Geometric and kinematic modeling of detachment folds with growth strata based on Bézier curves

In our geometric modeling, the inner layer of competent rock units in a detachment fold is approximated by several quadratic Bézier curves, which are joined together according to some geometrical rules, whereas the outer layers are calculated from the inner layer by parallel folding mechanism. In this model, an ideal fold is determined by three parameters: w, half of the distance between the locations of the fold's two limbs where their dips are measured, θ1 and θ2, the dip angles of its two limbs. Two more parameters, respectively the axial lift-up ratio (u) and the limb elongation (E) may be used to change the ideal fold shape, if necessary. In kinematic modeling, downward deflection in the two synclines flanking an anticline is assumed to maintain constant area during folding. Both mathematical derivation and numerical simulation show that the reduction of the balanced area (Ar) is directly proportional to the downward deflection angle (φ), and there is an approximate linear relationship between the detachment depth and the downward deflection angle. Based on this relationship, an iterative method is used to find the approximate value of φ. Furthermore, syn-folding growth strata can be modeled through calculating the velocity field above a detachment fold during folding. The method is applied to three seismic interpretation cross-sections of detachment folds respectively in Tarim Basin (western China), Zagros fold-belt (Iran) and Niger delta (western Africa). Variations of φ with the development of the detachment folds indicate the transfer of rock materials between the synclinal and anticlinal areas of the folds.