## 2 Materials and Methods
### Rationale  
An open natural system transforms the input water energy into outgoing energy through water discharge and energy loss to overcome the resistance to flow based on the principle of minimum energy expenditure in any link of the water system (surface and subsurface) and in the system as a whole [21]. Climate defines the input, whereas system structure defines the water retention capacity, potential energy transformation, and dissipation of an unregulated water system in different time periods (recharge and after recharge). It is assumed that future climate change will not considerably change the energy loss processes in the system to overcome the resistance to flow in a spatially self-organized river system because a stable structure corresponds to the minimum total energy dissipation in the system. In other words, future minor local changes in the channel properties will not considerably change the behavior of the entire river system. Thus, the system response to future energy input variations driven by climate change will be similar but will result in different discharge values (outgoing energy). The extent of these possible changes is difficult to estimate; therefore, the problem must be simplified. It is important to understand physical reality to assess the vulnerability of a system. In the river system, the streamflow at a certain locality is composed of a groundwater contribution equal to \(cS_{t-1}\) and surface runoff equal to \(dx_{t}\):  
\[Q_{t}=cS_{t-1}+dx_{t} \tag{5}\]  
The continuity equation for the groundwater storage is written as:  
\[S_{t}=(1-c)S_{t-1}+ax_{t} \tag{6}\]  
Combining Equations (5) and (6), streamflow \(Q_{t}\) can be written as:  
\[Q_{t}=(1-c)Q_{t-1}+dx_{t}-\left(d(1-c)-ac\right)x_{t-1} \tag{7}\]  
which is modeled when precipitation is an independent series, where \(x_{t}\) represents the precipitation in period \(t\), \(ax_{t}\) infiltrates, percolates, reaches the groundwater storage, and \(bx_{t}\) evaporates [22]. Hence, the equation:  
\[(1-a-b)x_{t}=dx_{t} \tag{8}\]  
Surface runoff is a fast flow driven by differences in gravitational potential energy caused by elevation differences. Groundwater contributes to the flow, and \(cS_{t-1}\) represents the base flow driven by differences in gravitational potential energy caused by groundwater table elevation, where \(c\) represents the aquifer coefficient and \(S_{t-1}\) is the groundwater storage at the beginning of period \(t\). The following conditions are necessary to validate the above:  
\[0\leq a,\,b,\,c,\,d\leq 1\text{ and }0\leq a+b\leq 1 \tag{9}\]  
Equation (5) holds for a karst hydrological system [23]. The presented conceptual hydrological model (precipitation-runoff) was used to analyze hydrological processes in water systems for the qualitative analysis and justification of the proposed vulnerability model. In a natural environment, all streams attempt to flow toward a state of dynamic equilibrium. More major changes in the stream network are the result of complex geomorphological processes over a long period (longer than 100 years). Therefore, the physical reality of the system will not change considerably in the near future. It can be concluded that the internal properties of the system and the sensitivity of the streamflow do not change substantially because the difference in gravitational potential energy in the watershed area and streams changes gradually with natural processes. However, occasional climatic extremes and the flows generated by internal properties, which occur more often, locally change the state and adjust the channel property toward the optimal state and can have a major impact on the local natural and socioeconomic system. Therefore, in the wet period (higher potential energy period), streamflow sensitivity is generally higher than that in the dry period because smaller changes in maximum inputs generate larger output oscillations than the same relative ratios of changes in the dry period.