Scott Painter

Evidence for Non‐Gaussian Scaling Behavior in Heterogeneous Sedimentary Formations

Vertical and horizontal fluctuations in permeability and porosity in sedimentary formations are analyzed and are found to be consistent with scaling models based on Levy-stable probability distributions. The approach avoids the assumption of Gaussian behavior and is supported by evidence from horizontal and vertical well logs and from permeability measurements on a sandstone outcrop and segments of core from a heterogeneous formation. The incremental values in these measurement sequences are accurately modeled as having Levy-stable distributions. The width of the distribution of increments depends on the spatial scale in a manner consistent with scaling behavior. The width of the distribution is smaller for horizontal increments than for vertical increments, reflecting the reduced variability in the horizontal direction. The scaling parameters are in the range associated with antipersistence and are roughly the same magnitude in the vertical and horizontal directions. The relationships between different physical properties are briefly studied, and it is suggested that they be quantified through off-diagonal terms in a multivariate Levy width matrix. Simulations designed to reproduce the observed statistical features are also described. These results have some fundamental implications, as Levy-stable distributions require a different set of statistical tools and theoretical methods compared to finite-variance distributions.

Fractional Lévy motion as a model for spatial variability in sedimentary rock

A new approach to the modeling of spatial variability in sedimentary formations is introduced. This approach avoids the assumption of Gaussian behavior. Specifically, borehole measurements of physical properties of sedimentary rock from contrasting geological settings are shown to have a statistical character consistent with fractional Levy motion. Successive increments in the measurement sequences are accurately modeled as having Levy-stable distributions. The Levy parameters are similar for boreholes in the same sedimentary basin, but vary from basin to basin. Thinly bedded formations have smaller values for the Levy index, suggesting that the Levy index may be a useful measure of heterogeneity. The measurement sequences are also statistically self-similar and have long-range negative dependence among the increments (antipersistence). In addition to reproducing the statistical properties of well logs, the new model also mimics the most striking visual features of sedimentary formations.

Stochastic Interpolation of Aquifer Properties Using Fractional Lévy Motion

This paper describes a new technique for conditional simulation of aquifer properties in unobserved regions. The technique utilizes a recently introduced model for heterogeneity based on the Levy-stable family of probability distributions to achieve a higher degree of realism than is possible with Gaussian-based techniques. Specifically, permeability and porosity variations are modeled using fractional Levy motion. A new algorithm for producing fractional Levy motion conditioned to match known data is introduced. Two-dimensional simulations using actual permeability measurements as input demonstrate that this algorithm produces random fields with the specified properties. Simulations using a subset of a detailed permeability map of an outcrop as input illustrate how the new method can be used to estimate important flow and transport properties and to quantify uncertainties in these estimates.

Random fractal models of heterogeneity: The Lévy-stable approach

Borehole measurements of petrophysical properties of sedimentary rock from contrasting geological settings are shown to be consistent with fractional Lévy motion. Specifically, successive increments in the measurements sequences are modeled accurately as having symmetric Lévy-stable distributions. The measurement sequences are statistically selfsimilar for a wide range of spatial scales, limited by the finite length of the measurement sequences at large scales and by the resolution limits of the logging devices at small scales. The measurements sequences display clear evidence of antipersistence (negative dependence in the increments). These results suggest that the fractional-Gaussian-noise model used in reservoir description is an inappropriate model for sedimentary rock, because the rock properties are not stationary and do not have a Gaussian distribution as required for fractional Gaussian noise. In contrast to those of Gaussian-based fractals, the sample paths of the new model increase and decrease in jumps of all magnitudes and mimic geological stratification. These results have fundamental implications for petroleum geostatistics, as many of the statistical techniques developed for Gaussian variables are not valid for Lévy-distributed variables.

Numerical Method for Conditional Simulation of Levy Random Fields

Stochastic simulations of subsurface heterogeneity require accurate statistical models for spatial fluctuations. Incremental values in subsurface properties were shown previously to be approximated accurately by Levy distributions in the center and in the start of the tails of the distribution. New simulation methods utilizing these observations have been developed. Multivariate Levy distributions are used to model the multipoint joint probability density. Explicit bounds on the simulated variables prevent nonphysical extreme values and introduce a cutoff in the tails of the distribution of increments. Long-range spatial dependence is introduced through off-diagonal terms in the Levy association matrix, which is decomposed to yield a maximum likelihood type estimate at unobserved locations. This procedure reduces to a known interpolation formula developed for Gaussian fractal fields in the situation of two control points. The conditional density is not univariate Levy and is not available in closed form, but can be constructed numerically. Sequential simulation algorithms utilizing the numerically constructed conditional density successfully reproduce the desired statistical properties in simulations.

Simulating residual saturation and relative permeability in heterogeneous formations

Network models are used to investigate the effect of correlated heterogeneity on capillary dominated displacements in porous media. Residual saturations and relative permeabilities are shown to be sensitive to the degree of correlation and anisotropy but not variability. The network models reproduce the experimental observation that relative permeability is greater in the direction parallel to the bedding compared to perpendicular to the bedding. The scatter commonly observed in core measurements of residual saturation is attributed to the presence of correlated heterogeneity in actual reservoir rocks.

Improved technique for stochastic interpolation of reservoir properties

Stochastic interpolation methods for generating input for reservoir simulations require accurate models for spatial variability. New models based on fractional Levy motion, a generalization of fractional Brownian motion, have strong empirical support. Stochastic interpolations based on the Levy model have a higher degree of spatial variability compared to Gaussian fractals. In two-dimensional waterflood simulations, the breakthrough curves for the Levy interpolation method are better clustered around predicted production behavior based directly on outcrop data.