Presenting and evaluating a new empirical relationship for estimating the rate of infiltration in trenches

Empirical formulas are indispensable tools in water engineering and hydraulic structure design. Derived from meticulous field observations, experiments, and diverse datasets, these formulas help to estimate water leakage in structures such as dams, tunnels, canals, and pipelines. By utilizing a few easily measurable parameters, engineers can employ these formulas to generate preliminary leakage rate estimates before proceeding with more detailed analyses. In this study, a physical model was developed, and a series of experiments were conducted, considering variables such as inflow rate, materials constituting the unsaturated medium, and variations in infiltration trench depth and width. As a result, a novel artificial recharge method was introduced, and an empirical equation, <inline-formula id="M1"> <math id="mathml_M1" display="inline" overflow="scroll"><msub><mrow class="MJX-TeXAtom-ORD"><mtext>Q</mtext></mrow><mrow class="MJX-TeXAtom-ORD"><mtext>out</mtext></mrow></msub></math></inline-formula>= 0.0066 ×<inline-formula id="M2"> <math id="mathml_M2" display="inline" overflow="scroll"><msup><mrow class="MJX-TeXAtom-ORD"><msub><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">D</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mn>50</mn></mrow></msub></mrow><mrow class="MJX-TeXAtom-ORD"><mn>0.64</mn></mrow></msup></math></inline-formula>× L ×<inline-formula id="M3"> <math id="mathml_M3" display="inline" overflow="scroll"><msup><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">P</mi></mrow></mrow><mrow class="MJX-TeXAtom-ORD"><mspace width="thickmathspace"></mspace><mn>0.36</mn></mrow></msup></math></inline-formula>, was proposed to estimate the infiltration capacity of the trench. This equation incorporates factors such as the wetted perimeter, mean soil particle diameter, trench length, and a coefficient. A comparative analysis between the observed data from nine Iranian earthen canals and the values calculated using the proposed equation revealed an average relative error of 15% between the two datasets. In addition, the Pearson correlation coefficient was determined to be 0.981 and the Root Mean Square Error (RMSE) was 0.381, demonstrating the strong predictive performance of the equation. The parameters considered in the proposed equation allow for its application across diverse regions. Given its accurate performance, this equation provides a reliable initial estimate of the leakage rate, thereby helping to reduce costs and save time.