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.
Petroleum reservoirs often exhibit heterogeneity over a variety of length scales. This multiscale heterogeneity has a considerable, often dominant, influence on hydrocarbon recovery. Traditional methods for interpolating or extrapolating reservoir properties into unobserved regions have tended to smooth data, thus suppressing the effect of heterogeneity and rendering predictions of reservoir performance inaccurate. In recognition of this smoothing problem, stochastic interpolation methods have been increasingly applied. These methods produce multiple equiprobable maps of the reservoir. Each map matches the measurements at the wells and has statistical properties governed by an imposed theoretical model. Success in reservoir performance prediction depends on the quality of the model in mimicking the spatial variability of sedimentary rock.
One approach to stochastic simulation treats the variations in rock properties as fractional Brownian motion, fBm, or fractional Gaussian noise, fGn. This approach is based on Hewett observation that fluctuations exist over a wide range of spatial scales, and that this is captured in the fBm or fGn models. This attempt to build in some understanding of the quantitative nature of sedimentary rock directly into the model is an appealing alternative to more mechanical approaches. However, the fBm and fGn models are based on an assumption of an underlying Gaussian distribution. This is rarely supported by empirical evidence, except perhaps in some carefully selected examples.
In this paper we summarize recent advances in the understanding of multiple-scale heterogeneity that go beyond the Gaussian-based fractal and achieve greater realism while retaining simplicity. We show how this new model for heterogeneity can be used to generate stochastic permeability maps of reservoirs, and compare the performance of the new method with that of the fBm model.
The new model for heterogeneity characterization is based on observations that incremental values in well logs and permeability sequences are accurately modeled as having Levy-stable probability distributions and that the width of the fitted distribution increases with separation distance in a manner consistent with scaling behavior. This suggests the use of a fractional Levy motion, fLm, model for heterogeneity. fLm can be regarded as a generalization of fBm.
Levy distributions have power-law tails Pr for large q and 0 < < 2, which leads to diverging theoretical moments, including the variance. These slowly decaying tails make the Levy distributions useful for modeling systems with a high degree of spatial variability. The slowly decaying tails in the distribution of incremental values correspond to the occasional sharp property contrasts or "big jumps" in property values associated with stratification. The use of this model does not require rock properties to actually have an infinite variance distribution. It is a useful approximation when power-law tails exist over a large but finite range, which for practical purposes gives the underlying distribution properties similar to those of an infinite variance distribution.
We conducted simulations of two-dimensional waterfloods to test the performance of the interpolation based on fractional Levy motion, and compared the results with other interpolation methods. The idealized waterfloods are intended to illustrate the effects of heterogeneity while avoiding complicating effects such as gravity, compressibility and viscosity contrasts. To further test our simulation technique, we make use of a detailed permeability map of a sandstone outcrop.