General modeling of karst spring hydrographs and development of a dimensionless karstic hydrograph concept

The response of a karstic-aquifer solution-cavity network subjected to a rainfall event leads to a hydrograph with rapidly increasing rising limb, but comparatively slower recession limb. In published studies, initial (early time) recession limb discharges have been modeled by consideration of the square root (nonlinear) response of the hydraulic head to karstic-sinkhole spring discharge. Late-time recession limb discharges are usually modelled by various mathematical methods such as exponential, quadratic and power functions, and straight line on semi-logarithmic plots. Hence, the recession-limb has been represented by different models, but without any model for the middle-time portion. In this paper, first, for early recession-limb times, a power model is developed by means of a convergence series. The result is compared with the literature models and it is observed that the presented model is significantly better than the previous ones. The literature models used the power 0.5, but this study uses 0.6 (a mathematical explanation for this adaptation is given elsewhere). The final portion of the recession limb is modelled by a straight line. Hence, the two-piece model used for the recession limb is examined. Then, a completely new approach is developed, where the whole karstic-sinkhole spring-discharge hydrograph is modelled mathematically by a single model through the logarithmic normal function. Finally, a dimensionless karstic-sinkhole spring-discharge hydrograph is suggested and its graphical and numerical forms are given for practical use by other researchers.