An Application of Geostatistics and Fractal Geometry for Reservoir Characterization

Abstract

Summary. This study presents an application of geostatistics and fractalgeometry concepts for 2D characterization of rock properties (k and) in adolomitic, layered-cake reservoir. The results indicate that lack of closelyspaced data yield effectively random distributions of properties. Further, incorporation of geology reduces uncertainties in fractal interpolation ofwellbore properties. Introduction Previous work showed that miscible flood performance is affected by the Previous work showed that miscible flood performance is affected by the degreeand scale of small-scale heterogeneities, information which, unfortunately, isnot obtained in conventional reservoir-characterization processes. Two methods-- geostatistics and fractal geometry -- may help processes. Two methods --geostatistics and fractal geometry -- may help to obtain such information andalso to estimate associated uncertainties in the reservoir characterization. This study evaluates how these two techniques characterize a reservoir unit(Unit A pilot area) while honoring the geological information (100 to 200values each of k and in each of 18 wells in Unit A, where distances betweenwells vary from 85 to 4,000 ft). Fig. 1 shows the Unit A well locations. Thisstudy does not extend to using reservoir descriptions to predict and/or tointerpret any field performance. The geostatistical technique was used areallywithin performance. The geostatistical technique was used areally withinindividual layers. More specifically, the correlation structure was obtainedfor individual layers based on data from the 18 wells in Unit A. Then, theconstraints of the spatial correlation structure and the available k and datawere used with a conditional simulation technique to create several possiblereservoir descriptions for the individual layers in the central five-spotpilot. The injector-to-injector distance is 288 ft for the five-spot pilot. Adverse mobility ratio miscible displacements (M=5) were simulated for thegenerated set of reservoir descriptions. The fractal geometry approach wasapplied across a vertical cross section between Wells 1 and 5. Available k anddata were used to create a fractal cross section between the two wells thatstill honored the geologic layers from the k/ plots. Single-phasefirst-contact-miscible displacements (M=5) for Npet less than 0.01 weresimulated across the fractal cross section. Background Geostatistics. Geostatistics is a branch of applied statistics that handlesspatially distributed variables. Recent interest in the petroleum industry hasbeen to use geostatistics concepts to characterize and to evaluate petroleumreservoirs. Basic geostatistical concepts can be found in petroleum reservoirs. Basic geostatistical concepts can be found in several publications. Ordinarykriging, a geostatistical method developed by Matheron, is often used as amapping technique in which the regionalized variable is estimated from anassessment of spatial continuity. The uncertainty of any estimated value can bedetermined. Kriging, however, results in a "smooth" realization of theregionalized variable. We used conditional simulation in this study because itgives a better visual idea of the uncertainties involved in the estimations. Spatial conditional simulation is a Monte Carlo simulation technique thathonors the available data, their location, and the variable's spatialstructure. Several realizations (sample outputs with different seed values)will indicate the uncertainties involved. These realizations, or reservoirdescriptions, can be used in reservoir flow simulators to provide a range ofpossible outcomes. Alabert developed a conditional simulator that allowsinformation from such different sources as core data, well logs, andgeophysical and geological data to be pooled. These data are classified aseither hard or soft. More significance is assigned to the hard data (morecertain data, such as core data) than to the soft data (less certain data, suchas geophysical data or qualitative interpretations). The simulator uses anindicator formalism and probability kriging to estimate variables. Applications of Geostatistics. The length of the permeability scalecontributes to the heterogeneity in a porous medium and significantly affectsmiscible displacements. Limited research has been conducted to establish thecorrelation structure of permeability and porosity in porous media. Two studiesare presented here. Da Costa e Silva estimated the spatial structure ofpermeability and porosity in a North Sea reservoir (26,247 × 65,617 ft) usingdata from 30 porosity in a North Sea reservoir (26,247 × 65,617 ft) using datafrom 30 wells. k and data were obtained from well tests and were assumed torepresent the reservoir's producing zones. Permeability and porosity were foundto have similar spatial structures, with anisotropy and scale lengths near13,000 ft. In some directions it was indicated that structures may exist on ascale smaller than the smallest lag length or shortest distance between thewells. Kriging results from the reservoir gave a smooth or "flat" reservoir description. Goggin et al. conducted outcrop studies of Eoliansandstone deposits near Page, AZ, to obtain a detailed description ofpermeability heterogeneity at various scales. About 2,000 permeabilitymeasurements were taken with a calibrated minipermeameter on concentric gridpatterns, and statistical analysis was done to infer the scale dependence ofpermeability variation. The spatial correlation structure of permeability wasobtained by variogram analysis of the measured data. These studies revealedthat the heterogeneity scales closely relate to the geologic scales. Thespatial correlation structure of permeability identified by the scale length, anisotropy ratio, and the inclination angle of measurement was found to bescale-dependent in Eolian systems. Fractal Geometry. Mandelbrot developed the concept of fractal geometry, which can be used to describe irregular and fragmented patterns in nature. Fractals are sets of continuous and non-differentiable shapes with a geometricdimension that strictly exceeds the topological (Euclidean) dimension. Animportant feature of fractals is their resemblance at various scales, whichmeans that the characteristics of the fractals can be identified at all scales. Both deterministic and random fractals exist, with random fractals being moreuseful for the characterization of the irregularities of natural lines andsurfaces. The characteristics of random fractals, however, are statistical whensubject to scaling. Various statistical models exist to describe naturalphenomena. One of the most useful mathematical models available for describingnatural random fractals (mountainous terrain, clouds, etc.) is the fractional Brownian motion, fBm, which is an extension of Brownian motion. Severalalgorithms exist for creating traces of fractional Brownian motion. SPEFE p. 11