T -distributed Random Fields: A Parametric Model for Heavy-tailedWell-log Data1

Histograms of observations from spatial phenomena are often found to be more heavy-tailed than Gaussian distributions, which makes the Gaussian random field model unsuited. A T-distributed random field model with heavy-tailed marginal probability density functions is defined. The model is a generalization of the familiar Student-T distribution, and it may be given a Bayesian interpretation. The increased variability appears cross-realizations, contrary to in-realizations, since all realizations are Gaussian-like with varying variance between realizations. The T-distributed random field model is analytically tractable and the conditional model is developed, which provides algorithms for conditional simulation and prediction, so-called T-kriging. The model compares favourably with most previously defined random field models. The Gaussian random field model appears as a special, limiting case of the T-distributed random field model. The model is particularly useful whenever multiple, sparsely sampled realizations of the random field are available, and is clearly favourable to the Gaussian model in this case. The properties of the T-distributed random field model is demonstrated on well log observations from the Gullfaks field in the North Sea. The predictions correspond to traditional kriging predictions, while the associated prediction variances are more representative, as they are layer specific and include uncertainty caused by using variance estimates.     

A Connectivity Index for Discrete Fracture Networks

Connectivity is an important measure for assessing flow transport in rock, especially through fractures. In this paper, rock fracture systems are modelled by a discrete fracture model simulated by a marked point process. A connectivity index is then introduced to quantify the connectivity between any two points in space. Monte Carlo simulation is used to evaluate the connectivity index for stationary cases and relationships between the connectivity index and the parameters of the discrete fracture model are analysed. The average number of intersections per fracture, X_f, and the fracture intensity, P12 (P32), are calculated and the relationships between these parameters and the connectivity index are investigated, concluding that X_f is the more suitable parameter for the classification of rock mass flow properties. The relationships between the connectivity index and the percolation state of the fractured medium are also discussed. An edge correction is briefly discussed and a practical example is used to demonstrate the method of computing the connectivity index.