Effect of Scaleup and Aggregation on the Analysis of Interference Tests
- Rajagopal Raghavan (Phillips Petroleum Co.) | Ralph R. Roesler (Southwestern Energy Co.) | O. Inanc Tureyen (Stanford U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- December 2004
- Document Type
- Journal Paper
- 392 - 398
- 2004. Society of Petroleum Engineers
- 5.5.8 History Matching, 2.2.2 Perforating, 5.1.5 Geologic Modeling, 5.6.9 Production Forecasting, 5.1 Reservoir Characterisation, 5.7.3 Deterministic Methods, 4.1.2 Separation and Treating, 5.6.4 Drillstem/Well Testing, 5.6.3 Deterministic Methods, 4.3.4 Scale, 5.5 Reservoir Simulation, 5.1.2 Faults and Fracture Characterisation, 5.6.5 Tracers, 4.1.5 Processing Equipment, 5.1.1 Exploration, Development, Structural Geology, 5.2 Reservoir Fluid Dynamics
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The 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 low-resolution 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.
This work provides guidelines concerning the evaluation of interference or pulse tests in reservoirs with complex geology. In the process, the paper demonstrates the role of small-scale heterogeneities on the responses of interference tests and how those heterogeneities may affect the analysis of a test. In the past, it has been suggested that interference and pulse tests provide a better measure of reservoir heterogeneity.1 Our work also permits us to examine the consequences of working with a low vertical resolution, as is normally done in most analyses. It is important to recognize the choice of this kind of model because the consequences of such a choice have a significant bearing on the spatial and petrophysical relationships we wish to determine. The importance of not ignoring the vertical resolution is demonstrated. The choice we make regarding the scale at which we analyze data influences the characteristics of the model we use to describe the reservoir; this choice, in turn, will have a significant bearing on our evaluation of fluid movement in injection/production schemes. In doing so, we also illustrate the consequences of aggregation and scaleup. The results of this study were obtained by the model given in Christie and Blunt2 because it provides information on a very fine scale and is thus particularly suited to meet the goals of this study.
The issue of heterogeneity as it relates to well performance and pressure tests may be traced to Cardwell and Parsons.3 Using deterministic methods, these authors considered steady, radial flow to a well in a reservoir in the form of a circle and showed that the equivalent transmissivity, Teq(khµ), is bounded by the volume-weighted arithmetic and harmonic means. The earliest work on statistical methods that is most often quoted is that of Warren and Price.4 On the basis of numerical experiments of pressure-buildup tests, they concluded that a porous medium that is assumed to consist of a quilt or patchwork of porous elements that are randomly distributed in 3D may be represented by a porous rock with a permeability equal to the geometric mean of the permeability of the individual elements. Although Warren and Price4 do not specifically recognize that the properties of rocks are usually correlated, at least weakly, on all scales, it has been shown that the effective permeability, keq , of a 2D system under the assumption that the porous body is a log-normal, subisotropic and ergodic medium is equal to the geometric mean of the permeability of the individual elements. For 3D systems, however, their result is only approximately correct.5 Technically, one should expect the equivalent permeability to be tensor.6
To better represent conditions in groundwater testing, Vandenberg7 conducted a series of evaluations of interference tests in a manner analogous to that of Warren and Price and concluded that interference tests in heterogeneous formations analyzed by the Theis8 method appear to yield transmissivity, T, estimates that reflect the arithmetic mean of the individual elements of the heterogeneous porous medium. Estimates of storativity, S(fcth), varied over a much wider range than estimates of T. Similar observations are made in reports by Herweijer and Young,9 Schad and Teutsch,10 and Meier et al.11 These studies are essentially restricted to 2D systems. The information in Refs. 8 through 11 also indicates that transmissivity estimates appear to depend primarily on the distance between the pumping and observation wells and may be reasonably constant (all other things being equal); however, the variation in the estimate of storativity, S, is quite significant. It is to be noted that the variations in the estimates of S are not the result of fitting the measured drawdowns to the Theis solution or to its asymptotic (semilog) approximation given by Jacob.12 The observation of these authors that T is reasonably constant may be understood by the results given in Oliver.13,14 Large variations in the estimate of S, however, are not a surprising result. As will be shown here, this result is a direct consequence of analyzing the well responses in a heterogeneous system by a homogeneous model. In particular, if interference responses are analyzed by the method of Theis,7 then any property of the reservoir involving the distance variable will usually be in significant error.15,16 Also, when interference tests in heterogeneous systems are analyzed in the conventional way, one must at least consider anisotropy (in magnitude and azimuth) and if rigor is to be reflected in the analysis, the possibility that the estimate of transmissivity reflects a tensor must be recognized. If no such recognition is made, then estimates of S may not be representative for significant variations in S obtained even in relatively homogeneous but anisotropic reservoirs whenever reservoir anisotropy is ignored; see Ramey.17
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