*4.4. Approximants*

The trial of approximants was mostly successful with good matching with an exponential function (Figure 7). The worst fit was for Experiment 4 (1D) (Figure 7a). For this experiment, the ponded head reached the upper boundary condition by about year 17. This appears to change the temporal pattern of recharge, which is not captured by the approximant.

**Figure 7.** Approximants against numerical solutions for which perching occurs. Solid lines represent numerical solutions and dashed lines approximants. (**a**) 1D modelling experiments 3, 4 and 6. The approximants are respectively: 1 − exp(− 0.07 × (*t* + 10)), *t* < 12; 4: min(0.486,1 − exp(−0.18 × (*t* − 14))), *t* > 15; and 6: 1 − exp( − 0.22 × (*t* + 4.4)), *t* > 5. (**b**) 2D Modelling Experiments 2, 3 and 3a. The approximants are respectively: 2: 1 − exp(− 0.12 × (*t* + 2.5)), *t* > 6; 3: 1 − exp( − 0.13 × (*t* − 0.8)), *t* > 4; and 3a:1 − exp(−0.15 × (*t* + 1)), *t* > 5.

The coefficient in the exponential is similar for all the 2D modelling experiments. The theory is predicting change in the exponential for experiment 3a. On the other hand, the 1D modelling is showing considerable variation, which was not expected. However, Figure 4a shows that the analytical expressions, which have similar coefficients for all the experiments, adequately fit the numerical simulation. The 1D modelling is trying to fit over small variations of the transfer function and then becomes sensitive to issues such as numerical dispersion.

Perhaps the largest difficulties are 1) the estimation of the time delay until recharge occurs and 2) the reference time in the exponential. However, the analytical model appears to be adequately estimating time delays and becomes an issue of finding the simplest form of estimation.

Overall, the collection of results shows promise for using approximants, in addition to using a PerTy3 or numerical models.
