Incorporating spatial distribution into stochastic modelling of fractures: multifractals and Levy-stable statistics

A new approach to the stochastic modelling of fractures is developed using data acquired in horizontal wells and applied to one-dimensional simulations. It differs from previous studies in the use of both Levy-stable statistics to describe the fracture attributes and of a (multifractal) strain-based model to create the spatial distribution. The structure of the multifractal strain is generated with a multiplicative cascade and used as a template or guide in the simulation. Resulting simulations have scale-invariant structure. Spacing distribution functions depend on the spatial partitioning of strain. For the case where strain distribution is homogeneous an approximate negative exponential spacing distribution results. Heterogeneous strain, exhibiting intermittency, leads to power-law (fractal) spacing distributions and spatial clustering. When plotting spacing as log-log cumulative frequency, the slope (fractal dimension) quantifies the degree of clustering. Small fractal dimensions are indicative of more clustering than large ones, the degree of clustering decreasing with an increase in fractal dimension.