A Stochastic Model of a Fractured Rock Conditioned by Measured Information

A method for the modeling of a fractured rock is developed which takes into account the uncertainty in the fracture network geometry as well as actually measured information, such as that obtained from cores. Based on certain simplifying assumptions, of which the most important are planar and independent fractures, a stochastic a priori model is formulated. The real fracture network is assumed to be a realization of this a priori model. Since measurements are performed on the real network, two different kinds of information are made available. The first kind is of deterministic nature and expresses the actual location of intercepted fractures; by inference the other part is probabilistic, given as the probability of observing a fracture intersecting the model region. This observation probability is shown to depend on the a priori model, on the geometry of the region and on the measurements only. A conditional model is formulated where each realization consists of the actually observed fractures and an additional number of stochastically generated fractures obtained by employing the probabilistic information. The number of stochastically generated fractures is a stochastic variable, the distribution of which depends on the number of fractures observed and the observation probability. Even if the rock is penetrated with only a few cores, it is possible to quantify the statistics of properties, such as the total leakage into a tunnel or the concentration of pollutants close to a waste repository. By increasing the number of cores, the uncertainty in these values is reduced. The amount of uncertainty reduction can be quantified by applying the model of the investigated domain taking into account the information from the additional measurements. As a demonstration, the proposed model is applied to a simple problem of steady state two-dimensional flow in a vertical plane. It appears that the technique presented may serve as a powerful tool for quantifying uncertainties in flow problems and in providing guidance on how to acquire additional information of the fractured network in a given domain of fractured rock.