In this paper, we present a field example where a streamline simulator was used to rank multi-million cell geostatistical reservoir descriptions and to find the optimum level of vertical upscaling for finite-difference simulation. During geostatitstical reservoir characterization, it is a common practice to generate a large number of realizations of the reservoir model to assess the uncertainty in reservoir descriptions and performance predictions. However, only a small fraction of these models can be considered for comprehensive flow simulations because of the high computational costs. A viable alternative is to rank these multiple ‘plausible’ reservoir models based on an appropriate performance criterion that adequately reflects the interaction between heterogeneity and the reservoir flow mechanisms. In this study, we explore the use of ranking based on streamline time of flight connectivity derived from a streamline simulator. The time of flight reflects fluid front propagation at various times and its connectivity at a given time provides us with a direct measure of volumetric sweep efficiency for arbitrary heterogeneity and well configuration. The volumetric sweep efficiency is the simplest measure that reflects the interaction between heterogeneity and the flow field. It is a dynamic measure that can be easily updated to account for changing injection/production conditions. We show that the proposed connectivity criterion can also be used to evaluate the effects of vertical upscaling in the dynamic performance and to determine the optimal level of upscaling for numerical simulation purposes. Our field study involves a Middle Eastern carbonate reservoir under a moderate to strong aquifer influx. The reservoir is on primary depletion and has no injectors. Multiple geostatistical reservoir descriptions were generated using a hierarchical approach whereby the larger level of uncertainty is defined first followed by smaller levels. The aquifer is modeled with a constant pressure boundary and for each time update, the location of the boundary was modified to account for the water encroachment. Using the field-wide sweep efficiency as a performance measure, the realizations were ranked, and used for flow simulation to assess risks associated with various development strategies. Subsequently, three selected realizations were upscaled for the purpose of comprehensive history matching and performance prediction. Background With the wide-spread use of geostatistics, it has now become a common practice to generate a large number of realizations of the reservoir model to assess the uncertainty in reservoir descriptions and performance predictions. Most commonly, these multiple realizations account for spatial variations in petrophysical properties within the reservoir and thus, represent a very limited aspect of uncertainty. For reliable risk assessment, we need to generate realizations that capture a much wider domain of uncertainty such as structural, stratigraphic, as well as petrophysical variations. From a practical point of view, we want to quantify the uncertainty and at the same time keep the number of realizations manageable. In this study, we will adopt an approach that is based on hierarchical principles. Thus, the uncertainty having the most potential impact is identified first. For example, with limited well control, the structural uncertainty derived from the seismic interpretations can have the most impact on the flow performance. Or, for faulted reservoirs, the uncertainty with respect to locations of faults can have the most impact. Then, the next level of uncertainty is identified and so on. The last level of uncertainty is the multiple geostatistical realizations of reservoir properties for a given set of input parameters. The petrophysical uncertainties generally tend to have a much lower impact on the reservoir performance compared to factors affecting large-scale fluid movements.
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Geological knowledge is an important ingredient in a successful reservoir characterization process. Geoscientists and engineers have used variogram extensively as the tool to quantify the spatial relationship of various attributes, e.g., facies/rock type, porosity, and permeability. Proper variogram modeling is a key factor to obtain a geologically-sound reservoir characterization model. This paper discusses the difficulty that is commonly encountered by many practitioners in modeling the variogram and proposes a way to incorporate geological knowledge as the soft information to improve variogram model. Common difficulty in variogram modeling is the calculation of horizontal variogram. The averaging technique that uses combination of geological knowledge and analogy in geophysical literature about frequency data analysis is implemented to solve the difficulty in calculating horizontal variogram. This technique has produced results that are agreeable with geology of the reservoir. The art of incorporating the geological knowledge in variogram modeling lies in the fact that geological knowledge is a qualitative measure whereas variogram is a quantitative measure. The methodology to combine these two measures presented in this paper is as follows. First, interpreting various geological aspects of the reservoir in detail. These include, but not limited to, the interpretations of geological environment, sequence stratigraphy, pore-space characteristics, iso-chores, iso-porosity and iso-permeability maps. From these interpretations, a summary table, that includes the major continuity direction, lateral extension and anisotropy index of each attribute, is prepared. Second, calculating experimental variogram using the Averaging Technique. Third, modeling the experimental variogram considering the information obtained from the first step. The procedure presented above has been implemented as a routine procedure in several reservoir characterization studies for both carbonate and sandstone reservoirs in the Middle East and in the USA. For illustration purposes, comparison of the realization results, taken from carbonate field study, between the model with variogram derived purely from the hard data, i.e., well log data, and the variogram derived from both hard and soft data, i.e., geological knowledge, is presented. It is concluded that the incorporation of geological knowledge has improved the confidence level of the results and should always be part of any reservoir characterization study. Introduction Geoscience data sets are distinguished from other types of data sets in one important aspect: they exhibit spatial relationship.1 In simple terms, neighboring values are related to each other. This relationship gets stronger as the distance between two neighbors becomes smaller. In most instances, beyond a certain distance the neighboring values become uncorrelated. This type of qualitative information needs to be defined in a suitable form so that it can be used to estimate values at unsampled locations. The most common statistics used to describe spatial relationship is variogram. It is the most widely used tool to investigate and model spatial variability of various reservoir attributes.2 The success in modeling the spatial relation, via the variogram, will provide higher chance of a successful reservoir characterization study. The main role of variogram is to reflect our understanding of the geometry and continuity of reservoir properties, which can have an important effect on the predicted flow behavior and reservoir management decisions. It is the measure of "geological variability" versus distance; it increases, as samples become more dissimilar. Therefore, it is clear that geological knowledge in reservoir charactersization is incorporated through variogram model. Therefore, thorough interpretation of geological knowledge should be done prior to modeling the variogram relationship. The interpretation that is of interest to the variogram modeling is anything that can lead to the information of lateral extent and major/minor continuity directions.
Summary Reservoir studies performed in the industry are moving towards an integrated approach. Most data available for this purpose are mainly from well cores and/or well logs. The translation of these data into petrophysical properties, i.e., porosity and permeability, at interwell locations that are consistent with the underlying geological description is a critical process. This paper presents a methodology that can be used to achieve this goal. The method has been applied at several field applications where full reservoir characterization study is conducted. The framework developed starts with a geological interpretation, i.e., facies and petrophysical properties, at well locations. A new technique for evaluating horizontal spatial relationships is provided. The technique uses the average properties of the vertical data to infer the low-frequency characteristics of the horizontal data. Additionally, a correction in calculating the indicator variogram, that is used to capture the facies' spatial relationship, is provided. A new co-simulation technique to generate petrophysical properties consistent with the underlying geological description is also developed. The technique uses conditional simulation tools of geostatistical methodology and has been applied successfully using field data (sandstone and carbonate fields). The simulated geological descriptions match well the geologists' interpretation. All of these techniques are combined into a single user-friendly computer program that works on a personal computer platform. Introduction Reservoir characterization is the process of defining reservoir properties, mainly, porosity and permeability, by integration of many data types. An ultimate goal of reservoir characterization is improved prediction of the future performance of the reservoir. But, before we reach that goal a journey through various processes must come to pass. The more exhaustive the processes, the more accurate the prediction will be. The most important processes in this journey are the incorporation and analysis of available geological information.1–3 The most common data types available for this purpose are in the form of well logs and/or well cores. The translation of these data into petrophysical properties, i.e., porosity and permeability, at interwell locations that are consistent with the underlying geological description is a critical step. The work presented in this paper provides a methodology to achieve this goal. This methodology is based on the geostatistical technique of conditional simulation. The step-by-step procedure starts with the work of the geologist where the isochronal planes across the whole reservoir are determined. This step is followed by the assignment of facies and petrophysical properties at well locations for each isochronal interval. Using these results, spatial analysis of the reservoir attributes, i.e., facies, porosity, and permeability, can be conducted in both vertical and horizontal directions. Due to the nature of how the data are typically distributed, i.e., abundant in the vertical direction but sparse in the horizontal direction, this step is far from a simple task, and practitioners have used various approximations to overcome this problem.4–6 A new technique for evaluating the horizontal spatial relationship is proposed in this work. The technique uses the average properties of the vertical data to infer the low-frequency characteristics of the horizontal data. Additionally, a correction in calculating the indicator variogram, that is used to capture the facies spatial relationship, is provided. Once the spatial relationship of the reservoir attributes has been established, the generation of internally consistent facies and petrophysical properties at the gridblock level can be done through a simulation process. Common practice in the industry is to perform conditional simulation of petrophysical properties by adapting a two-stage approach.7–10 In the first stage, the geological description is simulated using a conditional simulation technique such as sequential indicator simulation or Gaussian truncated simulation. In the second stage, petrophysical properties are simulated for each type of geological facies/unit using a conditional simulation technique such as sequential Gaussian simulation or simulated annealing. The simulated petrophysical properties are then filtered using the generated geological simulation to produce the final simulation result. The drawback of this approach is its inefficiency, since it requires several simulations, and hence, intensive computation time. Additionally, the effort to jointly simulate or to co-simulate interdependent attributes such as facies, porosity, and permeability has been discussed by several authors.11–13 The techniques used by these authors have produced useful results. Common disadvantages of these techniques are the requirement of tedious inference and modeling of covariances and cross covariances. Also, a large amount of CPU time is required to solve the numerical problem of a large co-kriging system. Another co-simulation technique that eliminates the requirement of solving the full co-kriging system has been proposed by Almeida.14 The technique is based on a collocated co-kriging and a Markov-type hypothesis. This hypothesis simplifies the inference and modeling of the cross covariances. Since the collocated technique is used, an assumption of a linear relationship among the attributes needs to be applied. The co-simulation technique developed in this work avoids the two-stage approach described above. The technique is based on a combination of simultaneous sequential Gaussian simulations and a conditional distribution technique. Using this technique there is no large co-kriging system to solve and there is no need to assume a relationship among reservoir attributes. The absence of co-kriging from the process also means that the user is free from developing the cross variograms. This improves the practical application of the technique.
One of the great challenges in reservoir modeling is to understand and quantify the dynamic uncertainties in geocellular models. Uncertainties in static parameters are easy to identify in geocellular models. Unfortunately, those models contain at least one to two orders of magnitude more gridblocks than typical simulation models. This means that, without significant upscaling, the dynamic uncertainties in these models cannot easily be assessed. Further, if we would like to select only a few geological models that can be carried forward for future performance predictions, we do not have an objective method of selecting the models that can properly capture the dynamic-uncertainty range.One possible solution is to use a faster simulation technique, such as streamline simulation. However, even streamline simulation requires solving a pressure equation at least once. For highly heterogeneous reservoir models with multimillion cells and in the presence of capillary effects or an expansion-dominated process, this can pose a challenge. If we use static permeability thresholds to determine the connected volume, it would not account for how tortuous the connection is between the connected gridblock and the well location.In this paper, we use the fast-marching method (FMM) as a computationally efficient method for calculating the pressure/ front propagation time on the basis of reservoir properties. This method is based on solving the Eikonal equation by use of upwind finite-difference approximation. In this method, pressure/front location (radius of investigation) can be calculated as a function of time without running any flow simulation. We demonstrate that dynamically connected volume based on pressure-propagation time is a very good proxy for ultimate recovery from a well in the primary-depletion process. With a predetermined threshold propagation time, a large number of geocellular models can be ranked. FMM can be scaled almost linearly with the number of gridblocks in the model. Two main advantages of this ranking method compared with other methods are that this method determines dynamic connectivity in the reservoir and that it is computationally much more efficient. We demonstrate the validity of the method by comparing ranking of multiple geocellular realizations (on Cartesian grid with heterogeneous and anisotropic permeability) by use of FMM with ranking from flow simulation. This method will allow us to select the geological models that can truly capture the range of dynamic uncertainty very efficiently.
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