Application of Neural Networks and Optimization Model in Conjunctive Use of Surface Water and Groundwater

Abstract

This study develops an optimization model for the large-scale conjunctive use of surface water and groundwater resources. The aim is to maximize public and irrigation water supplies subject to groundwater-level drawdown constraints. Linear programming is used to create the optimization model, which is formulated as a linear constrained objective function. An artificial neural network is trained by a flow modeling program at specific observation wells, and the network is then incorporated into the optimization model. The proposed methodology is applied to the Chou-Shui alluvial fan system, located in central Taiwan. People living in this region rely on large quantities of pumped water for their public and irrigation demands. This considerable dependency on groundwater has resulted in severe land subsidence in many coastal regions of the alluvial fan. Consequently, an efficient means of implementing large-scale conjunctive use of surface water and groundwater is needed to prevent further overuse of groundwater. Two different optimization scenarios are considered. The results given by the proposed model show that water-usage can be balanced with a stable groundwater level. Our findings may assist officials and researchers in establishing plans to alleviate land subsidence problems.

1 Introduction

The spatial and temporal variation in the rainfall distribution means that water resources in Taiwan can become scarce during the dry season. In addition, the demand for water is gradually increasing due to rapid urbanization and industrialization. In some regions, the local demand for surface water has already exceeded supply. Some irrigation zones tend to pump groundwater to fulfill their needs. However, over-drafting over a long period has seen the occurrence of severe disasters such as land subsidence, seawater intrusion, and groundwater pollution. Hence, the key issue addressed in this study is the efficient utilization of water resources with reasonable pumping.

A classification of groundwater management models has been presented that divides the approaches into two categories (Gorelick 1983)—hydraulic management models and policy evaluation and allocation models. The use of both simulation and optimization models together are termed a linked-simulation-optimization (LSO), and are a type of policy evaluation and allocation model. LSO uses the results of an external aquifer simulation model as the input to a series of economic optimization models. Because the simulation and economic management models are separate, complex economic problems can be considered. Singh and Datta (2006) developed a genetic algorithm (GA)-based LSO for optimal identification of unknown groundwater pollution sources. The main advantage of their methodology was the external linking of the numerical simulation model with the optimization model. They evaluated the performance for combinations of source characteristics, such as the location, magnitude, and release period, as well as for various data availability conditions and concentration measurement error levels. Bhattacharjya and Datta (2005) developed a saltwater intrusion management model that can be used to derive optimal and efficient management strategies for controlling saltwater intrusion in coastal aquifers. The GA-based optimization approach is particularly suitable for externally linking the numerical simulation model with the optimization model. An artificial neural network (ANN) model was trained as an approximator of the three-dimensional density-dependent flow and transport processes in a coastal aquifer. An LSO model was then developed to link the trained ANN with the GA-based optimization model to solve saltwater management problems. Models that solve the groundwater flow equations in conjunction with optimization models are widely used. Two different hydraulic management models, the embedding method (Willis and Yeh 1987; Peralta et al. 1991a, b) and the response matrix method, to couple a groundwater simulation model and a linear, single-objective optimization technique. In the embedding method, the numerical discretization of the partial differential equations is included as a constraint in an optimization model. The response matrix method is based on the principle of linear superposition. The drawdown matrix of the unit pumping rates is generated for specific locations. Louie et al. (1984) developed linear programming software to solve multi-objective optimization problems with linear constraints. Their study simultaneously considered three different objectives: (1) the location of the water supply, (2) water quality control, and (3) the prevention of groundwater overdraft. The response matrix then links the optimization and simulation models. Willis and Liu (1984) developed a bi-objective optimization model for the Yunlin area of southwest Taiwan, in which the response matrix was used to predict an inhomogeneous and isotropic aquifer system. Parametric linear programming was used to generate optimal planning policies (the set of non-inferior solutions), and determine the relationship of the total water deficit with the maximum pumping rate and minimum permissible head values in the aquifer system. Willis and Finney (1988) developed an optimization model for the Yunlin groundwater basin. The influence coefficient was predicted to simulate the hydraulic response of the groundwater system due to pumping. The planning model was then optimized to reduce water pumping and prevent localized land subsidence as well as salt water intrusion. Ejaz and Peralta (1995) developed a method for incorporating water quality constraints within a conjunctive water-use simulation–optimization model for a hydraulically connected stream/aquifer system. Their objective was to maximize conjunctive water-use. The response matrix technique was utilized to simulate the hydraulic response of a stream/aquifer system to various stimuli (pumping, diversion, and loading). Karamouz et al. (2004) developed a dynamic programming optimization model for conjunctive use planning, and used the response matrix method to represent the groundwater level in the Tehran metropolitan area. The nonlinear objective function of this model was developed to satisfy the demand for agricultural water, reduce pumping costs, and control groundwater table fluctuations. McPhee and Yeh (2004) developed a multi-objective optimization model of conjunctive use in the Upper San Pedro River Basin. The study considered three different nonlinear objectives: (1) minimization of the net present value of the mitigation costs, (2) maximization of aquifer yield, and (3) minimization of the drawdown at specified locations. The response matrix was used to link the simulation and optimization models, and a decision support system and fuzzy set theory were applied to assist decision-makers in finding the optimal solution to groundwater management problems. Pulido-Velázquez, Andreu and Sahuquillo (2006) simulated the groundwater flow in an optimization model using the response matrix method. The optimization model equations were solved using sequential linear programming in the unconfined case, and linear programming for a confined aquifer. The difference in the results was minor, suggesting that the use of the linear assumption may be valid in many basin-scale conjunctive management problems.

Emch and Yeh (1998) developed a nonlinear multi-objective management model for water-use within a coastal region. The SHARP flow model simulates groundwater flow using a quasi-three-dimensional finite-difference model based on the sharp interface assumption. The MINOS software was used to solve nonlinear problems in an iterative manner. Two conflicting objectives were considered: (1) the cost-effective allocation of surface water and groundwater supplies, and (2) the minimization of saltwater intrusion.

Recently, ANNs have been applied to simulate groundwater fluctuations at specific sites. The application of ANNs to hydrological problems has been discussed in detail by the ASCE Task Committee (2000). Many studies have trained ANNs to accurately predict transient water levels for several months in a multi-layered groundwater system (Ioannis et al. 2005; Coppola et al. 2003). Rao et al. (2004) proposed a nonlinear, nonconvex problem using simulated annealing algorithms to realize the conjunctive use of surface and groundwater for coastal and deltaic systems. Their model conjunctively allocates groundwater and surface water to each of the demands in the deltaic region, minimizing operational costs and maximizing groundwater reserves. They employed a trained ANN to replace SHARP in simulating a multilayered, two-fluid sharp interface. Karamouz et al. (2007) developed a methodology for the conjunctive use of surface water and groundwater resources that uses GAs to obtain optimal solutions. Groundwater fluctuations were represented by an ANN, and these were then incorporated into the conjunctive nonlinear optimization model as constraints. The training and validation data for the ANN were generated by the flow modeling software MODFLOW (U.S. Geological Survey 2005). The objective functions were set to minimize agricultural water shortages, minimize pumping costs, and maintain groundwater level fluctuations. Coppola et al. (2007) developed ANNs with simulation data from a numerical groundwater flow model of the study area. The ANN-derived state transition equations were trained and validated by the data generated by MODFLOW, and then embedded into a multi-objective optimization model. This enabled the Pareto frontier, or trade-off curve, between water supply and well-field vulnerability to be identified. Nikolos et al. (2008) proposed a combination of an ANN with a differential evolution algorithm to replace the classical finite-element model for water resources management problems. The objective of their optimization problem was to determine the optimal operational strategy for the productive pumping wells in the northern part of the island of Rhodes, Greece. The results showed that, with the use of an ANN as an approximation model, it was possible to (a) significantly reduce the computational burden associated with the accurate simulation of complex physical systems, and (b) provide near-optimal solutions for various constrained environmental design problems. Safavi et al. (2009) focused on the simulation and optimization of the conjunctive use of surface water and groundwater in the Najafabad plain of west-central Iran. The objective of their conjunctive model was to minimize shortages while meeting irrigation demands. The trained ANN simulated the surface and groundwater interaction, and was linked to the GA-based optimization model. The application of ANNs to replace the simulation model in groundwater management problems has also been discussed by Dhar and Datta (2009), Chu and Chang (2009), and Kourakos and Mantoglou (2009).

This study develops a network system for the large-scale conjunctive use of surface water and groundwater. The groundwater level at representative observation wells is calculated by an ANN. The model presented here can be used to optimize regional surface water allocation under different groundwater drawdown constraints to fulfill the target demand and maintain groundwater levels. Subject to pumping and infiltration, groundwater head fluctuations are simulated by the ANN and incorporated into the large-scale optimization model. To validate the developed methodology, a case study of the Chou-Shui alluvial fan is conducted.

2 Methodology

A flowchart of the proposed methodology is shown in Fig. 1. Three models are developed in this study: a MODFLOW simulation, the ANN groundwater model, and an optimization model for conjunctive use of surface water and groundwater. The ANN groundwater model will be incorporated into the conjunctive-use optimization model to simulate groundwater fluctuations at specific sites subject to pumping or infiltration. Data are generated by different pumping scenarios in the validated MODFLOW module to train and validate the ANN groundwater model. The steps are as follows:

Fig. 1

Flowchart of the study

• Step 1 Build and calibrate the groundwater flow model using MODFLOW.

• Step 2 Employ the calibrated MODFLOW to generate data for training and validating the ANN groundwater model.

• Step 3 Train and validate the ANN groundwater model, which is formulated as a linearized function, by comparing simulation results generated from the calibrated ANN groundwater model and MODFLOW.

• Step 4 Embed the validated ANN model into the optimization model for conjunctive use of surface water and groundwater.

The concept and mathematical formulation of MODFLOW, the ANN, and the conjunctive optimization model for surface water and groundwater are described as follows.

2.1 MODFLOW

MODFLOW (McDonald and Harbaugh 1988) is a modular finite-difference flow modeling program developed by the U.S. Geological Survey to solve groundwater flow equations. The governing equations are as follows:

$$ \frac{\partial }{\partial x}\left({K}_{xx}\frac{\partial h}{\partial x}\right)+\frac{\partial }{\partial y}\left({K}_{yy}\frac{\partial h}{\partial y}\right)+\frac{\partial }{\partial z}\left({K}_{zz}\frac{\partial h}{\partial z}\right)-W={S}_s\frac{\partial h}{\partial t} $$

(1)

where K _xx , K _yy , and K _zz are the hydraulic conductivity values along the x, y, and z coordinate axes (L/T); h is the piezometric head (L); W is the volumetric flux per unit volume of a water source or sink (T ^− 1) (where negative values are extractions and positive values are injections); S _s is the specific storage of the porous material (L ^− 1); and T is time.

There are several graphical user interfaces (GUIs) for MODFLOW. In this paper, the groundwater modeling system (GMS) (Aquaveo 2009) is employed as the GUI. More user information on MODFLOW can be found in the documentation of the U.S. Geological Survey (2005).

2.2 Optimization Model for Conjunctive use of Surface Water and Groundwater

2.2.1 Network Representation

Hsu and Cheng (2002) constructed a network system from a collection of nodes and a set of arcs. The nodes indicate the facilities, intersections, and transfer points of the network. The arcs show flows or movements from one node to another. In the configuration of the water supply system, a node can represent a source, junction, diversion structure, reservoir/storage, or a demand. An arc can represent a river, channel, aqueduct, pipeline, water treatment plant, pump station, or the carryover storage of a reservoir. A physical or conceptual system that can be represented as a directed graph can be described as a flow-based network. These straightforward characteristics make the network system useful.

2.2.2 Management Model

The management model aims to define a set of feasible groundwater pumping and surface water management policies for the region of interest. The management objective is considered in the linear formulation: maximize the supply of surface water and groundwater to demand nodes. The water balance of each node, the physical constraints of each arc, and the drawdown limit of the groundwater level should be included as the linear constraints in the management model. The physics-based MODFLOW is initially used to simulate the groundwater flow. However, the effectiveness and efficiency of the optimization model during the solving process leads to an ANN being used to simulate the hydraulic response at specific locations after the pumping procedures.

2.3 Artificial Neural Networks

This study uses back propagation networks to build an ANN groundwater model. A neural network with multilayer perceptrons (MLPs) is an interconnected set of nodes composed of three layers. The first is an input layer that represents the raw data fed into the network. The second is a hidden layer, determined by the activities of the input layers and the weights of the connections between the input and hidden layers. The third is an output layer, which reflects the activity of the hidden layer and the weights between the hidden and output layers. The learning algorithm is based on the concepts of dynamic tunneling and error back propagation, which prevents the algorithm becoming trapped at local minima. At each time step, the supervised learning algorithm is applied. The input is propagated in a standard feed-forward fashion. A unit in the output layer determines its activity according to a two-step procedure:

$$ ne{t}_j^n={\displaystyle \sum_i}{w}_{ji}^n{y}_i^{n-1}+{b}_n $$

(2)

$$ {y}_j^n=F\left( ne{t}_j^n\right) $$

(3)

where net ⁿ _j is the summation of the output value of the jth unit in the nth layer; w ⁿ _ji is the connection weight between the jth unit in the nth layer and the ith unit in the (n – 1)th layer; b _n is the bias of the nth layer; y ⁿ _j is the output value of the jth unit in the nth layer; and F(•) is the activation function of the hidden layer.

In supervised learning, the desired output result is required for each input vector. A supervised learning ANN, such as the MLP, uses the target result to guide the formation of the neural parameters. A commonly used cost function is the mean-squared error, which attempts to minimize the average squared error between the kth network’s output y _k and the target value r _k over all the sample pairs (Xu and Li 2002). The error function E can be defined as follows:

$$ E=\left(\frac{1}{2}\right){\displaystyle \sum_k}{\left({y}_k-{r}_k\right)}^2\kern2em $$

(4)

The activation function in our ANN can adopt a linear or nonlinear function. Input data includes the groundwater level at each control station and the net pumping rate at each site. The output describes the groundwater level of the designated observation well in the adjacent time step.

The steps for constructing the ANN groundwater model are as follows:

• A. Collect the pumping rate, the infiltration rate, and the groundwater level data from 2002 to 2006.

• B. Normalize the data from step A, and separate it into sets of training data and validation data.

• C. Use the set of training data collected from step B to train the ANN groundwater model.

• D. Calculate the head difference between the predicted and observed data of the ANN groundwater model. If the head difference is larger than the designated threshold, return to step C and try a different number of neurons in the hidden layer or another activation function to retrain the ANN.

3 Model Application

The methodology described in this paper is applied to the Chou-Shui alluvial fan, Taiwan. Below, we provide a brief introduction of the study area, model calibration, and validation.

3.1 Study Area

The Chou-Shui alluvial fan is the most important agricultural area in western central Taiwan, with a general elevation of between 0 and 100 m above sea level. This alluvial fan has a total area of 1,800 km². Figure 2 shows the geographical location of the Chou-Shui alluvial fan. There are four aquifers in the region. The first aquifer is unconfined, and the other three are confined aquifers (Central Geological Survey 2007). The terrain near the head of the alluvial fan mainly consists of gravel and coarse sand. In contrast, fine sand is found near the toe of the alluvial fan.

Fig. 2

The study area and boundary condition of the simulation model

The irrigation area of the Chou-Shui alluvial fan is 1,230 km², and the irrigation water represents around 90 % of the water resources in the region. The Chi-Chi weir is the most important intake structure in the Chou-Shui creek. There are two water intakes at the Chi-Chi weir. One diverts water to the northern part of the alluvial fan (Changhwa County), and the other diverts water to the southern part (Yunlin County).

In the 1980s, aqua cultural farming was intensively carried out in the coastal area of the alluvial fan. Large amounts of groundwater were withdrawn, resulting in serious land subsidence. According to Chiang et al. (2006), the quantity of groundwater pumping is around 1–2 billion m³ per year, which significantly exceeds the safe yield of the aquifers in the alluvial fan.

3.2 Model Calibration and Validation

3.2.1 MODFLOW

MODFLOW is used to simulate the groundwater flow in the Chou-Shui alluvial fan. The GMS GUI provides tools for depicting every phase of a groundwater simulation, including site characterization, model development, calibration, post-processing, and visualization. We use the validated MODFLOW to generate data for training and validating the ANN.

Hydrological data, geophysical data, and water resource conditions in the Chou-Shui alluvial fan are used to construct the MODFLOW model. Groundwater level data are used to calibrate and validate MODFLOW. The process of building the model is as follows:

• a. Apply a map generated by the Central Geological Survey, Taiwan, as the base map (shown in Fig. 2). The relative hydrological and geophysical characteristics can be marked on this map, such as the recharge area and the zones of various hydrological characteristics.

• b. Collect the stratification and coordinate data of each borehole analyzed by the Central Geological Survey, Taiwan. Develop the 3D-stratification of the aquifers using the interpolation function of the GMS.

• c. The dimension of each grid cell is 2.60 km from north to south, and 1.76 km from east to west. The distribution of the model grid is shown in Fig. 2. If a cell is outside the domain of the model, it is considered inactive.

• d. Three types of boundary conditions are considered, as shown in Fig. 2. These are the specified flow boundary, specified head boundary, and no-flow boundary.

• e. Precipitation will also infiltrate into the groundwater. The vertical leakance can be calculated by the soil properties, which have been investigated by Council of Agriculture (COA), Taiwan. There are four soil layers: 0–0.3 m, 0.3–0.6 m, 0.6–0.9 m, and 0.9–1.5 m. The distribution of the soil is shown in Fig. 3. The vertical leakance is calculated by the harmonic mean of the soil property over the four layers.

  Fig. 3

  

  Soil properties in the Chou-Shui alluvial fan

• f. Several model parameters must be set, including the horizontal and vertical hydraulic conductivities, specific yield of the first aquifer, storage coefficients of the second–fourth aquifers, and pumping rate.

• g. All parameters are initially set according to the report of Yang and Yu (2008), and are manually calibrated to match the observed values.

• h. The initial model conditions are set using the groundwater level from 18 observation wells in the alluvial fan on 01/01/1998. Each observation well gives groundwater level data for the four layers. The distribution of the initial head is then obtained by the interpolation function of the GMS.

• i. The spatial distribution of the pumping rates is calibrated first, because there is no actual record of this value. While adjusting the distribution of pumping rates, the total pumping rate in the study area remains the same. Secondly, our model makes minor adjustments to the hydraulic conductivity, specific yield, and storage coefficient to fit the observed head. Once the simulated head is within 1 m of the recorded value, the calibration procedures are stopped.

• j. The river package of MODFLOW is used to simulate the interaction between surface water and groundwater. This package also simulates head-dependent flux boundaries. If the head in a particular cell falls below a certain threshold, the flux from the river to the model cell is set to a specified lower bound.

• k. A comparison of the simulated and observed heads is shown in Fig. 4. They exhibit similar trends. The root mean square error (RMSE) between the computed head and the observed head is less than 0.2 m.

  Fig. 4

  

  Plots of observed and simulated water levels during validation period for a Yuan-Lin observation well (RMSE = 0.20 cm); b Gang-Ho observation well (RMSE = 0.07 cm)

3.2.2 Optimization Model of Surface Water and Groundwater

We construct a large-scale optimization model based on a network system. The objective function of the optimization model maximizes the water supply. Our intention is to maximize the water supply across different sources, including surface water and groundwater. The objective function of the optimization model can be expressed as follows:

$$ \max B={\displaystyle \sum_{t=1}^{N_t}}\left({\displaystyle \sum_{j\in \left\{{N}_D{\displaystyle \cup }{N}_A\right\}}}{\displaystyle \sum_{h_i=j}}w{t}_i{X}_{i,t}\right) $$

(5)

where B is the objective function; wt _i is the weight associated with arc i; X _i,t is the decision variable associated with arc i that conveys water to a demand node in time period t; h _i   is the end point of arc i; j is the irrigation or public demand node; N _t   is the number of time steps; N _D is the set of demand nodes for public water supply; and N _A is the set of demand nodes for agricultural water use. The constraints of the optimization model are as follows:

• a. Inflow nodes

  The constraints associated with junction node j are

  $$ \begin{array}{cc}\hfill {\displaystyle \sum_{f_i=j}}{X}_{i,t}={Q}_{j,t}^{in}\hfill & \hfill \forall j\in {N}_P\hfill \end{array} $$

  (6)

  where  Q ⁱⁿ _j,t is the flow rate from outside the system into node j at time period t; f _i   is the starting point of arc i; and N _P is the set of inflow nodes.

• b. Junction nodes or diversion nodes

  The continuity constraints associated with junction node or diversion node j are

  $$ \begin{array}{cc}\hfill {\displaystyle \sum_{h_i=j}}{X}_{i,t}={\displaystyle \sum_{f_i=j}}\ w{l}_i{X}_{i,t}\hfill & \hfill \forall j\in \left\{{N}_C{\displaystyle \cup }{N}_T\right\}\hfill \end{array} $$

  (7)

  where X _i,t is the decision variable associated with arc i at time period t; N _C is the set of junction nodes; N _T is the set of diversion nodes; and  wl _i is the ratio of water loss associated with arc i.

• c. Demand nodes for public water supply or agricultural water use

  The constraints at demand node j for a public water supply or agricultural water use are

  $$ \begin{array}{cc}\hfill {\displaystyle \sum_{h_i=j}}{X}_{i,t}\le {D}_{j,t}\hfill & \hfill \forall j\in \left\{{N}_D{\displaystyle \cup }{N}_A\right\}\hfill \end{array} $$

  (8)

  where D _j,t is the demand for public water supply or agricultural water use at node j in time period t.

• d. Groundwater node for limitation of drawdown

  First, the groundwater level and net pumping rate should be normalized, and then they can be used as the input terms of the ANN. After computing the hidden and output layers, the groundwater level in the next time step H _j,t + 1 can be predicted after denormalization at each site. The calculation steps for the groundwater level are expressed as follows:

  $$ \begin{array}{cc}\hfill {X}_{l,t}=w{a}_l\times {\displaystyle \sum_{h_i=j}}{X}_{i,t}\hfill & \hfill \forall j\in {N}_A,{f}_l=j\hfill \end{array} $$

  (9)
  $$ \begin{array}{cc}\hfill {P}_{j,t}^N={\displaystyle \sum_{h_i=j}}{X}_{i,t}-{\displaystyle \sum_{f_i=j}}{X}_{i,t}\hfill & \hfill \forall j\in {N}_G\hfill \end{array} $$

  (10)
  $$ \begin{array}{cc}\hfill {\overset{-}{H}}_{j,t}=\frac{H_{j,t}-{\mu}_j^H}{\sigma_j^H}\hfill & \hfill \forall j\in {N}_G\hfill \end{array} $$

  (11)
  $$ \begin{array}{cc}\hfill \overset{-}{P_{j,t}^N}=\frac{P_{j,t}^N-{\mu}_j^P}{\sigma_j^P}\hfill & \hfill \forall j\in {N}_G\hfill \end{array} $$

  (12)
  $$ \begin{array}{cc}\hfill ne{t}_{k,t}={\displaystyle \sum_{j\in {N}_G}}{w}_{j,k}^H\cdot {\overset{-}{H}}_{j,t}+{\displaystyle \sum_{j\in {N}_G}}{w}_{j,k}^P\cdot \overset{-}{P_{j,t}^N}+{b}_H\hfill & \hfill \forall k\in {N}_H\hfill \end{array} $$

  (13)
  $$ \begin{array}{cc}\hfill {\overset{-}{H}}_{j,t+1}={\displaystyle \sum_{k\in {N}_H}}{w}_{k,j}\cdot ne{t}_{k,t}+{b}_G\hfill & \hfill \forall j\in {N}_G\hfill \end{array} $$

  (14)
  $$ \begin{array}{cc}\hfill {H}_{j,t+1}=\left({\overset{-}{H}}_{j,t+1}\times {\sigma}_j^H\right)+{\mu}_j^H\hfill & \hfill \forall j\in {N}_G\hfill \end{array} $$

  (15)

  where  wa _l is the ratio of agricultural infiltration associated with arc l; P ^N _j,t is the net pumping rate in the district controlled by observation well j at time t; and H _j,t is the groundwater level at observation well j at time t. ${\overset{-}{H}}_{j,t}$ and $\overset{-}{P_{j,t}^N}$ are the normalized groundwater level and normalized net pumping rate, respectively. μ ^H _j and σ ^H _j are the mean and standard deviation, respectively, of the groundwater level at each observation well j for the control station. μ ^P _j and σ ^P _j are the mean and standard deviation, respectively, of the net pumping rate in each district controlled by observation well j. net _k,t denotes node k in the hidden layer at time t. w ^H _j,k (resp. w ^P _j,k ) is the weight between input term j of the groundwater level (resp. net pumping rate) and node k of the hidden layer. w _k,j is the weight between node k of the hidden layer and the output term j. b _H is the bias of the hidden layer, and b _G is the bias of the output layer. N _H is the set of nodes in the hidden layers, and  N _G is the set of groundwater observation wells for the control stations.

  The constraints at ground water node j that describe the limitation of the drawdown are

  $$ \begin{array}{cc}\hfill {H}_{j,0}-{H}_{j,E}\le d\hfill & \hfill \forall j\in {N}_G\hfill \end{array} $$

  (16)

  where H _j,0 is the groundwater level at observation well j at the beginning of the time horizon at node j; H _j,E is the groundwater level at observation well j at the end of the time horizon at node j; and d is the limit of the groundwater drawdown.

• e. Physical constraints

  The physical constraints for arc i are

  $$ 0<{X}_{i,t}\le {X}_{i,t}^{max} $$

  (17)

  where X ^max _i,t is the upper bound associated with X _i,t .

The decision variables in the optimization model are the flows associated with the arcs and the unknowns associated with the ANN groundwater model. The optimization model is coded in FORTRAN and solved by the LINGO linear solver package, which uses the revised simplex method with an inverse product form. The time step of the optimization model is 10 days.

The system of water resources in the Chou-Shui alluvial fan is illustrated in Fig. 5. There are four nodes for public demand, including the industrial and public demand nodes in Yunlin County, the dry irrigation node, and the industrial demand node for the fourth stage of expansion of the Central Taiwan Science Park. In addition, there are eight nodes for agricultural demand. The surface water system is composed of the Wu and Chou-Shui creek systems. The source of the Wu creek system is the Mao-Lou creek and the Wu creek itself; the source of the Chou-Shui creek system is the Ching-Shui creek and the Chou-Shui creek itself. Each irrigation district has its own groundwater source.

Fig. 5

Water resources system of Chou-Shui alluvial fan

The weights of the surface water and groundwater in the objective function are 5.0 and 1.0, respectively. The capacity limitations of the Chi-Chi weir, the north-side water intake, the Dou-Liu channel water intake, and the Chou-Gang channel water intake are 108.0 cms (cubic meters per second), 77.0 cms, 17.4 cms, and 122.0 cms, respectively. The maximum limit of the Lin-Nei diversion water intake is 2.89 cms. The weight of the river infiltration is assumed to be 0.25. The infiltration weights of each irrigation district are shown in Table 1.

Table 1 Weights of infiltration arcs to groundwater modes

The 7-year record of hydrological and irrigational data from 2001 to 2007 is used in this study (Changhwa and Yunlin Irrigation Association 2008), and all inflow sources are estimated using data from the nearest hydrological station.

3.2.3 ANN Groundwater Flow Model

The input data generated by MODFLOW are separated into training and validation sets, and applied to the ANN groundwater flow model. The steps are as follows:

• a. Divide the management zones of the GMS and select observation wells.

• b. Randomly generate the pumping rate and infiltration rate of each zone, and then calculate the net pumping rates as MODFLOW inputs.

• c. Collect the groundwater level data at each observation well and the net zone pumping rates for ANN training.

• d. Separate the input data into a set of training data and a set of validation data.

The management zones of the GMS are shown in Fig. 2. This study selected the How-Shiu, Yuan-Lin, and Pi-Tou observation wells in Changhwa County, and the Gang-Ho, Hu-Shi, Hu-Wei, and Chiu-Chang observation wells in Yunlin County. Although there are four aquifers in the Chou-Shui alluvial fan, only two are included in the ANN groundwater flow model, because most pumping behavior is observed in the upper two aquifers. The groundwater levels generated by the ANN groundwater flow model are compared to determine whether the simulation results are acceptable. The time step of the ANN groundwater model is 10 days. The structure of the ANN groundwater flow model for the Hu-Shi observation well is shown in Fig. 6. A comparison of the head difference between the ANN and MODFLOW is shown in Fig. 7.

Fig. 6

Structure of ANN at Hu-Shi observation well

Fig. 7

Comparison of head differences between ANN and MODFLOW

4 Case Study

A previous study revealed that ground subsidence and layer compression were consistent with the variation of groundwater level (Liu 2004). The Chou-Shui alluvial fan has been designated as a groundwater control zone by Water Resources Agency (WRA), Taiwan. However, to maintain the productivity of agriculture and industry in that area, the groundwater overdraft condition continues. Therefore, the issue of groundwater management under drawdown constraints is important. Liu (2004) developed a decision support system for the management of groundwater resources in the Chou-Shui alluvial fan, and used MODFLOW to determine the permissible yield in this region. Lin et al. (2013) applied the water balance method to estimate pumping rates. The potential recharge zones were assessed based on the simulated recharge rates from Soil and Water Assessment Tool (SWAT). Chen et al. (2010) developed a linear programming model to maximize groundwater extraction subject to a minimum water-table elevation requirement for the prevention of saline intrusion. Based on Darcy’s law, a set of equations was derived to describe the groundwater flow regime. Until now, no literature has discussed a management model under different drawdown limitations with the aim of optimizing water distribution in the Chou-Shui alluvial fan. The physical-based MODFLOW is substituted with the ANN to simulate the hydraulic response at specific locations due to pumping behavior. Therefore, two scenarios are discussed and analyzed in this study. Case I is the scenario where the groundwater drawdown is limited, and Case II is the scenario where the water demand of the Central Taiwan Science Park (fourth stage) is satisfied under some limit on groundwater drawdown. The results from these two scenarios are as follows:

4.1 Case I

This study analyzes the results of different drawdown limits, as well as hydrological datasets of different lengths. Each sub-scenario is summarized in Table 2.

Table 2 Results of case I simulation (10⁴ m ³)

• A. Case I-1: This sub-scenario applies two different groundwater drawdown limits from 2002 to 2003, i.e., 1 m/year (I-1a) and 2 m for every 2 years (I-1b). The results of sub-scenarios I-1a and I-1b are presented in Table 2. The water shortage in I-1b is less than that in I-1a, because more groundwater is pumped in I-1b than in I-1a. As shown in Fig. 8, the groundwater level in I-1b is lower than I-1a at Gang-Ho, Hu-Shi, and Chiu-Chang at the end of 2002, because I-1a must satisfy the drawdown limit of 1 m/year. However, I-1b only satisfies the drawdown limit of 2 m every 2 years, which could lead to more groundwater pumping in the first year. In addition, both sub-scenarios face significant water shortage at the end of 2003 because the drawdown limit must be satisfied. The groundwater levels in both scenarios are the same at the end of 2003 to satisfy the drawdown constraint.

  Fig. 8

  

  Case I-1 head difference of observation well

• B. Case I-2: This sub-scenario uses hydrological data from 2001 to 2004 and two different groundwater drawdown settings, i.e., 1 m/year (I-2a) and 4 m for every 4 years (I-2b). The results of these sub-scenarios are shown in Table 2 and Fig. 9. The water shortage in I-2a is slightly less than that in I-1a, and there is no water shortage in I-2b because the supply of groundwater is abundant. However, a water shortage condition occurs in I-2a in 2002 and 2003 because of the drawdown limit.

  Fig. 9

  

  Case I-2 head difference of observation well

• C. Case I-3: This sub-scenario uses hydrological data from 7 years (2001–2007) and two different settings for the groundwater drawdown, i.e., 3 m/year (I-3a) and 1 m/year (I-3b). The results (Fig. 10) show that sub-scenario I-3a satisfies the demand in each 10-day period over 7 years, but will result in dramatic groundwater fluctuations. In contrast, sub-scenario I-3b will cause water shortages during dry periods. However, the groundwater fluctuations in I-3b are smoother than in I-3a.

  Fig. 10

  

  Case I-3 head difference of observation well

4.2 Case II

The Wu-Shi weir is the water intake of the Central Taiwan Science Park (fourth stage), scheduled to begin operation in 2015. Prior to 2015, the Changhwa Irrigation Agency are contracted to supply 160,000 m³ per day to the Science Park. The source of this water is the north-side channel of the Chi-Chi weir.

Case II-1 simulates the water resource conditions after the Changhwa Irrigation Agency supplies the required amount of water to the Science Park. In contrast, Case II-2 simulates the condition in which no water is supplied to the Science Park. Both cases satisfy the drawdown limit of 1 m/year, and both use hydrological data from 2002 to 2006. Table 3 presents the water shortage conditions from 2002 to 2006. Case II-1 has more irrigation water shortages than Case II-2. Therefore, supplying water to the Science Park will result in an additional yearly average water shortage of 25.2 million m³ for the Changhwa Irrigation Agency.

Table 3 Yearly water shortage in Case II (10⁴ m ³/yr)

Finally, the simulation results show that the optimization model of conjunctive use developed in this study could be a useful tool for simulating the large-scale water resources system in the Chou-Shui alluvial fan. Because no reservoir operates in this region, year-by-year calculations will produce the best optimization results. With 154,085 decision variables and 25,870 constraints over a 5-year period, the year-by-year simulation requires only 1 s with an Intel® T2400 1.83 GHz processor.

5 Conclusions

This study reports the development of an optimization model for large-scale conjunctive use of surface water and groundwater. The proposed model optimizes the distribution of a water resources system subject to the constraints of groundwater drawdown. The objective of the model is to maximize the water supply to public and agricultural demand nodes. The decision variables are the flows along all arcs and the unknowns associated with the ANN model. The groundwater flow in the Chou-Shui alluvial fan is simulated using MODFLOW, which is a physically based model that simulates the response to pumping or infiltration. The optimization model and groundwater flow simulation are linked by incorporating the ANN into the optimization algorithm. The ANN groundwater model is trained with data generated by MODFLOW, and then embedded into the optimization model to give the groundwater level constraints.

The methodology was applied to the Chou-Shui alluvial fan. Our objective was to optimize the distribution of surface water and groundwater under two scenarios: (1) consideration of different drawdown limits to prevent further land subsidence in this region, and (2) analysis of the effect on this region with and without surface water supply to the Central Taiwan Science Park (fourth stage). Compared to other scenarios, the fluctuation in groundwater level becomes considerably smoother under the scenario that permits a drawdown of 1 m/year. However, the water supply for irrigation or industrial demand will be compromised because of the limitation of groundwater drawdown. Under the drawdown limitation of 1 m/year, the water shortage increase by an average of 25.2 million m³ per year owing to the water usage contract with the Central Taiwan Science Park. The results show that the optimization model developed in this study can effectively simulate different limitations on groundwater drawdown, and can be a useful analysis tool for strategies related to conjunctive use and water resources management in the Chou-Shui alluvial fan.