This paper focused on choosing the best design of subsurface land drainage systems in semiarid areas. The study presented three different soil layers with different hydraulic conductivity and permeability, all layers are below the drain level, and the permeability is increasing with depth. A mathematical model was formulated for the horizontal and vertical drainage optimal design. The result was a nonlinear optimization problem with nonlinear constraints, which required numerical methods for its solution. The purpose of the mathematical model is to find the best values of pipes and tubewells spacing, groundwater table drawdown, and pumps operating hours which leads to a minimum total cost of the subsurface drainage design. A computer code was developed in MATLAB environment and applied to the case study. Results show that the vertical drainage was economically better for the case study drainage network design. And the main factor affecting the mathematical model for both pipe and well drainage was the distance between pipes and tubewells. In addition, considering the lifespan of vertical drainage project, the optimal design involves the minimum possible duration of pumping stations. It is hoped that the proposed optimal mathematical model will present a design methodology by which the costs of all alternative designs can be compared so that the least-cost design is selected.
Subsurface drainage is widely used worldwide to remove excess water found below the earth’s surface [
This paper presents a novel strategy for the best design of subsurface horizontal and vertical drainage in an area of different saturated soil layers with different hydraulic conductivity and permeability, and the permeability is increasing with depth. With the use of modern optimization algorithms, we can find the suitable values of groundwater table drawdown and pipes/tubewells spacing that lead to the minimum cost of the total subsurface drainage system.
The least-cost design is that satisfying all design constraints with the minimum total cost. The objective function for pipe and well drainage was determined by considering all cost components that affect the drainage network design. Then constraints were formulated depending on the hydraulic study of the study area. The result was a nonlinear objective function with nonlinear constraints. A survey of modern optimization algorisms was conducted to find the one suitable for the solution of the formulated optimization problem. It was found that the interior-point optimization algorithm was adapted to the problem and produced satisfactory results. Two computer codes for both horizontal and vertical subsurface drainage were developed in MATLAB environment in order to derive the optimal solution for both types of drainage systems. The solutions then were compered to know how the lifespan and type of the project will affect the total cost of the network design.
The major cost components of pipe drainage system are drainage materials, installation, and operation and maintenance.
The costs of drainage materials and drainage pipe installation work and structures are major components of the total cost of a pipe drain project. Drainage materials include collector and lateral pipes, filters, pump set and pump house, outlet structures and manholes, and outlet pitching.
Collector and lateral pipes: the collector pipes have a transmission function to carry water to the outlet under the gravitational flow, so the UPVC (unplasticized poly vinyl chloride) corrugated nonperforated pipe is used for the collectors. The pipe diameter of the collectors is chosen depending on the expected flow. Single Wall Perforated Flexible Corrugated UPVC Pipes (outer diameters of 80–355 mm) are widely used for lateral pipes.
Artificial filters/envelopes: the filter or envelope material around the pipes plays an important role in preventing the fine particles of the soil from entering into the pipes with the flow. Nowadays, it is preferred to use artificial filters rather than gravel filters. The artificial filters are cheaper in cost, of better quality, and easier to handle and transport than gravel filters. For medium and light soil, it is preferred to use nonwoven polypropylene fibers with a thickness of more than 2.5 mm and an opening size of more than 300 microns, and a needle punched geotextile nonwoven fabrics are used for heavy soil. Perforated pipes precoated with industrial filter are more preferred than locally coated pipes for quality installation.
Pump set and structures: pump set and structure consists of diesel pump set and pump house, manholes (junction box), and outlet structures. Other cross drain structures may also be required in larger projects but these are avoided for simplicity. The outlet structure and manholes are precast Reinforced Cement Concrete (RCC) pipes of different lengths and diameters depending on the site conditions. In the case of pump outlets, 900 mm diameter manholes and 1200 mm outlet structures are used, where 900 mm diameter manholes are used in the gravity outlets. The bottom edge of the RCC pipe for outlet structures and manholes is closed by fixed plate at the bottom. Plastic coated iron bars are provided in the walls of outlet structures and manholes to help in inspection and cleaning. In the case of gravity outlets, instead of RCC pipes, plastic manholes and outlet pipes have recently been used. Plastic manholes and outlet pipes are relatively expensive, but they are easy to carry, handle, and install at the project sites [
Installing the drainage system is the function of the installation unit according to the specifications and design. Supervisors must specify that the installation (including drainage equipment) is carried out strictly in accordance with specifications and design. Various drainage machines such as hydraulic excavators, tractor-mounted trenchers, self-propelled trenchers with laser automatic control, and self-propelled machines with
The popular belief that subsurface drainage does not require any maintenance and operation is untenable. In the case of pump outlets, pump operation is required for at least the first few years of installation. The maintenance of subsurface drainage systems mainly involves removing sediment from outlets, manholes, and pipes, also repairing or replacing the damaged outlets, manholes, and pipes [
The most cost components that affect the subsurface pipe drainage design can be determined according to the total costs as shown in the following relationship:
The construction costs are calculated by using the following equation:
The cross-sectional area of excavation
Cross section of excavation.
Also, total length of all drainage pipes
So, the construction costs can be written as
Moreover, the annual investment costs are
The objective is to design the least-cost pipe drainage network. Thus, the objective function can be stated as
It is recommended to establish a proper strategy for vertical drainage system design, and it should be connected with the economic factors. For example, we can choose large numbers of tubewells with a small amount of discharge and a slight decrease in the groundwater level from each tubewell, or we can choose small numbers of tubewells with a larger amount of discharge and a larger decrease in the groundwater level from each tubewell and a larger spacing between the tubewells. There are a lot of choices, and these choices are controlled by Tubewell depth Tubewell spacing Tubewell discharge The decrease amount of the groundwater level
And the most suitable choice can be determined according to the total costs as stated in (
The construction costs for vertical drainage project are calculated by using the following equation:
Moreover, the annual investment costs are
So, the objective function for the design of vertical drainage tubewells can be stated as
A total area of 500 hectares is attended to apply an irrigation network to meet the crops need and help in washing the salinity that comes from underground water. So, a drainage network is needed also along with the irrigation network to serve in releasing exceed salty water out of the study area. The study area is located in Syria, as shown in Figure
The location and topography of the study area.
Table
General climate and crops indicators for the study area.
Factor | Month | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | The average annual |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Temperature | 7.3 | 9.1 | 12.5 | 16.4 | 21 | 25.9 | 27.9 | 28.1 | 24.5 | 19.1 | 13.4 | 8.6 | 17.8 | |
Rainfall (mm) | % | 18.7 | 14.1 | 13.3 | 8.6 | 3.2 | 0.7 | 0 | 0.2 | 1.6 | 8.6 | 8.5 | 22.2 | 99.7 |
Monthly rate | 121 | 91.6 | 84.9 | 55.9 | 21.2 | 4.8 | 0 | 1.9 | 10.6 | 54.9 | 54.9 | 145.6 | 649 | |
Evaporation from free water surface (mm/day) | Lambert | 1.2 | 1.7 | 2.4 | 2.6 | 5.7 | 7.9 | 10.5 | 9.5 | 6.5 | 3.8 | 1.4 | 1.1 | 4.8 |
Ivanov | 1.1 | 1.6 | 2.5 | 3.9 | 5.6 | 8.4 | 9.3 | 8.7 | 7.3 | 4.5 | 2.5 | 1.2 | 4.7 | |
Evatranspiration (mm/day) | 0.7 | 1.8 | 2.5 | 4.4 | 6.6 | 8.4 | 8.9 | 8.9 | 6.1 | 3.2 | 2 | 0.7 | 4.56 | |
Crop needs (mm/month) | Wheat and barley | 15.19 | 35.28 | 54.25 | 92.4 | 0 | 0 | 0 | 0 | 0 | 0 | 42 | 15.19 | 21.25 |
Cotton | 0 | 35.28 | 54.25 | 92.4 | 143.22 | 176.4 | 193.13 | 193.13 | 0 | 0 | 0 | 0 | 73.98 | |
Summer vegetables | 0 | 35.28 | 54.25 | 92.4 | 143.22 | 176.4 | 193.13 | 0 | 0 | 0 | 0 | 0 | 57.89 |
The study area contains three different soil layers (
Hydrogeology of the study area.
The groundwater table is about 3 m below the ground surface, the gradient of the groundwater surface at the beginning of the study area is
Unit cost of excavation is
We can determine the pipe spacing for subsurface pipe drainage network when we have three different layers and the permeability is increasing with depth by using the following formulations [
As
Case study pipe drainage geometry.
And we can obtain
Determination of
Calculation of
The drainage unit discharge (
For the hydraulic study of subsurface drainage pipes, we can use Manning formula which is as follows:
For better calculation of drainage pipe diameter, we can consider that the pipe is full of water, but we have to choose a pipe with an actual diameter greater than the calculated one in order to guarantee the free surface flow inside the drainage pipe.
The calculated velocity inside the drainage pipes must be between these limits:
And the critical pipe diameter must achieve the following formula:
The optimization problem for the pipe drainage design can be stated as follows.
Minimize
Subject to
According to Soviet Science Encyclopedia for calculating and design of drainage networks and land reclamation, we can determine the tubewells spacing when the permeability is increasing with depth by using the following formulation [
When the well is not reaching the impermeable layer (
For calculating
Case study vertical drainage geometry.
We can obtain the
0.1 | 0.15 | 0.2 | 0.25 | 0.3 | 0.4 | 0.5 | |
---|---|---|---|---|---|---|---|
2.33 | 1.07 | 0.49 | 0.17 | −0.01 | −0.19 | −0.22 |
And we can obtain
Calculation of
In Table
Protection vertical wells are placed at the edge of the study area along the feeding line
The spacing between surrounding wells can be calculated by using the following formula:
The total discharge that has to be released by all drainage wells can be determined by
The discharge of each drainage well is
Thus, the total number of drainage wells is
The number of surrounding wells can be obtained by
Thus, the number of investment wells is
The duration of pump operating required to maintain a favourable drainage depth is given by If the tubewells are placed in a rectangular pattern, If the tubewells are placed in a triangular pattern,
In our study, we will choose a rectangular pattern. Some researchers suggest operating the pumps only during the weeding period, but others prefer to operate the pumps in certain hours every day.
The optimization problem for the well drainage design can be stated as follows.
Minimize
Subject to
Table
General parameters calculated for the design of pipe and well drainage.
Parameter | ∆ | ∆ | ∆ | ||||||
---|---|---|---|---|---|---|---|---|---|
Pipe drainage | 16665 | 0.95 | 0.9 | 0.92 | |||||
Well drainage | 16665 | 1 | 0.95 | 0.95 | 0.911818 | 0.87667 | −0.01 |
By applying these values on the computer code in MATLAB environment, we derived the optimal solution for the pipe drainage design for a range of lifespan as seen in Table
Pipe drainage optimal solution for a range of lifespan.
Lifespan | Pipe spacing | Hydraulic head | Total cost |
---|---|---|---|
1 | 234.4577 | 1.3189 | 8.5823 |
10 | 234.8422 | 1.3211 | 1.1711 |
50 | 235.0565 | 1.3223 | 1.4698 |
100 | 235.0644 | 1.3223 | 1.4840 |
And the optimal solution for the vertical drainage design for a range of lifespan is shown in Table
Well drainage optimal solution for a range of lifespan.
Lifespan | Hydraulic head | Duration of pumping operation | Well spacing | Total cost | The discharge of each well | Total number of wells | Surrounding wells spacing | Number of surrounding wells | Number of investment wells |
---|---|---|---|---|---|---|---|---|---|
1 | 17.5848 | 184.9211 | 881.0597 | 2.9559 | 2220.8976 | 7.5 | 716.7423 | 3.4 | 4.1 |
10 | 9.9645 | 104.7862 | 673.7206 | 7.0352 | 1298.6063 | 12.8 | 515.0194 | 4.7 | 8.1 |
50 | 8.7810 | 92.3413 | 634.5266 | 1.0038 | 1151.9073 | 14.5 | 477.2781 | 5.1 | 9.4 |
100 | 8.7468 | 91.9817 | 633.3536 | 1.0174 | 1147.6523 | 14.5 | 476.1512 | 5.1 | 9.4 |
As we can see, for horizontal and vertical drainage networks, it is better to choose large distance between the lateral pipes and wells. These distances can be calculated by applying the optimization model on the study area. And for the case study described above, the vertical drainage will be a better solution as subsurface drainage design for the whole project lifespan. The cost for operating vertical drainage pumps plays an important role in determining the optimal design when considering the lifespan of the project.
In order to formulate an optimization problem for the design of subsurface drainage systems, cost equations have been introduced for both horizontal and vertical drainage. The cost equations contained the most cost components that affect the subsurface drainage networks design. Then the optimization problem constraints were derived from the hydraulic study of the case study. The case study contains three different soil layers with different hydraulic conductivity and permeability, and the permeability is increasing with depth. A mathematical model was formulated for the horizontal and vertical drainage optimal design in the case study. The result was a nonlinear optimization problem with nonlinear constraints, which required numerical methods for its solution. A survey of modern optimization algorisms was conducted to find the one suitable for the solution of the formulated problem. It was found that the interior-point optimization algorithm was adapted to the problem and produced satisfactory results. The results show that the proposed optimal mathematical model for both horizontal and vertical drainage networks was affected mostly by the distance between pipes and wells, and the optimal solution involved the maximum possible values of pipes and tubewells spacing. Also, for this case study, the model gave a lower cost for the designing of tubewells network compared with pipe network. And the total cost for the vertical drainage design involved minimum duration of pump operation when considering the lifespan of the subsurface drainage project. The study has shown that the pipes and tubewells spacing and the groundwater table drawdown cannot be selected randomly if we put the economic factor in consideration. Traditional pipes and tubewells design may lead to high costs compared with the optimal design. It is hoped that the proposed optimal mathematical model will present a design methodology by which the costs of all alternative designs can be compared so that the least-cost design is selected.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
The authors declare no conflicts of interest.
The authors would like to thank Hohai University for granting the scholarship which made the research possible; lab mates, Genxiang Feng and Wang Ce, for their suggestions and help; friends, Wael Alhasan and Saeed Assani, for their big support and help. This research was funded by the National Natural Science Foundation of China, under grant number 51879071.