Numerical and analytical modelling of bending-induced tensile fractures

Although the origin of outer-arc extension fractures in folded sequences is well-understood and documented in many natural examples, geometric and geomechanical factors controlling their spacing are hitherto unexplored. This study investigates the formation of bending-induced tensile fractures during constant-curvature forced folding using two-dimensional Distinct Element Method (DEM) numerical modelling. The DEM model comprises a central brittle layer embedded within elastic layers; the layer interfaces are cohesionless. Folding of this three-layer system is enforced by a velocity boundary condition at the model base, while a constant overburden pressure is maintained at the model top.

The models illustrate several key stages of fracture array development: (i) Prior to the onset of fracture, the neutral surface is located midway between the layer boundaries, consistent with pure bending; (ii) Once the outer-fibre stress equals the tensile strength of the layer, fractures nucleate and propagate through the brittle layer; (iii) The rate of fracture formation as a function of curvature decreases nonlinearly, with new fractures developing approximately midway between two existing fractures; (iv) Eventually no new fractures form, irrespective of any further increase in fold curvature, a state referred to as fracture saturation.

On the basis of these numerical model results, an approximate analytical solution for fracture spacing based on classic beam theory is developed. The predicted range of fracture spacing as a function of fold curvature is in good agreement with the numerical model results. Importantly, the analytical solution reveals which geometric and geomechanical factors control fracture spacing, namely layer thickness, radius of curvature, Young’s modulus, tensile strength and confining pressure. The fracture spacing to layer thickness ratio at saturation however depends only (nonlinearly) on the ratio of tensile strength to overburden pressure.

The numerical model results are qualitatively compared with field observations at outcrops located in the Montpellier Fold region. The folded lithologies are of Jurassic age and comprise (brittle) limestones, with (ductile) marl intercalations. Fracture-bound limestone blocks located within the fold hinges and observed on the fold-profile plane are laterally bound by V-shaped veins that thin towards the fold core. In the inner-arc of each vein-bound limestone block, the marl interbed thickens towards the veins, whereas in the outer-arc it thins towards the veins. Clearly, the thickness distribution of the marl interbeds reflects non-uniform loading, which is consistent with the loading conditions hypothesised on the basis of the theoretical models.