Summary An improved technique based on the concept of hydraulic flow units or hydraulic units (HU's) is presented to calculate permeability distribution in uncored wells. Graphical probability methods, nonlinear regression, and the Ward's analytical algorithm are presented to perform cluster analysis on core data and identify prevailing HU's in a formation. A Bayesian-based probabilistic approach is discussed next to estimate HU's and permeability distributions in logged wells. This is an inverse problem that requires constructing an a priori training database to capture implicit relationships between core-derived HU's and various well log measurements. Information from the database is drawn statistically to infer HU distributions at target logged wells. The method is applied to two heterogeneous reservoirs, a carbonate formation, and a laminated sandstone formation. Comparisons of permeabilities calculated by the HU approach and other conventional techniques are provided that demonstrate the usefulness of the HU method. Introduction Estimation of permeability in uncored but logged wells is a generic problem common to all reservoirs. Any field-scale reservoir characterization study inevitably requires knowledge of petrophysical properties at drilled wells for its starting point. Therefore, scientifically sound and geologically compatible procedures must be sought to allow for reliable calculation of permeability distributions in wells. Traditional approaches for estimation of permeability are based either on simple linear regressions or empirical inferences that use correlations among various well log responses. Usually, these empirically inferred models apply locally because there may be large differences in depositional characteristics at other locations. The regression method assumes that a linear relationship exists between core porosity and the logarithm of core permeability. Another linear regression establishes dependency between measured core porosity and well log-derived porosity. These two regression models, when combined, allow calculation of permeability at logged wells. The method explicitly ignores the scatter of data about the regression lines and implicitly attributes any scatter to measurement errors or second-order fluctuations in reservoir characteristics. A partial improvement to this method is achieved by first identifying lithological categories of the formation1–3 and then calculating linear regression lines for petrophysical core measurements that belong to each lithology class. Because the regression methods smooth data, the predicted permeability values from these regression models lack the variability seen on actual core data. For this reason, probability field simulations may be applied to add stochastic fine-scale variations to the predicted permeability values.4,5 A more reasonable approach for prediction of permeability is to address the development of permeability in reservoir rocks from fundamentals of geology and physics of flow at pore network scale. This is best achieved by attributing the nature of interdependency between permeability and porosity to geological variations in reservoir rock and by seeking functional relationships for permeability that capture geological controls on flow properties. This will require establishing causal relationships between core-derived pore-throat parameters and log-derived macroscopic petrophysical attributes.6 Such relationships are achieved best if rocks of similar fluid conductivity are identified and grouped together. Each grouping is referred to as an HU. The HU's do not necessarily correspond to lithofacies, which have been referred to as geological flow units.4 Petrophysical properties are controlled by both depositional characteristics, such as grain-size and sorting, and by diagenetic features, such as the amount and type of cement or clay minerals. Thus, an HU involves more than a genetic facies of a depositional system. Generally, the variability of petrophysical properties is large among the HU's and low within them. Permeability calculation by HU's offers an improved estimation over traditional regression-based averaged relationships by incorporating geology and fluid-flow principles. The extent of improvement depends mainly on the geological characteristics of the formation, where larger enhancements are usually achieved in more heterogeneous deposits. In any case, the method can describe permeability variations in a reservoir more comprehensively and realistically because it integrates geology, petrophysics, and engineering. This paper presents an improved technique for permeability estimation using HU's. The basic concept of HU's is first reviewed, and the theoretical basis for their classification in cored wells is provided. Next, a statistically based inference method is presented to identify HU's in uncored but logged wells and thereafter to calculate permeability distributions. Lastly, the applicability of the improved permeability estimation technique to a carbonate and a sandstone reservoirs is illustrated. Method Concept of HU's. An HU is defined as the representative volume of total reservoir rock within which geological properties that control fluid flow are internally consistent and predictably different from properties of other rocks. Thus, a flow unit is a reservoir zone that is continuous laterally and vertically and has similar flow and bedding characteristics.7 HU's are related to geological facies distributions but do not necessarily coincide with facies boundaries. The parameters that influence fluid flow are mainly pore-throat geometrical attributes. The pore geometry is in turn controlled by mineralogy (type, abundance, location) and texture (grain size, grain shape, sorting, packing). Various combinations of these geological properties can lead to distinct rock flow units that have similar fluid transport properties. Therefore, an HU can include several rock facies types, depending on their depositional texture and mineralogical content. The grouping of rocks based on their fundamental geological flow attributes is the basis of HU classification. If a porous medium is simulated as a bundle of straight capillary tubes, the following expression for rock permeability is obtained by combining Darcy's law for flow in porous media and Poiseuille's law for flow in tubes8:Equation 1 Concept of HU's. An HU is defined as the representative volume of total reservoir rock within which geological properties that control fluid flow are internally consistent and predictably different from properties of other rocks. Thus, a flow unit is a reservoir zone that is continuous laterally and vertically and has similar flow and bedding characteristics.7 HU's are related to geological facies distributions but do not necessarily coincide with facies boundaries. The parameters that influence fluid flow are mainly pore-throat geometrical attributes. The pore geometry is in turn controlled by mineralogy (type, abundance, location) and texture (grain size, grain shape, sorting, packing). Various combinations of these geological properties can lead to distinct rock flow units that have similar fluid transport properties. Therefore, an HU can include several rock facies types, depending on their depositional texture and mineralogical content. The grouping of rocks based on their fundamental geological flow attributes is the basis of HU classification. If a porous medium is simulated as a bundle of straight capillary tubes, the following expression for rock permeability is obtained by combining Darcy's law for flow in porous media and Poiseuille's law for flow in tubes8:Equation 1
Japan National Oil Corporation (JNOC) is conducting a survey on underground gas storage (UGS) using inert gas as cushion gas, which makes good use of assets of depleted gas or gas condensate reservoirs. Since the Japanese regulation for a heating value of city gas is extremely strict, a heating value and hence composition of gas withdrawn from those reservoirs must be carefully predicted to properly design UGS projects. JNOC has developed an UGS simulator, which can rigorously evaluate a heating value of withdrawn gas taking into consideration all the major phenomena that possibly affect composition of withdrawn gas. A conventional cubic equation of state (EOS) based 3-D compositional simulatorand a PVT simulator were improved to construct the UGS simulator equipped withthe functions for calculating physical phenomena:molecular diffusion andvelocity dependent dispersion,three-phase flash including dissolution of gaseous components into water and vaporization of water,adsorption of gaseous components onto a rock surface andturbulence and Klinkenberg effects. In addition, new functions of MPFA (Multi-Point Flux Approximation)and TVD (Total Variation Diminishing) scheme with local grid refinement in acorner point geometry were incorporated into the simulator to rigorously calculate a fluid flux with permeability tensors and to reduce numerical dispersion, respectively. Furthermore, functions to calculate temperature distributions both in a reservoir and wellbore were added to examine the effectof cooling caused by gas injection and adiabatic expansion on fluid flow and gas composition. Each function newly developed for the UGS simulator was then validated through test runs using laboratory test data. Field scale simulations for hypothetical reservoirs were also conducted to confirm the simulator's performance as wellas to examine the effect of cushion gas volume, working gas volume, reservoir heterogeneity and in situ gas composition on withdrawn gas composition. This paper describes the development and validation of the UGS simulator followed by the results of field scale simulation runs. Introduction Since natural gas was first stored underground in 1915, an increasing number of fields have been utilized for UGS for the purpose of peak shaving and stableenergy supply. According to 1991/1992 statistics, there are 550 UGS fields inthe world, 425 of which utilize depleted gas or oil reservoirs [1]. A conventional black oil type simulator or modified black oil type simulator that enables distinction between injection and in situ gas phases has been most commonly used for evaluation/prediction of UGS reservoir performances [2, 3,4]. A black oil type simulator, however, is not satisfactory to rigorously calculate composition of withdrawn gas and hence a heating value of it. Major phenomena that possibly affect composition of withdrawn gas are:*mixing between in situ fluids and injection gas,*diffusion/dispersion of cushion gas when inert gas is used as cushiongas,*dissolution of gaseous components into water phase and water vaporizationinto gas phase,*adsorption/desorption of gaseous components onto a reservoir rocksurface,*turbulent flow caused by high rate withdrawal/injection and Klinkenberg effect,*hysteresis of gas-liquid capillary pressure and relative permeability, and*reservoir temperature change that results from cooling by injecting vaporized LNG and adiabatic expansion of gas. In addition, numerical errors such as numerical dispersion and discretization errors should be minimized for accurately reproducing these physical phenomena.
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