## 4 Results and Discussion
### Stage Semiempirical Model  
To bring the water reservoir back to its original capacity, debris deposited on the bottom are dredged an pumped into a geotextile tube. The solids are retained inside the tube while the water flows out, therefore a certain number of passages are necessary to complete the drainage operation. Thus, overall process consists of a series of stages, each of which includes a filling step followed by the corresponding dewatering step. The total number of stages depends on the maximum capacity of the geotextile tube and on the cost of the entire purging operation. One of the essential requirements for a successful theory in technical models is judicious simplifications. The model presented here, starts from observation that that in the evacuation operations the dewatering time, \(t_{D}\) is always very long. Therefore, is reasonable to assume that at the end of each dewatering step solids in the geotube have attained the configuration they would have had in free settling process. This hypothesis allows to refine the model with experimental sedimentation results of material dredged by the reservoir. Let us consider the j-th filling step, hence the solid material balance takes the form  
\[Q_{F}^{(j)}t_{F}^{(j)}C_{s}^{F(j)}=V^{(j)}\!\left(t_{F}^{(j)}\right)C_{s}^{(j)} \!\left(t_{F}^{(j)}\right) \tag{12}\]  
where \(Q_{F}^{(j)}\) is the volume flow rate of clay material pumped to geotube, \(t_{F}^{(j)}\) is the time taken for the j-th filling, \(C_{s}^{F(j)}\) is the concentration of solids in the dredged material, \(V^{(j)}\!\left(t_{F}^{(j)}\right)\) is the geotube volume after j-filling steps, at the time, \(t_{F}^{(j)}\), \(C_{s}^{(j)}\!\left(t_{F}^{(j)}\right)\) is the concentration of solids in the geotube at the time \(\left(t_{F}^{(j)}\right)\). In each dewatering step the mass of the solids remains constant so that in j-th stage it must occur  
\[\frac{d}{dt}\!\left\{V^{(j)}(t)C_{s}^{(j)}(t)\right\}=0\;\;\;\;t_{F}^{(j)}\leq t \leq t_{D}^{(j)} \tag{13}\]  
Figure 6: Sedimentation rates. (**a**) Normalized sedimentation profiles of clay material from Occhito reservoir. The numbers indicate concentrations (kg m\({}^{-3}\)) of the solids. (**b**) Sedimentation rates computed for dispersion clays taken from Occhito’s reservoir and not mixed with flocculant solution. The numbers represent the concentration (kg m\({}^{-3}\)) of the solids. being \(t_{D}^{(j)}\) the time to complete j-th dewatering step. Obviously, the volume of the geotube, \(V^{(j)}(t)\), at j-th stage depends on the amount of water coming out. Let \(Q_{w}^{(j)}\) be volume flow rate of water, then we can write  
\[V^{(j)}(t)=V\dot{\left(t_{F}^{(j)}\right)}-\int_{t_{F}^{(j)}}^{t}Q(x)\mathrm{d} x\quad t_{F}^{(j)}\leq t\leq t_{D}^{(j)} \tag{14}\]  
The function \(Q_{w}^{(j)}\)(t) contains into itself a term about which little is known except that it depends strongly on the fabric of which is made. Herein, we model volume water-flow by means of a two-parameter function  
\[Q_{w}^{(j)}(t)=Q_{w}^{0(j)}\mathrm{erfc}\left(\frac{t-t_{F}^{(j)}}{\tau_{j}}\right) \tag{15}\]  
where \(\mathrm{erfc}(x)\) is the complementary error function, \(Q_{w}^{0(j)}\) is the maximum water-flow rate and \(\tau_{j}\) is the lifetime of j-th dewatering step. Using this equation, Equation (14) can be easily integrated to yield  
\[V^{(j)}(t)=V^{(j)}\left(t_{F}^{(j)}\right)-Q_{w}^{0(j)}\left[\left(t-t_{F}^{( j)}\right)\mathrm{erfc}\left(\frac{t-t_{F}^{(j)}}{\tau_{j}}\right)+\frac{1-e^{ \frac{\left(t-t_{F}^{(j)}\right)^{2}}{\tau_{j}^{2}}}}{\sqrt{\pi}}\tau_{j}\right] \tag{16}\]  
Mass of solids remains constant for each dewatering step, therefore Equation (13) provides  
\[C_{s}^{(j)}\left(t_{D}^{(j)}\right)=\frac{V^{(j)}\left(t_{F}^{(j)}\right)C_{s} ^{(j)}\left(t_{F}^{(j)}\right)}{V^{(j)}\left(t_{D}^{(j)}\right)} \tag{17}\]  
Equations (16) and (17) establish the concentration of solids and geotube volume after \(n\) stage, hence an economical evaluation of dredging project implemented can be carried out.