Exit Gradient Factors for Barrages

The question asks which factor does not influence the exit gradient of a barrage. The exit gradient is a critical concept in the design of hydraulic structures like barrages, primarily related to the phenomenon of seepage, which is the flow of water through the soil beneath the structure.

Understanding Exit Gradient

The exit gradient is defined as the hydraulic gradient (the loss of head per unit length of the seepage path) at the point where the seepage water emerges from the soil on the downstream side of the hydraulic structure. It is a crucial parameter because a high exit gradient can cause 'piping' or 'boiling', leading to the erosion of soil particles and potential failure of the structure. Theories like Bligh's creep theory and Khosla's theory help analyze seepage and exit gradients.

Factors Influencing Exit Gradient

Several factors affect the seepage pattern and the exit gradient. Let's analyze the given options:

• The Applied Head of Water ($H$): The total difference in water level between the upstream and downstream sides of the barrage drives the seepage flow. A greater applied head ($H$) generally leads to a steeper hydraulic gradient, including the exit gradient. The exit gradient is directly proportional to the applied head.
• The Horizontal Length of Floor ($L$): The floor (or impervious protection) extends horizontally downstream from the barrage to dissipate the seepage energy. A longer floor ($L$) increases the length of the seepage path, thereby reducing the overall hydraulic gradient, including the exit gradient. The exit gradient is inversely related to the seepage path length.
• The Depth of Downstream Cut-off ($d_d$): The downstream cut-off (a trench filled with impervious material) is installed below the floor on the downstream side to shorten the seepage path near the exit point and reduce the exit gradient. According to Khosla's theory, the exit gradient is highly sensitive to the depth of the downstream cut-off ($d_d$). A deeper downstream cut-off significantly reduces the exit gradient. The exit gradient is generally inversely proportional to $d_d$.
• The Depth of Upstream Cut-off ($d_u$): The upstream cut-off is installed on the upstream side to reduce the head acting on the floor and decrease uplift pressures. While the upstream cut-off affects the overall seepage flow quantity and the distribution of head loss along the seepage path, its direct impact on the *specific value* of the exit gradient (i.e., the gradient right at the exit point) is less significant compared to the downstream cut-off and the floor length. Khosla's theory uses correction factors involving the ratio of lengths to cutoffs ($L/d_u$, $L/d_d$), but the gradient calculation at the exit point is primarily governed by the immediate downstream conditions.

Analysis of Independence

The exit gradient is essentially the rate of change of hydraulic head with respect to distance ($dh/ds$) at the downstream exit point. Based on seepage theories:

• The exit gradient ($G_e$) is directly influenced by the applied head ($H$).
• It is inversely influenced by the effective seepage path length near the exit, which includes the floor length ($L$) and the depth of the downstream cut-off ($d_d$).
• The depth of the upstream cut-off ($d_u$) affects the overall seepage head distribution but does not directly alter the rate of head loss over the last unit length of the path in the same way the downstream conditions do. Khosla's theory accounts for the effect of upstream cutoff via potential theory and correction factors, but the fundamental gradient at the exit point itself is least sensitive to $d_u$ compared to $H$, $L$, and $d_d$.

Therefore, the exit gradient can be considered most independent of the depth of the upstream cut-off when compared to the other listed factors.

Conclusion

The factor that the exit gradient is independent of (or least dependent upon) among the choices is the depth of the upstream cut-off.