fluxes on a coarse scale. The least square method is utilized to convert the coarse scale pressures and fluxes into a block transmissibility. Palagi et al. presented an upscaling procedure for the permeability at the interface of Voronoi grid blocks. They used the power law averaging to calculate homogenized permeability. To find the optimum value for the power law coefficient, the simulation results of the initial permeability distribution were compared with the results of various upscaled distributions obtained with various values of . The that minimizes the difference between the fine grid and coarse grid results is accepted as the optimum . Ding presented an upscaling procedure to calculate the equivalent coarse grid transmissibility based on the results of fine grid simulation. To account for a well in a coarse grid block, the numerical productivity index for a coarse grid block was defined. The methods discussed above are based on the fine-grid numerical simulations. Numerical simulations are to be repeated for all coarse grid blocks and require a preprocessing task that usually take up several man-hours. To avoid lengthy numerical simulations, this study presents an analytical method to calculate upscaled well-block effective permeability. The method is based on the understanding that the effective permeability should preserve the ratio of the fluid flux and the potential drop realized across the fine grid blocks with the heterogeneous permeabilities. The method can be applied to either 2- or 3-D flow simulations. It is based on the incomplete-layers concept and the radial flow averaging laws that are applied to a Cartesian grid scheme.

Numerical simulations and laboratory experiments were used to study effects of heterogeneity and anisotropy on probe measured permeabilities. Simulation results show that the geometric factors for unsteady-state flow measurements are not different from the ones obtained for steady-state. Effects of a nearby flow barrier on probe measured permeability are studied experimentally and numerically, and it is concluded that the effect of a local flow barrier is negligible when the barrier is located farther than one-half of a probe-tip diameter from the probe.Probe measured permeability on a damaged formation is profoundly affected by the damaged zone thickness. Working with geometric factors, a methodology to obtain undamaged and damaged zone permeabilities, and damaged zone thickness is developed. To obtain such information uniquely three measurements with three different probe tip sizes are required.The effects of anisotropy on probe measured permeability are also investigated by introducing tensorial permeability for probe measurements. Several simulation runs were performed for different anisotropy ratios at various cross-bedding angles. Results are presented in the form of dimensionless parameters that can be used to determine depositional dip angle and permeabilities in the principal directions.

Computational costs limit large scale reservoir simulations to coarse grid systems. Determination of an effective permeability for a simulation grid block requires a proper scale-up of small scale permeability heterogeneities within that grid block. Conventional scale-up techniques are limited to a diagonal tensor representation of effective permeability. Therefore, such techniques cannot handle general permeability anisotropy (full tensor) exemplified by cross-bedded permeability structures that may be present on a smaller scale. An analytical method is developed to calculate an effective permeability tensor for a grid block by accounting for small scale heterogeneities within the grid block. The method honors both the location and the orientation of the small scale heterogeneities. Effective permeability tensors calculated using the analytical method and a numerical method show excellent agreement. Miscible displacement simulations show that the effective permeability tensor method outperforms conventional scale-up techniques in predicting flood front locations in cases of general permeability anisotropy. Introduction Simulation of fluid flow in a porous medium on a field scale require large scale numerical simulations. Although more powerful computers and simulation techniques are continuously being developed, the size of the grid blocks used in field scale fluid flow simulations is too large to explicitly account for the effect of small scale heterogeneities. Such heterogeneities may be comprised of interwell laminations and cross-bedding structures as well as sand/shale sequences. For the large scale simulator grid blocks, the effect of small scale heterogeneities can only be accounted for by calculating an effective permeability. An effective permeability preserves the ratio of the fluid flux and the potential drop across a heterogeneous block and an equivalent homogeneous block. Several methods are presented in the literature for calculating effective permeability. These methods can, in general, be divided into numerical and analytical methods. Numerical (simulation) methods may be used to handle complex heterogeneous systems, whereas analytical method usually are restricted by simplifying assumptions. Analytical methods have the advantage of being less expensive than numerical methods in terms of computational cost. In this study, an analytical effective permeability method is developed. This method combines the advantages of numerical and analytical methods. The proposed method is general in that it allows for full tensor representation of effective permeability. Both the location and orientation of permeability heterogeneities are considered. The method, in essence, captures the effect of the pressure distribution within the grid block. A directional search procedure in local areas (four quadrants of the grid block) identifies the principal axes of permeability and their orientation with respect to the simulation coordinate axes. Coordinate rotation yields effective permeability tensors. These tensors are combined into an effective permeability tensor for the entire grid block based on the coupling of cross-flow-averaged and no-cross-flow-averaged effective permeability tensors from the four quadrants. Darcy's law is manipulated throughout this process. P. 679^