Reservoir Variability and Modeling with Fractals

Abstract

An important factor in the simulation of reservoir performance is the input reservoir parameters. Layers are often modeled with Constant parameters (porosity, permeability, etc.) or with only minor variations. The failure to incorporate appropriate heterogeneity into reservoir performance predictions can be a significant problem. This paper examines the variability in reservoir parameters as measured with wireline logs. Both vertical and horizontal well data are utilized for this purpose. The nature of the log variability in both a sandstone and a carbonate reservoir are evaluated using rescaled range (R/S) analysis in conjunction with power spectrum analysis, similar to the methodology of Hewett et al. The R/S analysis yields a fractal dimension that can be used to model the reservoir parameters. Both vertical and horizontal well log data can be characterized as fractional Gaussian noise of similar fractal dimension. Fractal reservoir realizations are generated using Fourier transform techniques. This technique is straightforward and successfully maintains the desired spatial dependencies of the data. The fractal dimension is derived from the results of the statistical analysis. Some potential problems with this determination will be highlighted and discussed. Examples of simulation results demonstrate improved reservoir performance predictions using the fractal realizations. Reservoir sweep efficiencies are altered and a different recovery performance is obtained relative to linearly interpolated realizations. Introduction Improvements in reservoir characterization can yield major improvements in reservoir performance predictions. However it is usually not possible to completely describe fine scale reservoir variability. Statistical modeling is a way to incorporate heterogeneities consistent with observations into reservoir models. A number of approaches have been proposed to accomplish this goal. Stochastic modeling incorporates a variety of techniques to generate reservoir realizations (see Halderson for a good summary). For example, average bed thicknesses, widths, observed distributions, etc. can be used to produce Markov fields. Gathering and interpreting the data for input is normally quite labor intensive and requires a high skill level. This approach, if used properly, can provide good representations of the reservoir. Geostatistics provides a number of useful tools for modeling. Semi-variograms represent the variance as a function of lag spacing (separation between points). The assumption for semi-variograms, as usually applied in petroleum reservoir characterization, is of fields with no spatial correlation at long lag spacings. This can be a problem in the presence of long range cyclic structures. Improvements such as indicator kriging can more accurately represent the reservoir structure and variation. These geostatistical approaches also require a significant amount of data and effort. Fractal modeling can be used to include heterogeneities in a limited data environment. P. 803^

Fractals and Their Applicability in Geological Wireline Log Analysis

“…These results are slightly lower than the 1.145 reported from a porosity log for a deep submarine fan sandstone (Hewett, 1986), and the average of 1.10 for density logs of a braided stream sandstone (Crane and Tubman, 1990). It is not clear whether this may be due to the different type of wireline logs used, or the different geological background.…”

“…Mathews et al (1989) used a fractal method for fluid-flow prediction in a channel-sands reservoir, and decided that the results were better than from traditional approaches. Crane and Tubman (1990) applied fractals to both vertical and horizontal well logs, in order to examine the spatial variability of reservoir porosity, and discovered similar fractal dimensions. They also used Fourier transform techniques to generate fractal reservoir realisations.…”

Characterization and estimation of reservoir properties in a carbonate reservoir in Southern Iran by fractal methods

“…It is surprising that the logs analyzed by many authors in many different formations show similar to fGn; for example, Carne and Tubman analyzed three horizontal wells and four vertical wells in a carbonate formation. They found that all the logs behave as fGn (Crane and Tubman 1990). Moreover, Hardy found similar results by analyzing some wells in sandstone formations (Hardy 1993).…”

Integration of Fuzzy Methods into Geostatistics for Petrophysical Property Distribution Simulation of a Reservoir at Exploration Stage

“…He found that black and white area distribution vertically and laterally has the characteristics of fGn. Crane and Tubman [5] and Hardy [6] investigated the logging data on horizontal wells on sandstone and carbonate reservoirs and indicated that the porosity has the characteristics of fGn.…”