Nonclassical Stochastic Methods in Subsurface Modeling, 20-9117Printer Friendly Version
Inclusive Dates: 01/15/99 - 07/15/00
Background - The physical properties controlling mass and energy transport can vary greatly from point to point within the subsurface. In typical applications, these properties are sparsely sampled, leading to large uncertainties in predictions of fluid or contaminant movement. In recent years, groundwater hydrologists and petroleum engineers have acknowledged this unavoidable uncertainty and are adopting probabilistic frameworks for making predictions. These probabilistic approaches require realistic mathematical models for subsurface heterogeneity. However, subsurface properties are characterized by complex multiscaled spatial fluctuations that are known to be inconsistent with classical statistical models for spatially distributed systems. There is a broad need for more realistic alternatives to classical stochastic models for subsurface heterogeneity.
Approach - This focused IR&D project developed new stochastic subsurface modeling tools based on contemporary mathematical descriptions of nonlinear variability in complex systems. Emphasis was on fractal scaling models and other approaches that capture long-range spatial dependence and extreme variability. Existing and newly developed fractal models were evaluated with outcrop and borehole data. Practical simulation tools based on these models were also developed.
Accomplishments - Detailed analyses of borehole and outcrop data were undertaken. These analyses show that, while subsurface data clearly have long-range spatial correlation, this fractal-like behavior is not adequately represented in previous fractal models. In particular, the previous fractal models produce either too much or too little variability. A new fractal scaling model based on a subordination was developed to remedy this shortcoming of existing models. Subordination is the mathematical procedure of constructing a stochastic process by randomizing the variance in an existing process. Subordination of a fractional Brownian motion by a log-normal subordinator was shown to reproduce important features of these data, including long-range spatial dependence, highly non-Gaussian distributions, and clustered volatility (intermittency). Two new stochastic simulation algorithms based on the new approach were developed and tested. These stochastic simulation tools can be used to produce input for petroleum reservoir and groundwater aquifer modeling codes by simulating physical properties in unobserved regions.