ABSTRACT: Numerical modeling of rock masses requires the accurate estimation of material properties throughout a rock volume. No matter how dense and accurate the sampling scheme may be, most of the rock mass remains unsampled. Thus a key step in creating useful numerical models is to accurately estimate material properties in unsampled regions. Estimation accuracy has improved over the past decade through the use and improvement of methods based upon the spatial correlation among data, such as geostatistics and fractal geometry. However, the spatial pattern of a particular property in a rock mass is a function of more than its spatial correlation. A new method, Projection Onto Convex Sets (POCS), offers an alternative way to constrain the estimated values for numerical models. Its primary advantage is that it can simultaneously enforce a large variety of geologically useful constraints. This paper provides an overview of the theory and numerical implementation of POCS, and illustrates an application to flow through fractures.
OVERVIEW OF POCS
Geologic parameters in stochastic models are ot?en assigned properties throughout a large volume of rock based on interpolation from well tests, geophysical profiles or outcrop maps comprising a very small proportion of the rock to be modeled. Without substantial constraints, these stochastic models may have an unacceptably wide range of unity which would lead to major differences in design or predicted performance. Over the past decade, efforts to add spatial correlation constraints through geostatistics or fractal geometry have reduced the uncertainty in many ?. However, the spatial pattern of a particular property in a rock mass is a function of more than its spatial correlation. Having a model with the correct spatial correlation and conditioning it to known data points may not be sufficient to accurately estimate input values at unsampled locations (e.g. G-der, 1993).
Texture, the spatial pattern of properties, is a complex function of many parameters: the mean value, the standard deviation, the spatial correlation and its anisotropy, the presence of data discontinuities, minimum and maximum bounds, trends, and other quantities. A simple illustration of this is shown in Figure 1.1. These six images represent the value of hydraulic conductivity assigned to grid cells in a numerical model. The input value data sets appear more noisy or variable moving from top to bottom. However, the energy content, the population variance, and the population mean are all identical. In this case, the variability is a function solely of anisotropy and spatial correlation. These graphs are a series of self-affine fractal simulations conditioned to the same local data points with anisotropy increasing from lei? to right and the fraetal dimension D,., increasing from top to bottom. These simulations suggest that anisotropy is less important for higher fractal dimensions, and more important for lower. In other words, anisotropy plays a more important role as textural parameter when the degree of spatial correlation is greater. This simple example points out the complex interplay of textural parameters and the importance of incorporating as many textural constraints as possible.