Distribution of joints: probabilistic modelling and case study near Cardiff (Wales, U.K.)

Abstract

In this paper a new probabilistic two-dimensional model for simulating joint development in a single vertical set cutting horizontally layered rocks is presented. The problem of considering the probability of joints nucleating within limestone beds and the probability of joints propagating across mudstone interbeds is solved using matrix analysis. Sources of dispersion, relationships between joint spacing and bed thickness, and the influence of pre-existing joints on vertical propagation were taken into account. Simple assumptions were made concerning the sense of vertical joint propagation, poorly constrained by field data in most cases. We demonstrate that, with the input of just a few geometrical and statistical parameters, realistic cross-sections can be constructed based on the probabilistic modelling. Our field analysis aimed to: (1) provide real values for the statistical parameters; and (2) compare actual and simulated patterns in order to check our results. Input parameters include bed thicknesses, average numbers of joints and the proportion of joints cutting two adjacent limestone beds. Field data were collected from coastal exposures in Liassic rocks at Liantwit Major (Wales U.K.). At the sample sites, tabular layers comprise alternating decimetric limestones and centimetric mudstone interbeds. They are cut by two orthogonal sets of vertical joints. We studied the distribution of the dominant joint set striking N170 °. Both the models and the outcrops display similar characteristics e.g. a log-normal distribution of joint spacing and a negative exponential distribution of the number of joints vs the number of beds crossed. Our probabilistic modelling is applicable to other situations with a limited number of numerical constraints obtained from field or sub-surface observations.