## 4 Discussion  
Considering that the \(y_{2\Delta}(x)\) functions do not consist of functions of multiple non-periodic inflexions but involve two distinct trends along the \(x\)-axis, the way used to assume that such a mathematical expression is able to reflect these two trends has been to show a series of examples of which this double behaviour is known in geosciences and in which the fit obtained is sufficiently precise. Next, some possible concerns about the development of new empirical relationships are discussed. This discussion does not include debates on each of the presented examples, as the differences with existing relationships were added in each of the corresponding publications. A first possible objection might be that only in a relatively small number of cases the empirical relationship between two or more parameters is purely based on the observation of a statistical correlation. However, the above representative cases of empirical relation  
Figure 8: Friction factor of fluid flow in smooth-walled capillaries since [PERSON] et al. [4] (red line), according to the Hagen–Poiseuille [10; 31] equation for laminar flow (dashed grey line), and according to [PERSON] [32] for turbulent flow (dotted grey line). Figure 7: Velocity profile of fluid flow in capillaries for different turbulence exponents \(\tau\) according to [PERSON] et al. [4] (red lines), [PERSON] [30] data (grey lines), and velocity profile for laminar flow since the Hagen–Poiseuille [10; 31] equation (blue dashed line). ships between two quantities have already shown, based on the observation of a statistical correlation and the criteria for its development detailed in Section 2, to improve certain existing relationships, and nowhere in this study is it claimed that this occurs in all cases. In some phenomena that occur in geological environments, the number of variables may be very high. Theoretically, quantifying the effect of each is then very complex, making it necessary first to consider the existence of relationships that comprise the entire set of data (see [3]). A possible second reproach might be that deviations between empirical data and existing relationships derived from well-known theoretical models are attributable to the lack of knowledge of some involved parameters. However, it would be lax to use this argument to dismiss the new empirical relationships rather than the existing theoretical equations unless it were true that this is the case in the new relationships rather than in the existing ones. By way of example, it might be thought that there is a one-to-one relationship between permeability and porosity in fractured rocks such that the deviations between empirical data and existing relationships would be due to a lack of knowledge of the statistical distribution of fractures and their extent and connectivity. However, it is known that there is no unique dependence between permeability and porosity, so it is that assumption that fails in this case. Conversely, if in the development of a new empirical relationship, the formation factor of a fractured rock is considered to comprise these overlooked variables, this should lead to the acceptance of this new law, even if it is empirical in origin. Indeed, it may be that such dependence has not been analysed to date for fractured rocks, as is the case of the porosity/permeability relationship in granular porous media shown in this study. The third possible disapproval might be that empirical laws allow comparing substantially different phenomena from a physical point of view, such as the correlation between strength features and stress-strain behaviour. In the case of the relationship between strength and Young's modulus in soils, these may appear to be two substantially different phenomena to those that do not consider the degradation of this characteristic with respect to the load to which a soil is subjected.