Linear Reservoir
Linear Reservoir is a method and just assuming that the watershed behaves like a linear reservoir, where the outflow is proportional to the water storage within the reservoir.
\[Q_{out} = \frac{1}{K}S(t)\]
In addition to their relationship with output and storage, linear reservoir models also adhere to the continuity equation, often referred to as the water balance equation.
\[\frac{\mathrm{d}S(t)}{\mathrm{d}t} = Q_{in} - Q_{out}\]
By combining both equations, we obtain a differential equation (DGL).
\[Q_{in} = Q_{out} + K\frac{\mathrm{d}Q_{out}(t)}{\mathrm{d}t}\]
\[Q_{out}(t)=\int_{\tau=t0}^{t}Q_{in}(\tau)\frac{1}{K}e^{-\frac{t-\tau}{K}}\mathrm{d}\tau + Q_{out}(t_0)\frac{1}{K}e^{-\frac{t-t0}{K}}\]
Where:
\(Q_{in}\) is the inflow of the reservoir
\(Q_{out}\) is the outflow of the reservoir
\(S\) is the storage of the reservoir
\(K\) is the parameter that defines the relationship between $Q_out$ and $S$