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Fine scale heterogeneities can have significant effects on flow performances in subsurface formations. Fine-scale geostatistical realizations generated from well-log and seismic data are required to quantify the impact of such heterogeneities on flow. These realizations in many cases require number of grid cells on the order of 10 6-10 7 , hence making it too demanding CPU wise to directly history match geostatistical models. Non-uniformly gridding these models for emphasizing the heterogeneities does not necessarily present a solution since the practice of geostatistics on non-uniform grids is inefficient. In this paper we provide an entirely new approach to these problems by proposing a solution which is geostatistics based yet having the flexibility to work on non-uniformly gridded models. Hence coupling geostatistics with non-uniformly upscaled models results in an effective history matching technique incorporating all information at their scales. With the proposed approach we mainly target three advantages. Well and seismic data will still be effectively incorporated in the realizations. Flow simulations will be performed on non-uniformly upscaled models hence will be relatively fast. Finally the resulting field which is history matched will include the geology and will be on a non-uniformly gridded model for emphasizing heterogeneities or desired specifications, hence future predictions can be made fast, no posterior upscaling is required.

Rajagopal Raghavan, SPE, Philllips Petroleum Co. (retired); Ralph R. Roesler, SPE, Southwestern Energy Co.; and O. Inanc Tureyen, SPE, Stanford U. SummaryThe objective of this paper is to demonstrate the influence of detailed, small-scale heterogeneities on interference tests. Specific issues encountered when interference tests are analyzed in reservoirs with complex geological properties are discussed. These issues relate to questions concerning the use of low-resolution models, the degree of aggregation, the methodology of scaleup, and the reliability of conventional methods of analysis. This paper demonstrates the importance of capturing fine-scale heterogeneities to replicate the true transient behavior of interference tests at both active and observation wells. The paper shows the effects of aggregation and scaleup as used routinely in the industry on evaluating transient responses. The consequences of using low-resolution models in systems with complex geology is also demonstrated. If low-resolution models are used, reservoir properties may be adjusted unrealistically to match the transient behavior observed in high-resolution models. Though scaleup preserves pore volume, estimates of storativity predicted by lowresolution models will have a significant effect on reservoir behavior and resource management. If porosity values are not regressed, significant changes in vertical permeability values are observed. This is an important observation with potentially dramatic effects on reservoir performance, especially in processes involving mobility differences. Regression on a single-layer model (homogeneous, or based on aggregation) was also shown to yield totally different geological outcomes. This also shows the need to use geological constraints during inversion, aggregation, and scaleup.

With the advance of CPU power, numerical reservoir models have become an essential part of most reservoir engineering applications. These models are used for predicting future performances or determining optimal locations of infill wells. Hence in order to accurately predict, these reservoir models must be conditioned to all available data. The challenge in data integration for numerical reservoir models lies in the fact that each data has its own resolution and area of coverage. The most common data for reservoir characterization are; well-log/core data, seismic data and production data. Most current approaches to data integration are hierarchical. Fine scale models are used for integrating well-log/core and seismic data while coarse models are used to integrate mostly production data. The drawback of such a hierarchical approach is such that once the scale is changed, data conditioning, maintained in the previous scale, is lost. In this paper, we review a general algorithm as a solution to the multi-scale data integration. Instead of proceeding in a hierarchical fashion, a fine model and a coarse model is kept in parallel throughout the entire characterization process. The link between the fine scale and the coarse scale is provided by non-uniform upscaling. An optimization procedure determines the optimal gridding parameters that provide the smallest possible mismatch between fine and coarse scale reservoir models. A synthetic example application is given and demonstration of the methodology. The upgridding is accomplish by a static gridding algorithm, 3DDEGA. This algorithm aims at preserving geology by minimizing heterogeneity within a coarse grid block. The coarse grids are provided in a corner-point geometry fashion, hence this allows for accurate description of the reservoir with fewer number of grid blocks.

In this paper, new explicit analytical equations for stabilized and transient deliverability coefficients in terms of formation/fluid parameters are presented. These equations are derived for fully and partially penetrating vertical wells and for fully penetrating horizontal wells. The proposed equations use the relationship between the empirical (Rawlins and Schellhardt) deliverability equation and rigorous (or Forechheimer's) deliverability equations for both stabilized and unstabilized (transient) flow conditions. Based on these new equations, we also present new methods for determining not only the performance coefficients, but also formation parameters from stabilized and transient gas deliver ability data. Unlike the existing methods, our new methods do not require geometric mean of pressure or rate data. This is a distinct advantage of the methods presented in this paper over the existing methods because determination of geometric mean of pressure or rate data may be quite difficult, and the mean value is quite sensitive to the measurement errors in such data. Thus, the methods given in this paper should prove very useful for determining formation parameters accurately from deliverability tests in vertical and horizontal wells. The accuracy and applicability of the methods given in this study are verified by analyzing two test data published previously in the literature. Results are compared with those obtained from the existing methods in the literature and indicate that combination of our new methods with the Hinchman-Kazemi-Poettmann method for isochronal test types provides more information about the formation and reliable deliverability forecasting. Introduction Analyzing well-test data obtained from gas wells is of great importance to natural gas industry. Multipoint tests for gas wells, in general, can be divided into two groups; "stabilized" and "unstabilized" flow tests. The conventional backpressure tests for stabilized flow conditions and isochronal tests for unstabilized flow conditions are designed primarily to determine deliverability of gas wells. Transient data from these multipoint tests are often not considered for determining formation parameters (e.g, permeability-thickness product, non-Darcy flow coefficient, and skin factor). Thus, one of the main objectives of this work is to develop methods to determine deliverability characteristics of gas wells from transient data. As is well known, the deliverability characteristics of gas wells can be described by either the empirical U.S. Bureau of Mines (USBM) back-pressure equation developed by Rawlins and Schellhardt1, or the more general (i.e., theoretically more exact) equation known as the Forchheimer equation.2,3 Rawlins and Schellhardt's1 equation is given by Equation (1) where C and n are constants empirically derived from a log-log plot of the pressure squared differences vs. gas flow rate (qsc) data. C in Eq. 1 is called the stabilized-performance coefficient and, in general, depends on the flowing time. It becomes constant at long times. Taking the logarithm of both sides of Eq. 1, we find that a log-log plot of (Equation) vs qsc will be a straigt line with a slope of 1/n where 0.5 = n =1. When turbulent flow becomes important, the value of n becomes smaller than 1 and reaches the value of 0.5 for complete turbulence.