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Earth-science data exhibit spatial connectivity to greater and lesser degree. As the distance between two data points increases, the similarity between the two measurements decreases. Geostatistics is a rapidly evolving branch of applied statistics and mathematics that offers a collection of tools for understanding and modeling spatial variability. Spatial variability includes scales of connectivity (heterogeneity) and directionality within data sets. Geostatistical methods also allow us to quantify and assess the reliability of the models we generate.
Origin of geostatistics
Geostatistics originated in the mining industry. In the early 1950s, when “classical” statistics were found unsuitable for estimating disseminated ore reserves, D.G. Krige, a South African mining engineer, and H.S. Sichel, a statistician, developed a new estimation method. The French engineer Georges Matheron expanded on Krige’s innovative concepts and formalized them within a single framework, and coined the word “kriging” in recognition of Krige’s work. Although the kriging technique originally was developed for solving ore-reserve estimation problems, with the advent of high-speed computers in the 1970s, it spread to many other areas of earth science; however, it was not until the mid-to-late 1980s that geostatistical techniques were used to any extent in the petroleum industry, though their acceptance and use has grown steadily and significantly since.
Role of geostatistics in reservoir characterization
The enormous upfront expense of developing heterogeneous reservoirs, and the desire to increase ultimate recovery has spurred oil companies to develop and use innovative reservoir-characterization techniques. Geostatistics is one of the many recent technologies often incorporated into the reservoir-characterization process.
Since the late 1980s, geostatistical techniques have become an accepted technology for characterizing petroleum reservoirs, especially when incorporating 3D seismic data. The resultant numerical descriptions often are put into a fluid-flow simulator. Use of geostatistics necessitates the cooperation between the geoscience and reservoir-engineering disciplines, allowing each to contribute fully in the process of building the reservoir model. This is quite different from past approaches, in which mathematical formalization often was left to the reservoir engineers. The multidisciplinary approach, coupled with improved technology for reservoir modeling, ensures that important geologic characteristics are not overlooked in the process.
Traditional geology is qualitative, based soundly on classification schemes and descriptions associated with physical phenomena. In the normal course of reservoir modeling, such qualitative geologic models are transformed into numerical models, though often by a reservoir engineer, rather than by a geologist. If the geologic model is precise, such a transformation presents no problem; however, in the past, the numerical models tended to bear little resemblance to the geologic models on which they were based. The differences commonly were caused by discipline-related interpretation and typically were economically pragmatic. Reservoir models were and continue to be expensive to produce, such that simulating a reservoir at a very fine resolution is impractical. To reduce computer simulation time (ergo cost), the geologic model is coarsened to a more manageable number of grid nodes.
But drastically reducing the size of a reservoir model has ramifications. If the heterogeneity, or complexity, of the geology is oversimplified, the integrity of simulation results can be affected. A coarser initial representation may be appropriate and adequate for a relatively simple reservoir, but with a complex reservoir, it can yield misleading simulation results.
To prevent this problem, history-matching techniques are used to fine-tune the coarser engineering model. Porosity, permeability, and other parameters may be adjusted until the fluid-flow simulation matches the observed well performance, pressures, and flow rates from production tests. If any of these three conditions is matched, the model is assumed to be reasonable, although not unique. Although this model may match history up to a point in time, it can be a poor predictor of future production.
Reservoir simulation brings a kind of closure to a study by providing the economics and development plan, but the production forecast it provides often is inaccurate. Fortunately, the geostatistical approach to reservoir modeling has been in place long enough that the efficacy of the geostatistical procedure can be assessed in terms of production history after the simulation. The improvement to flow-simulation results from using geostatistically built models has been noted not only personally by the authors, but also by others, as well.    Frequently, these predictions were overly optimistic, largely because reservoirs were considerably more heterogeneous and compartmentalized than their reservoir models presumed them to be.
Such consistent deviation from prediction indicates the presence of an undefined parameter whose exclusion from the model created a bias. In pointing out that it frequently has been the case that individual wells have not performed as predicted and infill-drilling patterns have not behaved as predicted, we are not suggesting that there are no good reservoir models, nor do we mean to be overly critical. Instead, we are challenging our industry to be more innovative by taking advantage of current technology to preserve heterogeneity in the upscaled models that are input to the reservoir simulator. Compared to the early use of the geostatistical approach, more-recent efforts have included progressively more geology and concern for the consistency of the geology and the observed physical parameters. The results of these efforts certainly validate the need to incorporate more characteristics on the basis of our understanding of the depositional environments represented in the reservoir.
The idea that reservoirs are heterogeneous is not new. Using a relative scale of heterogeneity tied to the original depositional environments, Tyler and Gholston  and Tyler and Finley  have shown that a substantial amount of mobile hydrocarbons often are left behind in reservoirs of varying heterogeneity. Weber,  Srivastava,  King and Mansfield,  Botton-Dumay et al.,13 and Srinivasan and Caers were pioneers in bed-level evaluation of the effects of heterogeneity on hydrocarbon recovery.
Geostatistically derived reservoir modeling is perhaps the most successful means of improving performance predictions in heterogeneous reservoirs. It is successful because understanding the heterogeneity that exists in the interwell space inherently is a statistical problem that can be quantified. It is not the only approach, nor is it useful in all cases, but it is a rigorous approach that has proved beneficial in the face of many real conditions and practical considerations involved in modeling petroleum reservoirs.
The goal of geostatistically derived modeling is to construct a more realistic model of reservoir heterogeneity using methods that do not simply average reservoir properties. Like the traditional deterministic approach, it preserves both the measured (hard) data where they are known and the interpretative (soft) data whenever they are informative; however, unlike the deterministic approach, geostatistics provides scientists with numerous plausible results (realizations). The degree to which the various models differ is a reflection of the unknown, a measurement of the uncertainty. Some of the realizations may challenge the prevailing geologic wisdom and almost certainly will provide a group of economic scenarios ranging from optimistic to pessimistic.
Having more than one result to upscale and analyze in the flow simulator changes the paradigm of traditional reservoir analysis, though, and such a change is necessary because heterogeneity in dynamic data is not readily apparent when using the traditional method. Srinivasan explains the problem of dynamic data in this way, “Only limited information pertaining to the heterogeneity of the permeability field is present in the dynamic data. Such information must be extracted using calibration methods, so that reservoir models then can be constrained to that calibrated information. The methodology matches the history data in a probabilistic sense (acknowledging other unknowns, such as relative permeability) and can be used to make robust predictions of the future production.”
Personal communication between Jeffery Yarus and Sanjay Srinivasan, U. of Texas (2002).*
Classical statistics and its role in geostatistical modeling
A fundamental step in any scientific investigation is the quantitative-description stage. This is particularly true today of the geologic sciences, which in the past had depended largely on qualitative description. Until the facts are gathered and described quantitatively, analysis of their causes is premature. Statistics works with quantities of data, not with a single datum, and so requires those data to be in manageable form. Organized data are the clearest data. Thus, much of statistics deals with the organization, presentation, and summary of information.
Both the computation of classical statistical measures (e.g., mean, mode, median, variance, standard deviation, and skewness), and graphic data representation (e.g., histograms and scatter plots) commonly are used to understand the nature of data sets in a scientific investigation—including a reservoir study. A distinguishing characteristic of earth-science data sets (e.g., for petroleum reservoirs), though, is that they contain spatial information, which classical statistical descriptive methods cannot adequately describe. Spatial aspects of the data sets, such as the degree of continuity—or conversely, heterogeneity—and directionality are very important in developing a reservoir model. Analysis of spatially rich data is within the domain of geostatistics (spatial statistics), but a foundation in classical statistics and probability is prerequisite to understanding geostatistical concepts.
Sampling also has proved invaluable in thousands of studies, but it, too, can lead to statistical insufficiencies and biases. So, when can a sample be trusted? The answer depends on how the sample was selected. As discussed more fully later in this section, classical statistics assumes that each observation in the data set is independent of the others or is random. That is, it assumes that the samples (e.g., porosity measurements from a core or from logs) are from a larger, theoretical parent population in which each selected sample has the same chance as any other of being included in the sample group. Petroleum-geologic samples (e.g., well data) are not random, primarily for two reasons. First, they are oversampled in areas that are conducive to oil and gas production. Second, the samples themselves are tied to a coordinate system and so are related in geographic space. Thus, the use of a classical-statistical approach is problematic because usually samples are biased and have an underlying dependence. For a sample to be trusted, the bias must be adjusted and the spatial dependency accounted for.
- Krige, D.G. 1951. A Statistical Approach to Some Basic Mine Evaluation Problems on the Witwatersrand. J. Chem. Metall. Min. Soc. South Africa 52: 119.
- Sichel, H.S. 1952. New Methods in the Statistical Evaluation of Mine Sampling Data. Trans. Inst. Min. Metall. 61 (6): 261.
- Matheron, G. 1962. Traite de Geostatistique Appliquee, tome 1, 111. Paris, France: Editions Technip.
- Deutsch, C.V. and Meehan, D.N. 1996. Geostatistical Techniques for Improved Reservoir Management: Methodology. Hart’s Petroleum Engineer Intl. (March): 21.
- Beattie, C.I., Mills, B.R., and Mayo, V.A. 1998. Development Drilling of the Tawila Field, Yemen, Based on Three-Dimensional Reservoir Modeling and Simulation. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, Louisiana, 27-30 September 1998. SPE-49272-MS. http://dx.doi.org/10.2118/49272-MS
- Lia, O. et al. 1997. Uncertainties in Reservoir Production Forecasts. AAPG Bulletin 81: 775.
- King, M.J. and Mansfield, M. 1999. Flow Simulation of Geologic Models. SPE Res Eval & Eng 2 (4): 351-367. SPE-57469-PA. http://dx.doi.org/10.2118/57469-PA
- Tyler, N. and Gholston, J.K. 1988. Heterogeneous Submarine Fan Reservoirs, Permian Spraberry Trend, West Texas. Report of Investigations No. 171, Texas Bureau of Economic Geology, Austin, Texas, 37.
- Tyler, N. and Finley, R.J. 1991. Architecture Controls on the Recovery of Hydrocarbons from Sandstone Reservoirs. SEPM Concepts in Sedimentology Paleontology 3 (1–5).
- Weber, K.J. 1986. How Heterogeneity Affects Oil Recovery. Reservoir Characterization, 487-544, ed. L.W. Lake and H.B. Carroll Jr. Orlando, Florida: Academic Press.
- Srivastava, R.M. 1994. An Overview of Stochastic Methods for Reservoir Simulation. Stochastic Modeling and Geostatistics, 3, 3-16, ed. J.M. Yarus and R.L. Chambers. Tulsa, Oklahoma: AAPG Computer Applications in Geology.
- King, M.J. and Mansfield, M. 1997. Flow Simulation of Geologic Models. Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 5-8 October 1997. SPE-38877-MS. http://dx.doi.org/10.2118/38877-MS
- Botton-Dumay, R., Cogrel, Y.M., Massonnat, G.J. et al. 1997. Realistic Methodology for Permeability Modelling Used for Conserving Heterogeneity During Assisted History Matching-Applied to a Turbiditic Reservoir Field Case. Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 5-8 October 1997. SPE-38677-MS. http://dx.doi.org/10.2118/38677-MS
- Srinivasan, S. and Caers, J. 2000. Conditioning reservoir models to dynamic data - A forward modeling perspective. Presented at the SPE Annual Technical Conference and Exhibition, Dallas, Texas, 1-4 October 2000. SPE-62941-MS. http://dx.doi.org/10.2118/62941-MS
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