Optimal Design of Combined Weir–Orifice Tail Escape Structures Using Graphical Methods and the TAILOPT Tool

Abstract

Dual-inlet tail escapes, combining an orifice and a weir, are key hydraulic structures that evacuate excess water from canal termini during maintenance and protect berms by discharging surplus irrigation flows. Conventional sizing methods typically depend on trial and error, which is time-consuming and may yield suboptimal design. This study introduces a graphical design approach and a MATLAB-based tool, TAILOPT, developed to streamline tail escape design. The tool incorporates both the Fanning and Darcy–Weisbach friction formulations for head loss estimation and can automatically generate an “.inp” file for EPA-SWMM, enabling direct unsteady-state hydraulic assessment. This integration reduces design effort and supports evaluation of alternative hydraulic and drainage scenarios within a single workflow. Two applications illustrate the framework. The first shows that overly steep drainage slopes (Sp > 2%) are impractical, while vertical drops may require larger pipe diameters. The second application applies TAILOPT to a distributary canal, determining the optimal pipe size and verifying its performance in EPA-SWMM under emergency surplus flow and routine dewatering conditions. The results demonstrate that the method yields economical, robust, and practitioner-friendly designs; however, modeling simplifications, such as assuming continuously submerged orifice flow, can introduce minor deviations in the predicted channel emptying times.

1. Introduction

1.1. Background

Water scarcity has emerged as a paramount challenge for sustainable development worldwide, with agriculture being the single largest consumer of freshwater. Nearly 70% of global freshwater withdrawals are used for irrigation, yet up to 60% of this water is wasted due to leaky distribution systems, inefficient application methods, and other losses [1]. This wasteful use of water exacerbates stress on water resources in many regions. Indeed, roughly half of the world’s population experiences severe water scarcity during at least part of each year [2]. In light of these trends, improving water-use efficiency in irrigation and eliminating losses have become urgent priorities to enhance sustainability and resilience against water shortages.

A major inefficiency in canal-based irrigation systems is the loss of water during periods of low demand, particularly at night. In many countries, canals operate on continuous-flow, rotational schedules, such as the “warabandi” system in South Asia and rotational deliveries in Egypt, where water is delivered day and night regardless of actual field demand. When farmers do not irrigate during their allotted night-time turns, the canal often conveys more water than is being used, and the surplus typically spills from the system through drainage outlets or escapes at the canal tail, effectively going to waste. Without intervention, these nightly surplus outflows can be considerable and represent missed opportunities for productive water use. Several measures have been proposed to mitigate such losses. For instance, on-farm storage reservoirs can capture excess night-time flow for reuse in the morning [3], while surplus canal water can also be directed to groundwater recharge basins to augment aquifers. Another promising strategy focuses on the canal escapes themselves: by adjusting the design or operation of tail-end escapes, managers can minimize unnecessary spillage. Reyad et al. [4] demonstrated in Egyptian distributary canals that dynamically raising the tail escape weir during low-demand hours, combined with optimized offtake scheduling, can significantly reduce night-time water wastage. By integrating approaches such as intermediate storage, groundwater recharge, and improved escape operation, irrigation systems across different regions can substantially reduce non-productive drainage losses and conserve more water for crop use.

Tail escapes (sometimes called end escapes or surplus weirs) are special outlet structures constructed at the downstream end of irrigation canals to dispose of excess water into a drain or natural watercourse [5,6]. They function as safety valves for the canal network, preventing overtopping of canal banks and damage to infrastructure when the inflow exceeds what is drawn by farms [7,8]. If upstream releases are not reduced quickly enough after demand drops, or if unexpected inflows occur (such as storm runoff entering the canal), the tail escape automatically diverts the surplus flow out of the canal to protect it from overflowing. Given this critical role, design standards mandate providing escapes at canal termini (and sometimes at intermediate points along long canals), with capacities large enough to handle sudden surpluses, on the order of about half the canal’s design discharge for major canals and around 15% for minor canals [8,9].

Beyond their safety function, tail escapes have also been leveraged to deliver environmental flows in emergency situations. For example, during a large hypoxic “blackwater” event in Australia’s Murray–Darling Basin, water with higher dissolved oxygen was released via irrigation canal escapes into downstream rivers [10]. These releases raised the mean downstream dissolved oxygen by 1–2 mg/L for over 40 km and prevented fish deaths in the reaches receiving them, demonstrating an ecological benefit from infrastructure originally designed for surplus disposal.

Tail escapes are generally of two types: weir-type escapes and regulator (gated) escapes [8]. A weir-type escape has a fixed crest set at the canal’s full supply level. When the water level in the canal exceeds this crest, the excess automatically spills over into the escape channel, thereby keeping the upstream water level from rising further. This type of escape works passively and continuously, requiring no operator intervention. In contrast, a regulator-type escape features a gated orifice near the canal bed that can be opened or closed as needed. This gated orifice allows operators to deliberately release water, for example, to empty a canal reach for maintenance or to flush out sediment, which is not possible with a fixed crest. Thus, the weir-type escape provides automatic level control under normal conditions, while the gated orifice type provides operational control for extraordinary situations.

Modern canal systems often incorporate a combined weir–orifice tail escape to leverage the advantages of both types in one structure. A typical design is the well-type escape, essentially a vertical shaft or chamber that has an overflow weir around its rim and a sluice-gated orifice at its bottom. Under routine conditions, any surplus flow in the canal simply overtops the rim weir and is conveyed safely through the well and its outlet pipe, maintaining the canal water level at the full supply level. When a rapid drawdown is needed, for instance, to drain the canal for repairs or remove sediment, the bottom gate is opened, allowing water to be released quickly from the canal bed through the orifice. This dual-function design offers significant operational flexibility and efficiency. The overflow weir ensures passive, automatic regulation of the canal level without attention, whereas the gated orifice allows on-demand control to achieve full or partial canal emptying. The high-velocity jet produced by opening the bottom orifice also helps scour and evacuate sediments from the canal. Moreover, by optimizing the weir crest height and the gate operation schedule (for example, keeping the gate closed during the night to force more water over the crest and into storage), canal operators can minimize unutilized spillage and improve overall water conservation.

Figure 1 illustrates a longitudinal cross-section of a combined weir–orifice tail escape, showing its overflow crest, the lower orifice leading to a drainage pipe, and the energy dissipation (head loss, H_L) in the outlet conduit.

Relatively few studies have addressed the hydraulic behavior of tail escapes that incorporate both weir and orifice elements.

One notable exception is the work of Vatankhah and Khalili [11], who developed stage–discharge relationships for a combined sharp-edged weir–gate plate installed at the end of a circular channel. Using the energy principle alongside dimensional analysis and calibrating with 626 laboratory runs, they achieved a general stage–discharge equation with an average error of only about 1.9%. Their results also showed that purely empirical (dimensionless) modeling was not sufficient by itself—theoretical considerations were needed to obtain an accurate stage–discharge model [11]. In general, however, most prior research has examined these components separately. For instance, using unsteady-flow modeling, one study [4] found that night-time water loss over a tail escape weir can be greatly reduced by increasing the weir crest height and by allowing longer periods of water abstraction at farm turnouts. Longer, milder-sloping canal reaches were also shown to provide more temporary storage, further decreasing spillage. In another investigation, the effect of the escape weir’s geometry was explored: circular weir crests were observed to pass about 7.5–15% more flow per unit length than square crests under the same head conditions [12]. A related experiment [13] showed that the discharge coefficient of a circular weir inlet increases as the weir crest height is raised and the diameter of the well (shaft) is reduced, indicating that taller, narrower well configurations improve efficiency.

Downstream of the escape, the erosive impact of the discharged water has been studied [14]. As the head of water above the weir crest increases, the resulting scour hole in the channel bed becomes deeper and longer, whereas enlarging the escape well’s diameter has only a minor effect on the scour extent. Researchers have derived empirical formulas from these results to predict the scour depth and length for given flow conditions, aiding in the design of scour protection measures. Flow pattern analyses have also been carried out to understand the hydraulics within the escape structure. For example, computational and experimental work [15] using Fluent software showed that the tail escape weir’s shape and orientation significantly influence flow streamlines and vortex formation. Shorter crest lengths were associated with higher velocities above the weir and more intense downstream vortices in the simulations. Additionally, a broad laboratory evaluation of canal terminal outfalls [16] compared different tail escape configurations and assessed the effect of various design parameters on drainage performance, resulting in practical recommendations to ensure high discharge efficiency during canal shutdown and emergency release scenarios.

Another domain where combined weir–orifice hydraulics have been applied is eco-hydraulic design for fish passage. Liu et al. [17] introduced a differential weir–orifice (DWO) structure in a fishway transition channel to gradually dissipate energy along a non-prismatic reach. In their design, each sequential weir is slightly lower than the previous one and contains bottom orifices, causing the proportion of flow passing through the orifices to increase from about 13% up to 40% along the channel. As a result, the drop in water surface elevation at each weir is reduced by roughly 35–50%, which in turn lowers the velocity of flow over the weir (to about 1.2–2.1 m/s) while keeping the orifice jets at a moderate 0.8–1.3 m/s. The velocity profile in the DWO thus exhibits a favorable two-layer (bimodal) distribution, with different flow layers that improve the overall suitability for fish migration. Moreover, the DWO configuration was found to substantially decrease turbulence in the flow. Measurements showed that the vertical turbulent intensity at the weir crest was about one-third lower than in a comparable conventional design, and the turbulent kinetic energy in the bottom orifice outflow was only ~28% of that at the weir crest. This indicates that some of the energy is redistributed from the surface overflow to the submerged orifice flow, yielding a smoother water surface profile through the transition. The net effect of the DWO is a more fish-friendly hydraulic environment—with gentler drops, reduced velocities, and lower turbulence—which facilitates upstream fish passage [17].

Lessons from analogous hydraulic structures have informed tail escape design as well. Studies on vertical drop shafts and spillway systems highlight relevant flow behaviors. For instance, when water falls through a vertical shaft and is diverted by a horizontal elbow, it can form two distinct flow zones accompanied by air entrainment and vortex motion [18]. Numerical simulations of such elbow flows (using RNG k–ε turbulence modeling) confirmed these phenomena and matched observed water surface profiles, emphasizing the importance of accounting for air and vortex dynamics. In another line of research, a novel circular “piano-key” inlet for shaft spillways was tested and found to increase discharge capacity by roughly 15% compared to a conventional vertical shaft inlet [19]. Similarly, classic experiments on shaft spillways [20] indicated that providing a curvature radius for bends that is more than about twice the pipe diameter yields smoother flow transitions and reduces energy losses.

Vortex drop shafts, which operate on principles comparable to well-type tail escapes, have been studied extensively. Investigations have detailed how swirling flow is generated and controlled in these shafts, and design refinements have been developed to maintain stable operation [21]. More recently, a new vortex drop shaft design without any ventilation shafts was proposed [22]. Experiments and accompanying CFD simulations (with an RNG k–ε model) demonstrated that this vent-less design could function effectively: the pressure and velocity distributions in the shaft were well predicted, suggesting that the flow itself entrains sufficient air to prevent cavitation without dedicated vents. Siphon spillways also offer insights pertinent to tail escapes, since they likewise must convey excess water safely but in a self-priming manner. An experimental comparison by Larbi et al. [23] between siphon spillways and ordinary overflow weirs established the relationships between their discharge characteristics and identified the range of operating conditions where siphons are advantageous. Building on that, physical and numerical studies of high-head siphon spillways [24] (including tests of different outlet bucket angles) found that a siphon equipped with a 45° outlet bucket achieves superior flow stability and capacity. In addition, computational fluid dynamics has been used to model submerged siphon flows [25], yielding results that agree closely with lab measurements of flow rates and pressure profiles. One critical factor in such systems is air entrainment: a three-dimensional simulation study [26] confirmed that providing ventilation (approximately 45% of the shaft diameter in that case) dramatically reduces the risk of cavitation damage by ensuring adequate aeration of the flow.

Finally, recent research has addressed the transition between weir flow and orifice flow in partially submerged conditions. In a thin-walled orifice study [27], the investigators showed that as long as the downstream water level is below the orifice’s midpoint, the orifice behaves like a broad-crested weir, with an effective crest length equal to the width of the orifice opening. Once the tailwater submerges the orifice beyond its mid-height, the flow regime shifts to that of a submerged orifice. The proposed modeling approach, treating the orifice as an equivalent weir until it is about half-submerged, was validated against experimental data, with predictions matching observations within a few percent. This provides a useful tool for analyzing the orifice component of combined tail escape structures under varying flow conditions.

1.2. Study Objectives

This study seeks to address the current knowledge gap by systematically investigating the hydraulic behavior and design characteristics of dual weir–orifice tail escapes for irrigation canals, with a particular focus on enhancing their design and optimization. The specific objectives are as follows:

• Develop a graphical design methodology for determining the optimal dimensions of the drainage pipe in a combined weir–orifice tail escape. This method provides engineers with a clear, visual tool for selecting pipe sizes that achieve the desired discharge performance under specified operational constraints, offering a robust alternative to conventional trial-and-error approaches.
• Create an optimization software tool (TAILOPT) that automates the design process for combined tail escapes. The tool systematically generates feasible combinations of weir, orifice, and drainage pipe parameters and identifies the optimal configuration based on user-defined constraints, such as allowable velocity ranges and head loss limits. By replacing iterative manual calculations, TAILOPT enables rapid and comprehensive exploration of design alternatives.
• Conduct a comprehensive sensitivity analysis using TAILOPT to quantify the influence of key design parameters, including channel reach length, drainage pipe slope, and pipe roughness, on the feasibility, efficiency, and robustness of the tail escape design. This analysis will identify the most critical parameters affecting performance and resilience.
• Integrate the optimized tail escape design with unsteady hydraulic simulation by enabling direct export of a compatible input file (.inp) for the EPA-SWMM modeling environment [28]. The selected design will then be evaluated under transient flow scenarios, such as nightly demand reduction and full canal drainage for maintenance, to validate its hydraulic adequacy and operational reliability within a dynamic canal network.

1.3. Paper Organization

The remainder of this paper is structured as follows: Section 2 outlines the theoretical background and methodology, describing the hydraulics of the individual components of tail escapes, the formulation of the graphical design approach, and the development of the TAILOPT tool framework. Section 3 presents a step-by-step design example demonstrating the application of the graphical method and the TAILOPT tool for a representative canal. The latter part of Section 3 details a real-world case study in which the proposed design methodology is applied to an operational irrigation canal; the optimized design is subsequently evaluated using EPA-SWMM unsteady-flow simulations, and the results are critically analyzed. Section 4 discusses the key findings from the sensitivity analysis and the case study, highlighting the advantages and potential limitations of the proposed approach. Finally, Section 5 summarizes the main conclusions, acknowledges the study’s limitations, and proposes directions for future research aimed at further enhancing the optimal design of canal tail escape structures.

2. Materials and Methods

This study adopts a dual-method framework to determine optimal tail escape structure dimensions. The first method employs a graphical approach for the preliminary, visual exploration of feasible designs. The second method utilizes a computational optimization tool (TAILOPT), developed in MATLAB 24b, to automate the design process. The methodology commences with a review of the governing hydraulics for key combined weir–orifice tail escape components, proceeds with a description of the graphical method, and concludes with the development of the TAILOPT tool.

2.1. Governing Equations of Tail Escape

The combined weir–orifice type is a drainage structure that includes both a weir to drain the surplus flow and an orifice with a sluice gate to drain the channel for maintenance and/or for sediment or silt removal. The combined tail escape (for simplicity, it will be called from now on just “tail escape”) works to drain excess water by flowing water over the crest of the weir and also through an opening at the bottom of the well equipped with an iron gate that has a width greater than the diameter of the orifice.

The drainage pipe is installed with a near-horizontal alignment, typically with a slope not exceeding 3%. This gentle inclination helps limit internal velocities, thereby reducing abrasion and the progressive deterioration of pipe roughness while still promoting self-cleansing velocities to minimize sediment deposition. Steeper slopes can generate excessive velocities, requiring additional energy dissipation to protect the receiving drain. The pipe outlet should be positioned slightly above the maximum water level of the receiving drain to maintain free discharge and prevent backflow or submergence.

To improve operational reliability, the tail escape well should include a horizontal trash screen to intercept aquatic weeds and floating debris, thereby preventing clogging of the well or pipeline.

2.1.1. Weir Inlet

Flow over the weir inlet, with proper nappe aeration, is described by the broad-crested weir equation:

$$Q_w={\displaystyle \frac{2}{3}}{C}_{dw}{B}_w\sqrt{2g}{h}^{1.5}$$

(1)

where $Q_w$ is the discharge over the crest, $B_w$ is the crest length, $C_{dw}$ is the discharge coefficient (≤0.6), $g$ = 9.81 m/s² is gravitational acceleration, and h is the head above the crest (m), limited here to 0.15 m.

The required weir crest length, governing water entry into the tail escape well, is determined using Equation (1). As illustrated in the horizontal projection of a circular well-type tail escape (Figure 2), the weir crest length ($B_w$) is conventionally established as three-quarters of the well circumference. This relationship is expressed mathematically as $B_w$ = (3/4) × π × D_w, with D_w representing the average diameter of the circular well.

2.1.2. Orifice Inlet

The gated orifice (Figure 2) conveys flow under submerged conditions according to

$$Q_o=C_d\times {\displaystyle \frac{{{\pi d}_o}^2}{4}}\times \sqrt{2g(h_o-h_2)}$$

(2)

where C_d is the orifice discharge coefficient, d_o is the orifice diameter, and (h_o − h₂) is the differential head on the orifice, as shown in Figure 1.

If the orifice is partially submerged, its flow can be approximated as weir flow using the top-width weir approach [27]:

$$Q_{op}=C_{dp}\times L_w\times h_w^{3/2}$$

(3)

where $Q_{op}$ is the partially submerged orifice flow, C_dp is the discharge coefficient when the orifice becomes partially submerged, and $h_w$ is the upstream head above the invert of the orifice. $L_w$ is the weir width, which can be determined following the top-width weir approach, as shown in Equations (4) and (5).

$$\mathrm{For} h_w/{\mathrm{d}}_{\mathrm{o}} < 0.5, L_w=d_o\mathrm{s}\mathrm{i}\mathrm{n}\left(\theta /2\right)$$

(4)

where $\theta =2\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(1-{\displaystyle \frac{2h_w}{d_o}}\right)$

$$\mathrm{For} h_w/d_o \ge 0.5, L_w=d_o$$

(5)

According to [27], C_dp could be related to the orifice discharge coefficient, as per Equation (6).

$$C_{dp}=2.46\times C_d$$

(6)

2.1.3. Time of Emptying the Channel

The end reach of the canal may be completely drained by lifting the sluice gate that controls the orifice at the base of the escape well. The time required to achieve full emptying can be estimated based on the top-width weir approach, as per the following equations:

$$T=T_1+T_2+T_3$$

(7)

where $T_1$ is the time required to empty the channel from the maximum depth d to a water depth equal to d_o. In this zone, the orifice flow Equation (2) is used. $T_2$ is the time required to empty the channel from a maximum depth equal to d_o to a water depth equal to d_o/2. During this zone, the weir flow Equation (3) with L_w defined as per Equation (5) is applied. $T_3$ is the time required to empty the channel from a maximum depth equal to d_o/2 to a minimum water depth equal to y_min << d_o/2. In this zone, the weir flow Equation (3) with L_w defined as per Equation (4) is applied.

Using conservation of mass, −∂Volume/∂t = Q (where negative sign means volume decreases with drainage time). This leads to a separable differential equation for depth y(t) assuming a prismatic channel:

$${\displaystyle \frac{dy}{dt}}=-\hspace{0.17em}{\displaystyle \frac{Q\left(y\right)}{L\left(b+ty\right)}}$$

(8)

Equivalently, we can write the differential time for a small drop dy as

$$dt=-{\displaystyle \frac{L\left(b\hspace{0.17em}+ty\right)}{Q\left(y\right)}}\hspace{0.17em}dy$$

(9)

After simple integration of dy from the initial channel depth, d, down to the final minimal depth, y_min, we obtain the emptying time for each phase, as shown in Equations (10)–(12):

$$T_1={\displaystyle \frac{L}{C_d{a}_o\sqrt{2g}}}\left[2\left(b+t{d}_o\right)\left(\sqrt{d-d_o/2}-\sqrt{d_o/2}\right)+{\displaystyle \frac{4t}{3}}\left({\left(d-d_o/2\right)}^{3/2}-{\left(d_o/2\right)}^{3/2}\right)\right]$$

(10)

$$T_2={\displaystyle \frac{2L}{C_{dp}{d}_o^{3/2}}}\left[\left(\sqrt{2}-1\right)\left(b+\sqrt{2}t{d}_o\right)\right]$$

(11)

$$T_3\hspace{0.33em}=\hspace{0.33em}{\displaystyle \frac{L}{2\hspace{0.17em}{C}_{dp}}}\left\{\hspace{0.33em}{\displaystyle \frac{b\hspace{0.17em}\sqrt{\hspace{0.17em}{d}_o-y_{\mathrm{m}\mathrm{i}\mathrm{n}}\hspace{0.17em}}}{d_o\hspace{0.17em}{y}_{\mathrm{m}\mathrm{i}\mathrm{n}}}}\hspace{0.33em}-\hspace{0.33em}{\displaystyle \frac{b\hspace{0.17em}\sqrt{\frac{d_o}{2}}}{\hspace{0.17em}{d}_o\hspace{0.17em}\left(\frac{d_o}{2}\right)}}\hspace{0.33em}+\hspace{0.33em}{\displaystyle \frac{\hspace{0.17em}b+4t\hspace{0.17em}{d}_o\hspace{0.17em}}{d_o^{3/2}}}\left[\hspace{0.17em}\mathrm{l}\mathrm{n}\left({\displaystyle \frac{\sqrt{d_o}+\sqrt{\hspace{0.17em}{d}_o-y_{\mathrm{m}\mathrm{i}\mathrm{n}}\hspace{0.17em}}}{\sqrt{\hspace{0.17em}{y}_{\mathrm{m}\mathrm{i}\mathrm{n}}\hspace{0.17em}}}}\right)\hspace{0.33em}-\hspace{0.33em}\mathrm{l}\mathrm{n}\left(\sqrt{2}+1\right)\right]\right\}$$

(12)

Theoretically, y_min should be set = 0 for full channel drainage, but to prevent the computational singularity associated with division by zero in $T_3$, a lower bound $y_{\mathrm{m}\mathrm{i}\mathrm{n}}$ > 0 was imposed to avoid singularity; a small value of y_min of the order of 0.01 m is practically sufficient. In the case of neglecting the partial submergence phase of the orifice, the emptying time could be greatly simplified using the orifice flow assumption, as given in Equation (13).

$$T={\displaystyle \frac{2\sqrt{d}L}{3C_d{a}_o\sqrt{2g}}}\times (B+2b)$$

(13)

where $T$ is the time required for emptying in seconds, L is the length of the last reach of the canal in meters, B is the top width of the canal at the highest water level in meters, b is the width of the canal bottom in meters, d is the depth of water in the canal in meters, a_o is the area of the circular opening in square meters, C_d is the coefficient of discharge (0.6), and g is the acceleration due to gravity 9.81 m/s².

The emptying time can be further approximated using the following equation, derived from the ratio of the water volume stored in the downstream canal reach to the average discharge through the escape opening.

$$T={\displaystyle \frac{2L{b}_{e}d}{C_d{a}_o\sqrt{2gd}}}$$

(14)

where b_e is the average width of the canal; [(B + b)/2].

The required area of the orifice opening ($a_o$) and, by extension, the orifice diameter (d₀) may be determined by using either Equation (13) or its approximate counterpart, Equation (14), with the design based on the target duration for emptying the terminal canal reach. In practice, a drainage period of approximately 24 h is frequently employed for sizing tail escapes to support maintenance operations [9]. This duration is generally regarded as practical, enabling timely drawdown of the canal section while minimizing disruption to overall water delivery schedules. It should be emphasized, however, that principal design standards, such as the U.S. Bureau of Reclamation’s Design of Small Canal Structures and Indian Standard IS 6936 [5,7], do not stipulate a universal drainage timeframe. Rather, these guidelines advocate for determining the escape capacity based on local operational needs and engineering judgment, such that the canal reach can be drained “within a reasonable time” to facilitate inspection and rehabilitation activities.

2.1.4. Drainage Pipe Outlet

In emergency operation when the discharge gate is raised, the flow that enters the overflow well must be transferred through the well drainage pipe to the d/s disposal drain. This flow is equal to the sum of the flow over the weir $Q_w$ and the flow through the lower orifice $Q_o$, as shown in Figure 1 and Figure 2.

$$Q_p=Q_w+Q_o$$

(15)

where Q_p is the discharge flowing through the drainage pipe (m³/s).

To compute the drainage pipe diameter ($D_p$), a flow velocity ($V_p$) of between 1.0 and 3.0 m/s is suggested [6,7]. This range mitigates risks of sedimentation (at low velocities) and pipe erosion (at high velocities). $D_p$ is then derived from the following continuity equation:

$$D_p=\sqrt{{\displaystyle \frac{4Q_p}{\pi V_p}}}$$

(16)

The pipe diameter obtained from Equation (16) should be checked to ensure it satisfies proper operation constraints for both the drainage pipe and the tail escape well. The two operational constraints are as follows:

A.

The first operational constraint requires that the ratio of the height h₁ to the diameter of the pipe should not be less than 0.20, i.e., (h₁/D_p ≥ 0.20), where h₁ is the height of the water inside the well above the inner edge of the upper point of the pipe, as shown in Figure 1. The reason behind this constraint is to ensure the drainage pipe inlet is submerged under design conditions, which greatly improves hydraulic efficiency and capacity. Moreover, inlet submergence helps ensure the drainage pipe barrel runs full or nearly full, making its performance more predictable and allowing the use of standard energy equations assuming the pipe is full.

B.

The second operational constraint requires that the distance between the water level inside the well and the weir’s crest level should not be less than 50 cm. This constraint is primarily intended to ensure adequate aeration beneath the nappe, the sheet of water flowing over the weir. Without sufficient air space, a partial vacuum can develop beneath the nappe, causing it to adhere to the downstream face of the weir. This adherence alters the flow regime and increases the effective discharge coefficient. Studies have shown that inadequate ventilation can cause the discharge coefficient to increase by approximately 5–15% in moderate cases and up to 25–30% under fully clinging conditions [29,30].

Figure 1 indicates that the vertical distance (H_L) between the drainage pipe outlet centerline and the tail escape well’s water surface can be determined by using the following equation:

$$H_L=h_1+L_p\times S_o+{\displaystyle \frac{D_p}{2}}$$

(17)

where L_p is the length of the discharge pipe in meters and S_o is the longitudinal slope of the discharge pipe.

Equation (17) is employed to determine the upstream head above the crest of the inlet of the drainage pipe, h₁, when the total head loss (H_L) is specified. Although H_L, which includes both frictional and local losses, can, in theory, be estimated via Equation (18), if the drainage pipe discharge (Q_p) is known, the process is complicated by the hydraulic interdependence within the system. Specifically, Q_p represents the aggregate of both the orifice discharge (Q₀) and the weir discharge (Q_w). Importantly, the orifice discharge (Q₀) is governed by two parameters: the maximum upstream water surface elevation (h₀), which is known, and the water level within the tail escape well (h₂), measured relative to the orifice centerline and not known in advance. Notably, h₂ is itself a function of the discharge, resulting in a coupled system where Q₀ and h₂ are mutually dependent. This coupling precludes direct analytical determination and instead necessitates an iterative computational approach. Iterative refinement of Q₀ and h₂ is required to ensure hydraulic consistency between the predicted discharge and the resulting well water level, thereby achieving a convergent and physically realistic solution for the drainage pipe design.

$$H_L=(1.5+{\displaystyle \frac{4f_F{L}_p}{D_p}})\times {\displaystyle \frac{V_p^2}{2g}}$$

(18)

where $f_F$ is the “Fanning” friction factor [dimensionless], and it practically ranges from 0.005 to 0.01. The discharge flowing through the drainage pipe can be correlated with the head loss through it, H_L, the drainage pipe diameter, D_p, the length of the drainage pipe, L_p, and the “Fanning” friction factor, $f_F$, through the following equation:

$$Q_p={\displaystyle \frac{\pi D_p^2}{4}}\times \sqrt{{\displaystyle \frac{2g{H}_L}{1.5+\frac{4f_F{L}_p}{D_p}}}}$$

(19)

Another approach for estimating the friction losses is to use the Darcy–Weisbach equation based on the Darcy friction factor $f_D$ rather than the Fanning friction factor as follows:

$$H_f={\displaystyle \frac{f_D{L}_p}{D_p}}\times {\displaystyle \frac{V_p^2}{2g}}$$

(20)

where f_D is the Darcy–Weisbach friction factor and is calculated using the following explicit formula for turbulent flow as a function of R_n and roughness height ratio e/$D_p$:

$$f_D={\displaystyle \frac{1}{{\left(-2\times Log(\frac{e}{3.7D_p}+\frac{5.74}{R_n^{0.9}})\right)}^2}}$$

(21)

where e is the roughness height, which depends on the pipe material, and R_n is the Reynolds number and is calculated using the following equation:

$$R_n={\displaystyle \frac{V_p{D}_p}{\upsilon }}$$

(22)

where ν is the kinematic viscosity of water. The total head losses are calculated using the following equation:

$$H_L=(1.5+{\displaystyle \frac{f_D{L}_p}{D_p}})\times {\displaystyle \frac{V_p^2}{2g}}$$

(23)

The traditional design of the tail escape drainage pipe requires some trial-and-error procedures in addition to some iterations. This takes a lot of time and effort. This design may be uneconomic. In this paper, a graphical solution is proposed to obtain all possible designs and select the best one (minimum diameter of the drainage pipe). Moreover, the TAILOPT MATLAB tool is created to find the optimum design. In addition, sensitivity analysis is performed to show the effect of some parameters on the optimum design.

2.2. Graphical Method

Graphical methods offer engineers immediate visual access to the complete design space (e.g., trade-offs between pipe diameter and head loss, alongside hydraulic constraints), enabling rapid preliminary screening of feasible solutions prior to optimization. Unlike code-based outputs yielding point solutions, graphical representations elucidate entire sets of viable solutions and clarify the rationale for optimality. This approach is particularly valuable for resource-constrained irrigation agencies with limited computational access: standalone graphical aids (e.g., nomographs) serve as field references for expedited design validation without software and as pedagogical tools for explaining complex hydraulic relationships to non-specialists. Furthermore, graphical techniques provide independent validation for computational tools like the TAILOPT algorithm discussed subsequently.

The graphical analysis for the combined tail escape structure—integrating an upstream orifice and weir discharging into a drainage pipe—extends the well-established pump operating point paradigm. Analogous to plotting pump performance curves against system curves, two rating curves define system equilibrium: Curve A (drainage pipe discharge vs. head loss for varying diameters) and Curve B (aggregated orifice–weir flow vs. head loss). Their intersection identifies the operating point, balancing inlet conveyance and pipe discharge capacity.

To enhance efficiency, Curve A is first generated across pipe sizes. Hydraulic constraints (e.g., allowable velocity, head loss) then refine the solution space by excluding non-viable diameters. Curve B is superimposed for feasible configurations, with intersections within the constrained domain denoting valid hydraulic equilibria. Solutions outside acceptable bounds are excluded, ensuring designs adhere to prescribed performance criteria through an intuitive visual framework.

2.2.1. Preparation of Base Charts (Representing Curve A)

To facilitate graphical solutions for the design of drainage pipes in tail escape systems, a series of fifteen Flow–Head Loss Feasibility (FHLF) diagrams with velocity constraint base charts has been developed (see Figure 3 and Supplementary Figures S1–S13). Each chart corresponds to a specific value of the roughness parameter, α [L], which is defined by the relation

$$\alpha =f_F\times L_p$$

(24)

where $f_F$ denotes the dimensionless Fanning friction factor, and L_p is the length of the drainage pipe. The practical analyzed values of α are 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, and 0.8 m. Each chart presents the relationship between the discharge rate (m³/s) and the associated head losses for various drainage pipe diameters, specifically, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, and 1.5 m. The head losses are computed according to Equation (18), in which H_L denotes the head loss (m), D_p is the drainage pipe diameter (m), and V is the mean velocity (m/s). For each curve, the analysis begins with a minimum velocity of 1 m/s and extends to a maximum of 3 m/s. As demonstrated by the charts, head losses increase with higher discharge rates and decrease with larger pipe diameters, reflecting the expected hydraulic behavior of the system.

Similar FHLF graphical charts could be also developed for the head losses in drainage pipes using Equations (21) and (23) based on the Darcy–Weisbach friction coefficient $f_D$ instead of the Fanning friction factor $f_F$. A sample of these charts is presented in Figure 3c, and a set of these charts is also given in the Supplementary Materials Figures S14–S23.

An additional chart has been developed to determine the orifice discharge, Q_o (m³/s), as a function of the differential head between the upstream (h_o, m) and downstream (h₂, m) sides of the orifice for various orifice diameters (see Figure 4). The discharge through the orifice is calculated using Equation (2).

2.2.2. Minimum Head Loss Constraint (H_Lmin)

For a specified drainage pipe diameter, length, and slope, Equation (6) directly relates the head loss in the drainage pipe (H_L) to the submergence head at the pipe inlet (h₁). According to several drainage design guidelines, the minimum recommended submergence head is h₁ ≥ 0.2 D_p, where D_p is the pipe diameter. Accordingly, the minimum head loss can be expressed as

$$H_{Lmin}=0.7D_p+L_p\times S_o$$

(25)

By substituting Equation (7) and performing the necessary algebraic simplification, the relationship between head loss and discharge under the minimum head loss constraint (for zero drainage pipe slope) is formulated as the following nonlinear implicit equation:

$$Q_p^2={\displaystyle \frac{c_{1}g.{\pi }^2{H}_{Lmin}{\left(H_{Lmin}-S_{p.}{L}_p\right)}^5}{\left[15\left(H_{Lmin}-S_{p.}{L}_p\right)+28\alpha \right]}}$$

(26)

where $c_1$ is a constant and equals 5.206164.

The constraint (H_Lmin) presented by the equation serves as a lower bound for the feasible design domain.

2.2.3. Maximum Head Loss Constraint (H_Lmax)

To ensure proper aeration for weir flow, a minimum freeboard must be maintained within the tail escape well. Common engineering practice recommends a minimum freeboard (F) of at least 0.5 m. Considering the geometric configuration (see Figure 1) and applying the energy equation, the following relation can be established:

$$drop+d+0.1=D_p+h_1+F=D_p+H_L-{\displaystyle \frac{D_p}{2}}-S_o{L}_p+F$$

(27)

which can be rearranged to yield the head loss as follows:

$$H_L=drop+d+0.1-{\displaystyle \frac{D_p}{2}}+S_o{L}_p-F$$

(28)

Thus, the scenario of minimum allowable freeboard corresponds to the maximum permissible head loss, given by

$$H_{Lmax}=drop+d+0.1-{\displaystyle \frac{D_p}{2}}+S_o{L}_p-F_{min}$$

(29)

By substituting Equation (18) to eliminate D_p, the upper bound for the head loss–discharge relationship is derived as

$$Q_p^2={\displaystyle \frac{4g.{\pi }^2{H}_{Lmax}{\left(C-H_{Lmax}\right)}^5}{\left[3\left({C-H}_{Lmax}\right)+4\alpha \right]}}$$

(30)

where C is given as

$$\mathrm{C}=drop+d+0.1+S_o{L}_p-F_{min}$$

(31)

The constraint H_Lmax therefore establishes the upper bound for the feasible hydraulic design domain.

A solved example will be presented in the next section to demonstrate the full process of using the graphical approach.

2.3. TAILOPT Tool

To streamline the traditionally laborious trial-and-error procedure in determining optimal dimensions for tail escapes in accordance with drainage manual standards, a graphical user interface (GUI) computational tool named TAILOPT (Tail Escape Optimizer) was developed using MATLAB environments.

Figure 5 presents a block diagram separating the GUI, core engine, and data output layers. It maps each function to its objective: GUI nodes (e.g., TAILOPT_GUI, createInputFields, readInputFields, callbacks, results/sketch tabs) acquire inputs and trigger computation; core nodes (runFanning, runDarcy, Q_HL_Lp_Dp_e, evaluate/rank) implement hydraulic calculations and design evaluation; and data outputs (createSWMMInput, results table, SWMM.inp) capture outputs and automatically export the selected tail-escape-relevant design given by the TAILOPT tool as an input EPA-SWMM file for further unsteady-flow assessment within the environment of EPA-SWMM. This step saves a lot of effort for a designer to build a model inside EPA-SWMM.

Figure 6 focuses more on the workflow of the core engine or the solvers, whereas Figure 7 shows snapshots of the different GUI tabs in the tool.

The program begins by prompting the user, through a user interface, to input key canal parameters, including the high water level (HWL), bed level, canal bed width (b), reach length (L), side slope (t), mean velocity (V_c), and discharge ratio (F_wcr), representing the proportion of canal discharge allocated to the tail escape (Q_w/Q_c).

In addition, the user specifies drainage pipe parameters, namely, the pipe length (L_p), pipe slope (S_p), and friction factor. The interface also accommodates orifice-related parameters: the required drainage time and orifice discharge coefficient (C_d).

Upon receiving these inputs, TAILOPT proceeds with a series of hydraulic computations, including the following:

Calculation of canal discharge:

Cross-sectional area of the canal:

$$A_c=b\times d+t\times d^2$$

(32)

where A_c is the canal flow area, b is the channel bed width, t is the channel side slope, and d is the water depth which can be taken as the normal depth of the channel.

The canal discharge can be roughly calculated using the continuity equation if the canal cross-sectional average velocity, V_c, is known or alternatively by applying the Manning equation while assuming the uniform flow assumption and knowing the channel Manning roughness n and channel slope.

Based on the design requirements and applied standards, the weir of the tail escape should be sized to receive/drain a specified portion of the canal flow; this portion is called ${\mathrm{F}}_{wcr}$.

$$Q_w=Q_c\times {\mathrm{F}}_{wcr}$$

(33)

The approximate head over the weir, $h_w$, can be calculated using Equations (34)–(36):

$$B=b+2\times t\times d$$

(34)

$$V_s=1.17\times V_c$$

(35)

where $V_s$ is the approximate average velocity of the surface flow that is expected to be drained by the surface weir.

$$h_w={\displaystyle \frac{Q_w}{B\times V_s}}$$

(36)

Knowing Q_w and h_w from Equations (33) and (36), respectively, the weir crest length (B_w) can be calculated using Equation (1), and thus, the average diameter of the circular well (D_w) can be obtained. The crest level of the weir can be estimated using Equation (37).

$$Crest level=Canal high water level+0.1$$

(37)

The size of the orifice can be selected by calculating the volume of water in the last reach of the channel that needs to be drained in a specified emptying time via the orifice inlet only of the tail escape. The area of the orifice can be calculated as per Equation (38):

$$a_o={\displaystyle \frac{A_c.L/T_{empty}}{0.5C_d\times \sqrt{2gd}}}$$

(38)

To facilitate hydraulic flexibility and design efficiency, TAILOPT automatically generates all feasible design combinations of drainage pipe diameters and well bed drops. This results in a total of 102 design alternatives, spanning pipe diameters from 0.4 m to 1.5 m (with 0.05 m increments) and well drops ranging from 0.0 m to 0.5 m (with 0.1 m increments). For each configuration, the program computes the orifice flow corresponding to the lowest water depth in well condition and aggregates it with the weir flow. The total flow capacity is then compared with the pipe discharge. If the pipe discharges exceed the combined orifice and weir discharges, the water level is considered too low, thus invalid, and other design combinations will be tried. Conversely, if the pipe discharge is less than the combined flows, then the orifice flow corresponding to the highest water depth in well condition will be computed and then aggregated with the weir flow. The total flow capacity is then compared with the pipe discharge. If the pipe discharge does not exceed the combined orifice and weir discharges, the water level is considered too high, thus invalid, and the current design combination would be disregarded, and other design combinations will be tried. Conversely, if the pipe discharge exceeds the combined flows, then the current design combination is considered viable.

Only those designs where the pipe discharge capacity matches the total aggregated inflow within an acceptable depth range are retained as viable solutions. This automated, algorithmic procedure eliminates the need for manual iteration, thereby expediting the design process while ensuring compliance with established hydraulic standards.

3. Results and Discussion

In this section, we will present two solved examples. The first example will focus on how to use the graphical approach to obtain the optimal size of the drainage pipe in the tail escape. The focus of the first approach is only directed to the drainage pipe in the tail escape, whereas in the second example, a more comprehensive and practical example is discussed where the TAILOPT program is initially used to obtain the optimal size of a tail escape for a distributary canal, and then the design given by TAILOPT is assessed by conducting unsteady-state analysis using the open-source freeware EPA-SWMM package.

3.1. Design Example No. 1

This example demonstrates the application of the graphical method to determine optimal drainage pipe dimensions for a tail escape structure. The same problem is subsequently resolved using the TAILOPT program. The design parameters are as follows:

• Channel water depth (d) = 2 m;
• Orifice diameter (d_o) = 0.6 m;
• Orifice head (h_o) = 1.745 m (for zero bed-level drop);
• Weir discharge (Q_w) = 1.17 m³/s;
• Bed-level drops = {0 m, 0.5 m};
• Drainage pipe slope (S_p) = 0;
• Roughness parameter length (α) = 0.25 m.

3.1.1. Tail Escape with No Drop

This part assumes that the invert level of the tail escape well has no drop and the slope of the drainage pipe is horizontal. The first step in the graphical method is to select the suitable Q-H_L base chart from Figure 3 or from the figures presented in the Supplementary Materials (Figures S1–S13) based on the given roughness data. Based on the given data, Figure 3a is selected and replotted as Figure 8.

The second step is to draw on the selected Q-H_L chart the Q-H_Lmin constraint curve. This line can be plotted using Equation (16) while assuming relevant values of H_Lmin and obtaining the corresponding Q values from the equation. The Q-H_Lmin constraint curve is plotted as a green dashed curve in Figure 8.

The third step is to draw again on the selected Q-H_L chart the Q-H_Lmax constraint curve. This line can be plotted using Equation (20) while assuming relevant values of H_Lmax and obtaining the corresponding Q values from the equation. The Q-H_Lmax constraint curve is plotted as a blue dashed curve in Figure 6. The Q-H_Lmin constraint curve generally forms the lower bound of the feasibility domain, whereas the Q-H_Lmax constraint curve forms the upper bound of the feasibility zone, and therefore, the feasibility zone is demarcated as the yellow-colored area, as shown in Figure 8.

Figure 6 shows that pipes of sizes greater than 1.3 m lie outside of the feasibility zone. It should be noted here that diameters of 1.4 and 1.5 m will result in maximum allowable head losses smaller than minimum allowable head losses. Therefore, they turned out to be infeasible; therefore, the focus will be directed to pipe sizes from 0.4 to 1.3 m only.

For each feasible diameter presented in Figure 8, the allowable maximum and minimum discharge for the orifice flow is determined. Based on Figure 1, Equation (39) can be derived, and thus, the differential head on the orifice can be obtained, as shown in Equation (40).

$$h_2=H_L+{\displaystyle \frac{D_p}{2}}-{\displaystyle \frac{d_o}{2}}-0.25-SL$$

(39)

$$h_{\mathrm{o}}-h_2=h_{\mathrm{o}}-(H_L+{\displaystyle \frac{D_p}{2}}-{\displaystyle \frac{d_o}{2}}-0.25-S_{o}L)$$

(40)

Therefore, the minimum and maximum orifice flow ($Q_{o,min}, Q_{o,max}$) can be calculated based on Equation (2), as shown in Equations (41) and (42), respectively.

$$Q_{o,min}=C_d\times {\displaystyle \frac{{{\pi d}_o}^2}{4}}\times \sqrt{2g(h_{\mathrm{o}}-(H_{Lmax}+{\displaystyle \frac{D_p}{2}}-{\displaystyle \frac{d_o}{2}}-0.25-S_{o}L\left)\right)}$$

(41)

$$Q_{o,max}=C_d\times {\displaystyle \frac{{{\pi d}_o}^2}{4}}\times \sqrt{2g(h_{\mathrm{o}}-(H_{Lmin}+{\displaystyle \frac{D_p}{2}}-{\displaystyle \frac{d_o}{2}}-0.25-S_{o}L\left)\right)}$$

(42)

In the fourth step and after identifying the feasible pipe sizes (shown in Figure 8), we construct the table (

Table 1. Feasibility analysis of drainage pipe sizes based on the overlap between the pipe drainage capacity range and the combined (Q_w + Q_o) discharge range (zero-drop case).

Dp (m) (1)	Qmin (m3/s) (2)	Qmax (m3/s) (3)	HLmin (m) (4)	HLmax (m) (5)	Qo,min (m3/s) (6)	Qo,max (m3/s) (7)	Qw + Qo,min (m3/s) (8)	Qw + Qo,max (m3/s) (9)	Feasibility? (Discharge Overlap?) (10)
0.4	0.15	0.32	0.28	1.4	0.63	1.01	1.8	2.18	X
0.45	0.215	0.43	0.315	1.375	0.63	1	1.8	2.17	X
0.5	0.28	0.54	0.35	1.35	0.63	0.98	1.8	2.15	X
0.55	0.365	0.67	0.385	1.325	0.63	0.96	1.8	2.13	X
0.6	0.45	0.8	0.42	1.3	0.63	0.94	1.8	2.11	X
0.65	0.575	0.95	0.455	1.275	0.63	0.92	1.8	2.09	X
0.7	0.7	1.1	0.49	1.25	0.63	0.91	1.8	2.08	X
0.75	0.85	1.29	0.525	1.225	0.63	0.89	1.8	2.06	X
0.8	1	1.48	0.56	1.2	0.63	0.87	1.8	2.04	X
0.85	1.195	1.675	0.595	1.175	0.63	0.85	1.8	2.02	X
0.9	1.39	1.87	0.63	1.15	0.63	0.83	1.8	2	✓
1	1.84	2.3	0.7	1.1	0.63	0.79	1.8	1.96	✓
1.1	2.38	2.76	0.77	1.05	0.63	0.74	1.8	1.91	X
1.2	3	3.28	0.84	1	0.63	0.69	1.8	1.86	X
1.3	3.72	3.8	0.91	0.95	0.63	0.64	1.8	1.81	X

Note: Highlighted rows indicate feasible designs that comply with all constraints. “X” means no overlap exits and the corresponding design is infeasible.

). In the first column, all the feasible pipe sizes are listed (according to Figure 6: from 0.4 m to 1.3 m). Next, in columns 2 to 5, the values of Q_min, H_Lmin and Q_max, and H_Lmax are identified either graphically (from Figure 8) for each pipe size or by using Equations (15), (16) and (19), and (20), respectively. Next, Equations (41) and (42) are applied to calculate the minimum and maximum values of the orifice flow, and the values are recorded in columns 6 and 7 (Table 1). Then, the orifice flow is aggregated with the weir flow to obtain the minimum and maximum values of Q_o + Q_w, and the values are recorded in columns 8 and 9. By comparing the discharge range of the pipe drainage presented in columns 2 and 3 with the discharge range of the aggregated weir and orifice flow (presented in columns 8 and 9), the pipes that show range intersection are identified as the most feasible pipe sizes that satisfy all constraints and hydraulic conditions (column 10).

Based on Table 1, two pipe sizes are found to be feasible: D_p = 0.9 m and D_p = 1.0 m. The next step is to construct the aggregated flow–head loss graph (Q_w + Q_o) − H_L (Curve B) within the feasible zone. For instance, from Table 1 and in the case of utilizing a drainage pipe of size D_p = 0.9, H_L ranges from a minimum value of 0.63 m to a maximum value of 1.15 m; therefore, we constructed

Table 2. Relation between minimum and maximum allowable head loss versus corresponding Q_w + Q_o for drainage pipe diameter of 0.9 m in the case of zero drop in well bed level.

HL (m) (1)	ho − h2 (m) (2)	Qo (m3/s) (3)	Qw + Qo (m3/s) (4)
0.63	1.215	0.83	2
0.65	1.195	0.82	1.99
0.7	1.145	0.8	1.97
0.75	1.095	0.79	1.96
0.8	1.045	0.77	1.94
0.85	0.995	0.75	1.92
0.9	0.945	0.73	1.9
0.95	0.895	0.71	1.88
1	0.845	0.69	1.86
1.05	0.795	0.67	1.84
1.1	0.745	0.65	1.82
1.15	0.695	0.63	1.8

, which presents the hydraulic performance of a drainage pipe of size “D_p = 0.9 m”. The second column represents the differential head across the orifice, as per Equation (40), the third column represents the orifice discharge according to Equation (2), and the fourth column aggregates the orifice and weir flow.

The last step in the graphical solution is to plot the aggregated weir and orifice flow versus H_L curve on the same FHLF diagram (Figure 8) using the same scale and to obtain the operating point, which is defined as the intersection of the H_L-Q_pipe curve with the H_L-(Q_o + Q_w) curve. The actual operating drainage point shown in Figure 9a gives the actual flow in the drainage pipe and the actual head loss for D_p = 0.9 m.

The same steps can be repeated for the second viable pipe size (D_p = 1.0 m, as shown in the feasibility analysis in Table 1), and Figure 9b presents the actual operating drainage point in the case of using a drainage pipe size (D_p) equal to 1.0 m. The comparison of the actual operating points presented in Figure 9a,b shows that as D_p increases, pipe capacity increases and head loss decreases.

3.1.2. Effect of Using a Drop in the Tail Escape Well

Suppose that a drop of 0.25 m is applied to the well bed level. The previous procedures presented in Section 3.1.1 are repeated. Based on Equation (15), the curve representing the H_Lmin constraint curve keeps the same as in the case of zero drop. However, the curve representing the H_Lmax constraint is shifted upward and to the right. A feasibility analysis table (like Table 1) for the 0.25 m drop is presented in

Table 3. Feasibility analysis of drainage pipe sizes based on the overlap between the pipe drainage flow capacity range and the combined (Q_w + Q_o) discharge range (0.25 m drop case).

Dp (m)	Qmin (m3/s)	Qmax (m3/s)	Qw + Qomin (m3/s)	Qw + Qomax (m3/s)	Overlap?
0.4	0.15	0.36	1.8	2.25	X
0.45	0.215	0.475	1.8	2.23	X
0.5	0.28	0.59	1.8	2.22	X
0.55	0.365	0.713	1.82	2.2	X
0.6	0.45	0.848	1.84	2.19	X
0.65	0.575	0.995	1.85	2.17	X
0.7	0.7	1.155	1.86	2.15	X
0.75	0.85	1.325	1.87	2.13	X
0.8	1	1.508	1.88	2.12	X
0.85	1.195	1.702	1.88	2.1	X
0.9	1.39	1.909	1.88	2.08	✓
1	1.84	2.356	1.88	2.04	✓
1.1	2.38	2.851	1.88	2	X
1.2	3	3.393	1.87	1.96	X
1.3	3.72	3.982	1.86	1.92	X
1.4	4.58	4.618	1.85	1.87	X

Note: Highlighted rows indicate feasible designs that comply with all constraints. “X” means no overlap exits and the corresponding design is infeasible.

. It was also found that the only feasible pipe sizes are diameters of 0.9 and 1 m. Figure 10a,b show the actual operating drainage points for both feasible pipe sizes on the FHLF diagram.

In the case of using a 0.5 m drop, the previous procedures presented in Section 3.1.2 for the 0.25 m drop case are repeated. In the FHLF diagram presented in Figure 11, it has been noted again that the curve representing the H_Lmin constraint does not change. However, the curve representing the H_Lmax constraint is shifted more upward and to the right. Based on the FHLF diagram, drainage pipes of D_p = 1.5 m lie out of feasibility; therefore, they are excluded from the feasibility analysis table. The feasibility analysis for the 0.5 m drop is presented in

Table 4. Feasibility analysis of different drainage pipe sizes based on the overlap between the pipe drainage flow capacity range and the combined (Q_w + Q_o) discharge range (0.5 m drop case).

d (m)	Qmin (m3/s)	Qmax (m3/s)	Qw + Qomin (m3/s)	Qw + Qomax (m3/s)	Overlap?
0.4	0.15	0.126	1.83	2.31	X
0.45	0.215	0.477	1.87	2.3	X
0.5	0.28	0.589	1.9	2.28	X
0.55	0.365	0.713	1.92	2.27	X
0.6	0.45	0.848	1.94	2.25	X
0.65	0.575	0.995	1.95	2.24	X
0.7	0.7	1.155	1.96	2.22	X
0.75	0.85	1.325	1.97	2.2	X
0.8	1	1.508	1.97	2.19	X
0.85	1.195	1.702	1.97	2.17	X
0.9	1.39	1.909	1.97	2.15	X
1	1.84	2.356	1.98	2.12	✓
1.1	2.38	2.851	1.97	2.08	X
1.2	3	3.393	1.97	2.04	X
1.3	3.72	3.982	1.96	2.01	X
1.4	4.58	4.618	1.95	1.96	X

Note: Highlighted rows indicate feasible designs that comply with all constraints. “X” means no overlap exits and the corresponding design is infeasible.

. Based on the discharge range intersection, it was also found that the only feasible drainage pipe is of a diameter equal to 1 m, and the 0.9 m pipe size turned out to be infeasible. This could be justified, as increasing the drop from 0.25 m to 0.5 m resulted in an increase in the drainage velocity for the 0.9 m size, exceeding the maximum velocity constraint. Therefore, it turned out to be infeasible.

Figure 11 shows the actual operating drainage point for the 1 m drainage pipe diameter.

3.1.3. Effect of Drainage Pipe Slope

The objective of this part is to study the effect of increasing the drainage pipe slope on the feasible sizes of drainage pipes.

In the case of 1% slope of the drainage pipe, the H_Lmin constraint curve is shifted upward and to the right, as per Equation (15) and as shown in Figure 12. Also, the second H_Lmax constraint curve which represents the condition F >= 0.5 m is shifted upward and to the right, according to Equation (19). Based on the FHLF diagram, Figure 10, drainage pipes of D_p = 1.3–1.5 m lie out of feasibility; therefore, they are excluded from the feasibility analysis table. The feasibility analysis for zero drop and S_o = 1% is presented in

Table 5. Feasibility analysis of different drainage pipe sizes based on the overlap between the pipe drainage flow capacity range and the combined (Q_w + Q_o) discharge range (zero drop and S_o = 1%).

Dp (m)	Qmin (m3/s)	Qmax (m3/s)	Qw + Qomin (m3/s)	Qw + Qomax (m3/s)	Overlap?
0.4	0.2	0.36	1.84	2.18	X
0.45	0.285	0.475	1.83	2.17	X
0.5	0.37	0.59	1.83	2.15	X
0.55	0.475	0.713	1.84	2.13	X
0.6	0.58	0.848	1.84	2.11	X
0.65	0.725	0.995	1.84	2.09	X
0.7	0.87	1.155	1.83	2.08	X
0.75	1.035	1.325	1.81	2.06	X
0.8	1.2	1.508	1.79	2.04	X
0.85	1.4	1.702	1.76	2.02	X
0.9	1.6	1.909	1.73	2	✓
1	2.15	2.356	1.6	1.96	X
1.1	2.7	2.851	1.4	1.91	X

Note: Highlighted rows indicate feasible designs that comply with all constraints. “X” means no overlap exits and the corresponding design is infeasible.

. Based on the discharge range intersection, it was also found that the only feasible drainage pipe is of a diameter equal to 0.9 m.

Figure 13 presents the FHLF diagram in the case of utilizing a drainage pipe of 3% slope and zero drop in the tail escape’s well. The FHLF diagram initially shows that the feasibility zone (the yellow-colored zone) is significantly reduced and pipes of sizes > 0.75 m are initially infeasible.

To investigate the feasibility of the remaining pipe sizes, a feasibility assessment table is constructed (

Table 6. Feasibility analysis of different drainage pipe sizes based on the overlap between the pipe drainage flow capacity range and the combined (Q_w + Q_o) discharge range (zero drop and S_o = 3%).

Dp (m)	Qmin (m3/s)	Qmax (m3/s)	Qw + Qomin (m3/s)	Qw + Qomax (m3/s)	Overlap?
0.4	0.28	0.377	1.95	2.18	X
0.45	0.39	0.477	1.97	2.17	X
0.5	0.5	0.589	1.99	2.15	X
0.55	0.625	0.713	2	2.13	X
0.6	0.75	0.848	2	2.11	X
0.65	0.925	0.995	1.99	2.09	X
0.7	1.1	1.155	1.98	2.08	X
0.75	1.315	1.325	1.95	2.06	X

Note: “X” means no overlap exits and the corresponding design is infeasible.

). The table shows that no flow range overlap exists for any pipe size. Therefore, there is no solution or accepted diameter in the case of adopting a 3% slope for the drainage pipe.

3.1.4. Solution of Example 1 Using TAILOPT

The TAILOPT program was used to obtain all accepted designs for the different cases discussed before for example 1 in the case of using the Finning friction factor and adopting α = 0.25 m, and the results are summarized in

Table 7. Feasible drainage pipe designs using TAILOPT program in the case of α = 0.25 m.

Slope (%)	Drop (m)	Dp(m)	Pipe Velocity (m/s)	ho (m)	h1 (m)	h2 (m)	F (m)
0	0	0.9	2.864	1.745	0.642	0.992	0.558
0	0.1	0.9	2.908	1.845	0.676	1.026	0.624
0	0.2	0.9	2.951	1.945	0.709	1.059	0.691
0	0.3	0.9	2.993	2.045	0.741	1.091	0.759
0	0	1	2.458	1.745	0.271	0.721	0.829
0	0.1	1	2.494	1.845	0.293	0.743	0.907
0	0.2	1	2.529	1.945	0.315	0.765	0.985
0	0.3	1	2.563	2.045	0.337	0.787	1.063
0	0.4	1	2.596	2.145	0.359	0.809	1.141
0	0.5	1	2.628	2.245	0.38	0.83	1.22
1	0	0.9	2.971	1.745	0.476	0.826	0.724
3	0	None	N/A	N/A	N/A	N/A	N/A

.

The TAILOPT program was also used again to obtain all accepted designs for example 1 in the case of using the Darcy–Weisbach friction factor (instead of the Fanning friction factor) while assuming that the length of the drainage pipe is L_p = 25 m and the pipe roughness height is e = 0.6 mm. All the feasible results are summarized in

Table 8. Feasible drainage pipe designs using TAILOPT program in the case of e = 0.6 mm and L_p = 25 m.

Slope (%)	Drop (m)	Dp (m)	Pipe Velocity (m/s)	ho (m)	h1 (m)	h2 (m)	F (m)
0	0	0.9	2.9826	1.745	0.4577	0.807	0.742
0	0.45	1	2.676	2.195	0.207	0.657	1.342
0	0.5	1	2.6916	2.245	0.216	0.666	1.383

.

3.2. Example 2 (A Case Study of A Tail Escape at the End of a Distributary Canal)

3.2.1. Site Description and Operational Data

A real-world case study was selected to evaluate the performance of the developed Tail Escape Optimizer (TAILOPT). This case, originally presented in the literature [12], involves a typical distributary canal with the following characteristics, as shown in Figure 14a: bed width b = 1.0 m, side slopes of 1.5(H):1.0(V), and Manning’s roughness coefficient n = 0.013 (corresponding to a lined canal). The canal serves a total command area of approximately 1000 feddans and supplies five uniformly distributed side offtakes, each commanding about 200 feddans. A head regulator is located at the upstream end of the distributary canal, while a tail escape weir is situated at the downstream end. According to the drainage standard, the design height of the weir is generally taken as the normal water depth plus 0.10 m. The canal operates on a rotational schedule comprising three fifteen-day terms, with water available for five days followed by a ten-day interval without inflow. Therefore, the actual water duty during the irrigation period for rotational operation should be WD_rot = 150 m³/fed/day. Under continuous, 24 h inflow conditions, the total average canal discharge (Q_in) is 1.736 m³/s.

3.2.2. Optimal Design via TAILOPT

Using the Manning equation, the corresponding normal depth (y_n) was calculated as 1.05 m, and the downstream tail escape crest weir height (H_w) was determined to be 1.15 m. The TAILOPT solver was applied to optimize the dimensions of the tail escape structure, yielding the following results: weir crest level (height) of 1.25 m, maximum top width of 4.45 m, inner well diameter (D_win) of 2.25 m, average well diameter (D_wave) of 2.5 m, outer well diameter (D_wout) of 2.75 m, and orifice diameter of 0.35 m.

Table 9. Feasible drainage pipe designs using TAILOPT program in the case of e = 0.6 mm and L_p = 25 m (Example 2).

Slope (%)	Drop (m)	Dp (m)	Pipe Velocity (m/s)	ho (m)	h1 (m)	h2 (m)	F (m)
0	0.5	0.5	2.6687	1.3364	0.6726	0.7226	0.5274
0	0.2	0.55	2.2037	1.0364	0.3247	0.4247	0.5253
0	0.25	0.55	2.233	1.0864	0.3417	0.4417	0.5583
0	0.3	0.55	2.2623	1.1364	0.3577	0.4577	0.5923
0	0.35	0.55	2.2904	1.1864	0.3743	0.4743	0.6257

presents the first five feasible drainage pipe sizes along with the corresponding well drop heights and associated hydraulic heads, as determined by the optimization process.

3.2.3. Assessment of TAILOPT Design

To evaluate the optimal design, the smallest feasible drainage pipe size identified in Table 2 was selected to minimize costs. The hydraulic performance of the entire system, incorporating the tail escape at the downstream end of the distributary canal, was subsequently analyzed using the EPA-SWMM modeling package. The simulation utilized the geometric dimensions of the tail escape structure, as determined by the TAILOPT program.

The input file of the EPA-SWMM model was developed via the TAILOPT tool to represent the distributary canal, including all side offtakes and the end escape structure. Figure 14b presents the model idealization, which includes six segments of upstream open-channel reach that drain to a tail escape junction that bifurcates to a 0.35 m diameter of orifice flow reach and an upper weir flow reach adjusted at a height = 1.2 m + drop size (i.e., 1.7 m). The last two reaches then merge again into the well of the tail escape. Based on TAILOPT’s recommendation, a drop of 0.5 m was applied to the tail escape, which means the invert of the well is lowered by 0.5 m from the upstream tail escape junction. The water is transmitted from the well to the outfall via a horizontal drainage pipe of 0.5 m diameter (as per Table 9).

Assessment Criteria for Tail Escape Design

To comprehensively assess the effectiveness of the proposed tail escape configuration, two operational scenarios were simulated:

Scenario 1: During the initial 24 h period, it was assumed that the fifth side offtake becomes inoperative, resulting in its allotted water share being unutilized. Under this condition, the tail escape structure is required to safely convey the surplus flow to the disposal drain.

Scenario 2: In the subsequent 24 h period, the head regulator is assumed to be closed for maintenance immediately following the first day, necessitating that the tail escape efficiently drains the remaining water in the canal reach within 24 h.

Performance Evaluation

Figure 15a to e summarize the results obtained by the EPA-SWMM model. Figure 15a shows the temporal variations in the water depth inside and upstream of the tail escape well. It is noted that the channel maintains a minimum freeboard of 0.2 m with the berm, and the freeboard F inside the well up to the weir crest is usually maintained above the minimum value (F_min = 0.5). Figure 15b presents the temporal variations in the orifice, weir, and drainage pipe flow. The weir flow exists only when the upstream water depth exceeds the weir crest (refer to Figure 15a), and this took place mainly in the first 24 h. It is also clear that the drainage pipe discharge is equal to the combined weir and orifice flow discharge, which means that the changes in the volume storage in the tail escape well are negligible. Figure 15c presents the time variation in the water velocity in the drainage pipe. The velocity stays within the limit of higher than V_min and lower than V_max with the combined contribution of the weir and orifice flow within the first 24 h; however, during the drainage period, the velocity is off course and expected to go lower than V_min. It is also of interest to notice that based on Figure 15d, the average depth in the drainage pipe was running partially full despite the inlet being submerged, and this means that the open-channel and gradually varied flow equation should be applied rather than the pipe flow equation used in the TAILOPT tool. Figure 15e presents the temporal variation in the average water depth in the six reaches of the distributary channel. The sixth reach is the reach located just upstream of the tail escape. The figure indicates that after the closure of the head gate and the offtakes, the tail escape was able to empty the majority of the distributary channel within 24 h. However, reach No. 6 seemed to take a longer time; this could be justified by the fact that TAILOPT assumes that the orifice flow will run submerged all the time, but this is not true during the last 24 h. Studying the water depth decline rate in the different channel reaches clearly shows that the decline rate was steep when the orifice was working as a submerged orifice; however, at about T = 36 h and later, the orifice became unsubmerged and thus worked as weir flow, and therefore, the water depth decline rate reduced significantly.

3.3. Sensitivity Analysis

The TAILOPT tool is used to examine the sensitivity of the following design parameters for the selected optimal drainage pipe size: the roughness parameter α, the drainage pipe slope S_o, and the channel’s reach length L. The parameter α is changed from 0.25 to 0.5 and 0.75 m while keeping S_o and L constant. Then, the drainage pipe slope, S_o, is changed from 0 to 0.001, 0.005, and 0.01 while keeping both α and L constant. Moreover, the canal reach length is changed from 3000 m to 2000 m while keeping the other parameters constant. It can be noticed that there are multiple solutions for each case. The most economical solution selected is that with the minimum diameter of the drainage pipe and then the minimum applied drop in the tail escape’s well.

Figure 16, Figure 17 and Figure 18 illustrate the results of sensitivity analysis conducted using the TAILOPT optimization tool to evaluate the influence of the design parameters α, S_o, and L on the optimal design of the drainage pipe for a combined tail escape structure.

Figure 16 focuses on studying the sensitivity related to changing the design parameter α. In this analysis, α was varied across three values, 0.25 m, 0.5 m, and 0.75 m, while the channel slope (S_o) and pipe length (L) were held constant. The figure employs a dual-y-axis format, where the x-axis represents the required drop height (m) across the tail escape’s well. The primary (left) y-axis displays the feasible drainage pipe diameters (m), indicated by solid lines, and the secondary y-axis (right) presents the corresponding pipe velocities (m/s), represented by dashed lines. The results clearly demonstrate that as α increases, the hydraulic conditions demand either a higher drop or a larger pipe diameter to maintain system feasibility. This trend aligns with fundamental hydraulic principles: a greater α implies higher flow resistance that could be in the form of a longer drainage pipe length or rougher pipe material which results in higher upstream water levels. Consequently, the system compensates either by enhancing the gravitational head through additional drop or by selecting a larger pipe size to counteract the increased resistance. The figure shows that increasing α from 0.25 m to 0.75 m shifts the Pareto-optimal solutions toward larger pipe diameters (D_p), greater hydraulic drops, or both. For instance, at α = 0.25 m, the optimal solution exists at D_p = 0.9 m with zero drop. When α increases to 0.5 m, the optimum design shifts to D_p = 0.9 m with a 0.4 m drop (which means that the pipe size is preserved by increasing the head). At α = 0.75 m, the solution requires both a slightly larger pipe (D_p = 0.95 m) and a 0.45 m drop (to balance the hydraulic constraints).

The dashed lines denote the corresponding velocities for each α scenario. As α increases, pipe resistance increases, and the velocity curves shift downward. This indicates a consistent decrease in velocity as α increases from 0.25 to 0.75 m. This outcome reflects the system’s adaptive response: to prevent velocity from exceeding limits, it trades off drop height and pipe diameter to reduce the velocity under increasing α.

Figure 17 focuses on studying the sensitivity related to changing the slope (S_o) of the drainage pipe. In this analysis, S_o was varied across three values, 0.001, 0.005, and 0.01, while the length–roughness parameter (α) and pipe length (L) were held constant. The figure employs a dual-y-axis format where the x-axis represents the required drop height (m) across the tail escape’s well. The primary (left) y-axis displays the feasible drainage pipe diameters (m), indicated by solid lines, and the secondary y-axis (right) presents the corresponding pipe velocities (m/s), represented by dashed lines. The results clearly demonstrate that increasing the drainage pipe slope does not have a significant effect on the optimal diameters. The most economical solution is 0.9 m, which is the same as in the case of the horizontal drainage pipe. Moreover, increasing the drainage pipe slope to a very steep slope value of 0.02 leads to unfeasible solutions due to exceeding the maximum permissible velocity limit. It can be also noticed from Figure 15 that increasing drop values lead to an increase in the required diameter from 0.9 to 0.95 m at drop = 0.30 m for slope = 0.001. Moreover, for slope = 0.005, the required diameter increases from 0.9 to 0.95 m at drop = 0.20 m. In addition, for slope = 0.01, the required diameter increases from 0.9 to 0.95 m at drop = 0.10 m.

Figure 18 focuses on studying the sensitivity related to changing the channel length. In this analysis, L varies across two values, 2000 and 3000 m, while the other parameters are kept constant. The results clearly demonstrate that a decrease in the drained reach length from 3000 m to 2000 m reduces the orifice diameter required to drain the reach in 24 h. Hence, the discharge passing through the drainage pipe is reduced, and the optimum diameter decreases. It can also be noticed in Figure 18 that increasing the drop makes the required diameter increase from 0.85 to 0.9 m for reach length = 2000 m and from 0.9 to 0.95 m for reach length = 3000 m when the velocity approaches 3 m/s. This happens at drop = 0.4 m for reach length = 2000 m and at drop = 0.35 m for reach length = 3000 m.

4. Conclusions

This study presents a rigorous yet practical methodology for optimal design of combined weir–orifice tail escape structures, integrating a hydraulically grounded graphical analysis with a computational optimization tool (TAILOPT). Two key performance curves, one for drainage pipe capacity vs. head loss and another for combined weir–orifice inflow, are employed to identify the hydraulic operating point that satisfies all design constraints (e.g., velocity limits, minimum submergence, nappe aeration). Building on this framework, the TAILOPT tool systematically evaluates all viable weir–orifice–pipe combinations to pinpoint the most cost-effective design, eliminating tedious trial and error and ensuring adherence to performance criteria. The tool’s ability to export a compatible SWMM input file streamlines the workflow, enabling designers to seamlessly transition from steady-state flow-based optimization to transient flow and dynamic assessment. Application to representative case studies demonstrates the approach’s robustness and yields clear design insights. For example, extremely steep pipe slopes (above ~2%) prove impractical due to excessive flow velocities, whereas moderate slopes (≤1%) coupled with slight well drops enable smaller, more economical pipes without sacrificing performance. By contrast, purely vertical drop configurations (zero pipe slope) often demand larger pipe diameters to remain feasible. Additionally, sensitivity analysis reveals that pipe roughness (α), canal reach length, and pipe slope are the dominant parameters influencing the optimal design: increases in roughness or reach length necessitate larger pipes or greater drop heights to maintain capacity. In summary, the proposed approach provides a unified and efficient framework that combines visual insight with computational rigor, streamlining tail escape design and clearly delineating the optimal solutions for managing surplus irrigation flows.

5. Limitations, Environmental Considerations, and Future Work

A key limitation of the current EPA-SWMM environment is the absence of a dedicated dual-drainage link element capable of directly simulating the combined weir–orifice behavior of tail escape structures. While SWMM includes separate elements for weirs and orifices, accurately representing a combined configuration requires manually linking both to a shared junction (representing the escape well) and connecting this node to a downstream conduit. Although feasible, this process is neither intuitive nor efficient and demands a higher level of modeling expertise.

Given that EPA-SWMM is open-source, a promising avenue for future development is the creation of a custom TailDrain link element that natively couples the hydraulics of the weir and orifice within a single well node, discharging through a drainage conduit. Integrating the TAILOPT optimization engine into such an element would allow users to perform design generation, optimization, and unsteady-state verification entirely within the SWMM interface, streamlining the workflow and providing a unified modeling environment for complex drainage scenarios.

Furthermore, since TAILOPT is implemented in MATLAB, future enhancements could leverage existing MATLAB–SWMM integration frameworks, such as MatSWMM [31]. This would enable simultaneous multi-parameter optimization, giving designers a powerful platform to identify solutions that balance hydraulic performance, cost-effectiveness, constructability, and operational reliability.

It should also be emphasized that the analyzed tail escape structures operate under environmental and agricultural conditions typical of irrigation networks in semi-arid and arid regions (e.g., the Middle East, North Africa, and South Asia), where rotational irrigation schedules often lead to night-time surpluses. While the present analysis assumes steady freshwater inflows under moderate climatic conditions, changes in environmental factors can influence performance. High sediment loads may accelerate pipe roughness deterioration, saline or debris-laden flows may increase clogging risks, and poor drainage or high water tables in the receiving environment may induce partial submergence of the outlet. Likewise, extreme rainfall events can increase the frequency of emergency releases.

An additional design-related limitation concerns the assumption of a minimum submergence head of h₁ ≥ 0.2 D_p, as recommended in several drainage design guidelines. This criterion ensures full-pipe flow, under which the Darcy–Weisbach equation remains valid. If the submergence head falls below this threshold, the pipe may run partially full, and the calculated discharge capacity may be underestimated, while the outlet velocities may exceed safe limits, increasing the risk of downstream erosion. Although such cases are uncommon, since most tail escape pipes are short (<40 m) and can feasibly satisfy the submergence requirement, off-design scenarios can still be evaluated using the EPA-SWMM model exported from TAILOPT, which explicitly accounts for partially full and tailwater conditions. These variations underline the importance of producing sets of feasible solutions, rather than single-point outcomes, through the graphical method and TAILOPT tool. By providing designers with a transparent design envelope that can be further stress-tested using EPA-SWMM, the methodology remains adaptable to site-specific environmental and operational conditions.