^{1}

^{*}

^{2}

The system dynamics technique is used as a decision tool for engineering problems. It is one of the object oriented approaches that study and manage complex feedback systems. In this paper, the system dynamics technique was used to simulate the performance of a drainage system under wheat crop in a clay soil. The model was calibrated and validated using observed experimental field data (drainage discharge and water table level) collected from Mashtul Pilot Area (MPA), Egypt. The results indicated that, the model is capable to predict hydrological parameters such as water table fluctuation, drainage discharge, upward flux, evapotranspiration, deep percolation, infiltration, runoff, soil moister content and unsaturated hydraulic conductivity on the basis of variation of soil moister content. The trends of the parameters found to be legible. Six statistical indexes were calculated to determine the agreement between the observed and simulated values of water table and drainage discharge. Results indicated that the system dynamics technique can be considered as a good decision tool to predict the subsurface drainage water precisely.

Artificial drainage has been known to be an important water management practice for farming of the most productive soils of the Midwest [

The field work (sampling and measurements) was carried out in MPA. MPA was constructed in 1980 in south-eastern part of the Nile Delta [

Irrigation applied m^{3}/feddan | No. of irrigations | Irrigation schedule | Crop |
---|---|---|---|

1500 - 1900 | 3 | 40 - 60 days after first irrigation then every 30 days | Wheat |

The southern and western boundaries are formed by the Mahmoudia Drain and its branch; the northern and eastern are bound by tertiary irrigation canals. It is characterized by a deep clay top layer and a sandy aquifer. The clay layer, which is approximately 6.0 m thick, contains about 35% silt and 65% clay. Irrigation water is delivered by gravity to the tertiary canals and lifted approximately 0.5 m to field level by pumps.

The area is drained through a subsurface drainage system that consists of parallel PVC lateral drains, which discharge into buried concrete collector drains through a manhole. The design of the subsurface drainage system was made according to the standard criteria of the Egyptian Public Authority for Drainage Projects (EPADP). Two types of criteria can be distinguished; namely, the agricultural and the technical ones. The agricultural criteria are an average depth of the groundwater table midway between the drains of 1.0 m and an average drainage rate of 1.0 mm/day to permit sufficient leaching. The technical criteria are a design discharge rate for the determination of drain pipe capacity of 4 mm/day for rice areas and 3 mm/day for other crops, a safety factor of 25% in the design of the collector drains to account for sedimentation and misalignment and change in diameters and maximum depth of 1.5 m for laterals and 2.5 m for collectors. The area was divided into eighteen drainage units with different drain depths and spacing. The units were cultivated with a single crop for each unit during each cropping season. Berseem (Egyptian clover) and wheat were cultivated as winter crops and cotton, rice, and maize as summer crops [

MPA was controlled under the existing farming conditions, aiming to apply the data measurement program for two years, from June 2005 to May 2007, and consequently two winter and two summer seasons. The area was surveyed with a Global Positioning System (GPS) to locate its boundaries, and with a Geographic Information System (GIS) for sampling locations, as shown in

Determination crop pattern, fertilizer amount, and time required to apply each crop for each unit.

Collected drainage water samples before cultivation, before and after applying fertilizers, and periodically every 10 days.

In order to quantifying analyze the subsurface drainage system in MPA, a dynamic modeling of the system is made (

the model. Furthermore, the model calculates the future drainage water flow. Solving the water and solute transport equations requires two soil water relations, namely the soil water content-water potential relation and the soil water potential-hydraulic conductivity relation. They were taken according to [

The conceptual model was applied on the study site to model the water balance predictions of the subsurface drainage which is employed to simulate the performance of drainage and related water management systems.

The conceptual model uses dynamic modeling to quantify subsurface drainage, deep seepage, infiltration, and evapotranspiration. Subsurface drainage flux is calculated based on the assumption that mainly lateral water movement occurs in the saturated region. The flux is determined by the water table elevation at the mid plane of the drains and the water level in the drains. The Hooghoudt’s steady-state equation [_{1}):

q H = 8 K e d e m d r + 4 K e m d r 2 c r f L d r 2 (1)

where q_{H} is the water flux [L T^{−1}], m_{dr} is the midpoint water table height above the drain [L], K_{e} is the effective lateral hydraulic conductivity [L T^{−1}] and L_{dr} is the horizontal distance between drains [L]. d_{e} is the equivalent depth from the drain to the restrictive layer [L] and is specified in the model to correct for convergence flux near the drains. It can be defined as a function of L_{dr}, the drainage tube radio r_{dr} [L] and d [L], the real depth from the drain to the restrictive layer [_{rf} is the ratio of the average flux between the drains to the flux midway between the drains and is controlled by the shape of the water table profile. c_{rf} is approximated as 1.0, which implies that Equation (1) corresponds to the ellipse equation that is often used to determine drain spacing [_{e} is defined as:

K e = ∑ i = 1 i = N ( ( K H ) i E i ) ∑ i = 1 i = N E i with E i = 0.0 and E P = d P for i ≤ P (2)

where N is the number of soil (horizontal) layers, (K_{H})_{i} is the horizontal hydraulic conductivity of the i-th horizontal layer and E_{i} is the thickness of the i-th layer [L]. This equation should be used observing the convention that soil layers are numbered progressively from up to down. If the water table is located in soil layer P (with 1 ≤ P ≤ N), then d_{P} is the distance between the midpoint water table elevation and the bottom of layer P.

In the event that the depth of water on the surface exceeds S_{1} (the surface storage), the assumption of a curved water table completely below the soil surface fails (Bouwer and van Schilfgaarde, 1963) and in this case, the flux is calculated as:

q H = 4 π K e ( z p n + z d r − r d r ) c p n L d r 2 (3)

where z_{pn} is the ponded depth [L], z_{dr} is the distance from the soil surface to drain [L] and c_{pn} [--] is a constant defined for a given depth of soil profile and drain-size, -depth and -spacing (Skaggs, 1981). The deep vertical seepage flux is computed using Darcy’s law to calculate the flux through the restrictive layer:

q V = K V ( h 1 − h 2 ) E r e s t (4)

where q_{v} is the deep vertical seepage flux [L T^{−1}], K_{V} is the effective vertical hydraulic conductivity of the restrictive layer [L T^{−1}], h_{1} is the average distance from the bottom of the restrictive layer to the water table [L], h_{2} is the hydraulic head in the groundwater aquifer referenced to the bottom of the restrictive layer [L], and E_{rest} is the thickness of the restrictive layer [L]. Infiltration is computed through a modified Green-Ampt procedure. The potential evapotranspiration (PET) is estimated using the Thornthwaite equation [

Despite the efforts of several alternative approaches to manage intangible factors, none has been sufficient to fully incorporate relationships between variables, delays and feedback, all of which characterize the behavior of intangible resources. So, Managers continue taking decisions only based (or support) on their experience, knowledge that constitute their mental models [

Causal loop diagram is an important tool for representing the feedback structure of systems. A causal diagram consists of variables connected by arrows denoting the causal influences among the variables. A feedback loop is a succession of causes and effects such that a change in a given variable travels around the loop and comes back to affect the same variable. If an initial increase in a variable in a feedback loop eventually results in an increasing effect on the same variable, then, the feedback loop is identified as a “reinforcing or positive” feedback loop. If an initial increase in a variable eventually results in a decreasing effect on the same variable, then the feedback loop is identified as a negative, counteracting or balancing’ loop [

The causal loop diagram in this study has shown in _{et}”, which in turn decreases evapotranspiration. The second feedback loop represents the interaction between evapotranspiration and upward flux: the larger the evapotranspiration, the larger the upward flux, then the larger the soil water content and k_{et}, which in turn increases evapotranspiration. The third feedback loop represents the interaction between soil water storage and percolation: the larger the storage, the larger the hydraulic conductivity, then the larger the percolation, which in turn decreases soil water storage. The fourth feedback loop represents interaction between water table and soil water storage: an increase in the percolation increases water table and upward flux and soil water content. In the fifth feedback loop as water table rises by deep percolation, the depth of water above the drain increases which increases the drain discharge, and in turn decreases the water table.

The model was calibrated by considering the time series of observed subsurface flux at the outlet of the drainage system (^{2}). The characteristic of the different statistical criteria is given in

Mean Absolute Error (MAE) | |
---|---|

MAE = 0 | model is perfect |

MAE = min | model is optimal |

0 < MAE | model is less perfect |

Relative Root Mean Square Error (RRMSE) | |

RRMSE = 0 | model is perfect |

RRMSE = min | model is optimal |

Model Efficiency (EF) | |

EF = 1 | model is perfect |

EF = max | model is optima |

EF < 1 | model is less perfect |

EF = −¥ | model has no prediction capability |

Coefficient of Residual Mass (CRM) | |

CRM = 1 | model has no prediction capability |

CRM < 1 | model has some at least prediction capability |

CRM close to 0 | model is optimal |

Coefficient of Determination (CD) | |

CD = 0 | model has no prediction capability |

0 < CD | model has some at least prediction capability |

CD = max | model is optimal |

Goodness of Fit (R^{2}) | |

R^{2} = 1 | model is perfect |

R^{2} = max | model is optimal |

R^{2} = 0 | model has no prediction capability |

The measured data and simulated results were compared in terms of subsurface drainage discharge (

MAE | RRMSE | CD | EF | CRM | R2 | |
---|---|---|---|---|---|---|

Measured & Dynamic system | 0.450 | 0.344 | 0.780 | 0.821 | 0.133 | 0.873 |

MAE: mean absolute error; RRMSE: relative root mean square error; CD: coefficient of determination; EF: model efficiency; CRM: coefficient of residual mass; R^{2}: goodness of fit.

A dynamic model was developed to predict drain discharge behavior for different drain systems (depth and spacing). The developed model was used to simulate the daily drainage water at the midpoint of drain spacing in a clay soil. The measured data and simulated results were compared in terms of subsurface drainage discharge. The statistical analysis implies a good fit between measured and simulated values. The comparative study reveals that the model performs well, reliable and accurate for predicting subsurface drainage water in clay soils. Results (predictions) can be used to design the drainage system geometry for better water management on both field and catchment scales. Results indicated that, the model can be used as a decision support tool to help policy makers in long term strategic management for irrigation projects. The developed model can potentially help in setting guidelines for using subsurface drainage water in agricultural sector in Egypt.

The authors declare no conflicts of interest regarding the publication of this paper.

El-Sadek, A. and Radwan, M. (2019) Using System Dynamics for Simulating Subsurface Drainage Systems in Clay Soils. Journal of Water Resource and Protection, 11, 529-539. https://doi.org/10.4236/jwarp.2019.115030