The model is based on the numerical solution of the partial differential equations that describe the water movement in a confined or unconfined aquifer

Ground water management by mathematical models Christos Babajimopoulos and Sofia Kavalieratou

Department of Hydraulics, Soil Science and Agricultural Engineering School of Agriculture

Aristotle University of Thessaloniki Thessaloniki, Greece, 54124 babajim@agro.auth.gr, kavalier@agro.auth.gr Abstract

Mathematical models, when properly calibrated and verified, can provide a very useful tool for the sustainable groundwater management. Many commercial models are available. The easiness of obtaining and using these models, very often leads to erroneous and dangerous results. This happens because many times the people who use these models are not familiar at all with their mathematical background and the required procedure for their calibration and verification.

In this paper a mathematical model, which has been developed in the Laboratory of the General and Agricultural Hydraulics and Land Reclamation of the School of Agriculture, is presented. The model is based on the numerical solution of the partial differential equations that describe the water movement in a confined or unconfined aquifer. The model can accept any type of boundary conditions and any type of recharge/discharge scheme. It is also accompanied by an automatic calibration routine. It has been applied at the aquifer of Pieria, Greece, with very good results.

Keywords: Ground water, water management, modeling Introduction

Irrigation in Greece is, by far, the greatest consumer of fresh water. Almost 85% of the consumed fresh water in the whole country covers irrigation needs. In connection to that, in many areas, ground water constitutes the main source of irrigation water. However aquifers are usually exploited without any real sustainable management plan, withdrawing water from them over the allowed renewable amounts. This results to a decrease of the ground water surface, which on many occasions has reached an alarming level. For example in some areas of the Thessaly plain the decrease of the ground water surface has exceeded 40 meters in the last 20 years.

The advent of computers and numerical methods has resulted in a very extensive use of mathematical models for the prediction of aquifer responses to changes in stresses. With these models complex questions such as quantitative studies of groundwater supplies, prediction of piezometric surface levels resulting from pumpage, optimum design of wells etc. are no longer hampered by inadequate or cumbersome methods of analysis. There are several types of mathematical models, very well summarized by Heijde et al. (1985). Typical model applications can be found in Brodie (1999), Dufresne and Drake (1999), Varni and Usunoff (1999), Ramireddygari et al. (2000), Reeve et al. (2000), Sophocleous and Perkins (2000), Abdulla et al. (2000), Batelaan et al. (2003).

In this paper the mathematical model that has been developed in the Laboratory of the General and Agricultural Hydraulics and Land Reclamation of the School of Agriculture, is presented. Special emphasis is given to the automatic calibration procedure and to its application to Pieria aquifer.

The mathematical model

The model is based to the numerical solution of the equations, which describe the water movement in a confined or unconfined aquifer. These equations are the following:

Confined aquifer

(1)

Unconfined aquifer

(2)

where Τx Τy are the transmissivities of the aquifer (L²/T), h is the piezometric head (L), q′(x,y,t) is a sink-source term (L/Τ), S(x,y) is the aquifer storativity, Κx, Κy are the hydraulic conductivities of the aquifer (L/Τ) and Η is the aquifer thickness (L).

Equation (1) and (2) are solved by the alternating direction implicit method (Babajimopoulos and Terzidis, 1980, Babajimopoulos, 2002). According to this method the solution proceeds in two steps:

1st

step

2 /

, ,

,

12 ¹₂

t h S h

y q T h y x

T h x

n j i n

j i j i n

y n

x

∆

= −

⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛

∂

∂

∂ + ∂

⎟ ⎠

⎜ ⎞

⎝

⎛

∂

∂

∂

∂

^{+ +}

(3)

2nd step

2 /

12

, 1 , , 2 1

1

t h S h

y q T h y x

T h x

n j i n

j i j i n

y n

x

∆

= −

⎟⎟ +

⎠

⎜⎜ ⎞

⎝

⎛

∂

∂

∂ + ∂

⎟ ⎠

⎜ ⎞

⎝

⎛

∂

∂

∂

∂

^{+ + + +}

(4)

where the superscript n denotes time step and the subscripts ί and j denote the nodal point indexing in x- and y- direction respectively.

Equations (3) and (4) result in tridiagonal systems of equations, which are solved by the Thomas algorithm (Smith, 1975).

Automatic calibration

A mathematical model can be calibrated by trial and error or by an automatic method. The first method is very time consuming requiring special sensitivity analysis of all the parameters involved in (1) and (2). Automatic methods are either

“direct” or “indirect”. The direct approach treats the model parameters as dependent variables. The indirect approach is an iterative procedure, which improves an estimate of the parameters until the model response is “sufficiently close” to that of the real system. In this work an automatic calibration procedure utilizing the indirect method of Rosenbrock with constraints is used.

The parameters which are calibrated are: the hydraulic conductivity of the phreatic aquifer in x and y direction, the transmissivity of the confined aquifer in x and y direction, the storage coefficient and the inflows from the lateral boundaries.

The problem is solved by minimizing the objective function F:

( ) ( )

[ ]

∑

∑ ∑

=

= =

⎟ ⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

=

m

m

N 1 n

n N

1 n

Nn 2 1

k c

n m k n k

N h h

F

(5)

where

( ) ^h

ⁿ_{k m}^and

( ) ^h

ⁿ_{k c} is measured and predicted piezometric head in each piezometer (k), Nm is the number of monthly piezometric head measurements, Nn is the number of piezometers used in the nth measurement.

Optimization of the objective function is achieved through a searching procedure, in which constraints of the hydrological parameters are taking into account.

Generally a parameter PM must obey the following constraint:

GM ≤ P_M ≤ H_M , M = 1,2,…,N (6)

where GM and HM denote the allowed minimum and maximum value of the parameter.

Application of the model

The model is applied to Pieria aquifer, Greece. The Pieria aquifer expands to about 20 Κm north of the Olympus Mountain and is surrounded by the Pieria hills to the west and the Thermaikos Gulf to the east. The total area of study is 207 km². Esson, which is a small river, the springs of which are located in the Pieria hills, divides the area of study in two parts, while Varikou and Litochorou drainage canals in the southern part of the area collect the water of the Dion springs and surface runoff. Α small part of the aquifer in the southern part of the area of study is unconfined, while overlying impervious formations of clay and sand in the northern part confines it.

There are more than 2000 private wel1s in the area of study but they are not evenly distributed throughout the whole area. While they are clustered in some parts of it

they are completely missing in others. Α total of 44 of these wells were used to monitor the water levels from September 1992 to August 1994 (Terzidis and Babajimopoulos, 1995).

A time step of 1 day and a 700 m square mesh (Figure 1) was used for the computation. Figure 1 shows the area of study and the grid system. The boundary between confined and unconfined aquifer is also shown in that figure.

Boundary conditions are: specified inflow, impermeable boundary and specified piezometric head boundary. Recharge and discharge of the aquifer is of critical importance. Recharge from the Olympus and the Pieria hills boundaries is estimated to be 65x10⁶ m³ the first year and 160x10⁶ m³ the second year. Direct recharge to the unconfined part of the aquifer is not very important, since the ground water table is more than 50 m from the surface and is computed by a mass balance approach (SOGREAH, 1979). The aquifer is discharged into the Thermaikos gulf at the

eastern boundary, by the Dion springs and by pumping wells. Since the quantity which discharges into the sea is not known, a known head boundary condition across the boundary to the sea is used. The head here was monitored throughout the whole year by piezometers across the beach. Pumpage is estimated by considering agricultural practices in the area of study and by using the Penman method.

Climatological data are collected in certain meteorological stations in and around the Pieria plain.

The model was calibrated by dividing the area of study in 15 zones (5 zones in the the phreatic and 10 zones in the confined part of the aquifer). It was assumed that in each one of these zones the hydraulic parameters had a constant value. The following parameters were optimized: x- and y- hydraulic conductivities of the freatic aquifer, x- and y- transmissivities of the confined aquifer, aquifer storativities of both the aquifers and boundary inflows from Olympus and Pieria hills.

Results

Figures 2 and 3 show the measured piezometric head contours at the 22 of March and the 23^rd of September of the year 1993. The piezometric head contours, which were computed during the calibration process, are also shown in the same figures. It is pointed out that the calibration process was terminated when the difference between two subsequent values of the objective function was smaller than a convergence criterion which was set equal to 10^-9. Convergence was reached after 4,058 iterations.

piezometric head contours

predicted measured

Figure 2. Predicted and measured piezometric head contours (22-3-93).

piezometric head contours

predicted measured

Figure 3. Predicted and measured piezometric head contours (23-11-93).

The calibrated model was validated by comparing computed and measured heads at several wells in the area of study. Figures 4 and 5 show two typical outcomes of the validation process. It is pointed out that the mean absolute difference between computed and measured values varies between 0.54 and 1.31 m and RMS values varie between 0.70 and 1.62 m.

0 2 4 6 8 10 12 14 16 18

23/9/1992 20/10/1992 24/11/1992 21/12/1992 25/1/1993 24/2/1993 22/3/1993 27/4/1993 25/5/1993 28/6/1993 26/7/1993 25/8/1993 21/9/1993 25/10/1993 23/11/1993 20/12/1993 25/1/1994 24/2/1994 27/3/1994 25/4/1994 30/5/1994 28/6/1994 27/7/1994 26/8/1994

piezometric head (m)

measured computed piezometric

head (m)

Figure 4. Well No 202 - Measured and computed piezometric head

0 1 2 3 4 5 6 7 8 9

23/9/1992 20/10/1992 24/11/1992 21/12/1992 25/1/1993 24/2/1993 22/3/1993 27/4/1993 25/5/1993 28/6/1993 26/7/1993 25/8/1993 21/9/1993 25/10/1993 23/11/1993 20/12/1993 25/1/1994 24/2/1994 27/3/1994 25/4/1994 30/5/1994 28/6/1994 27/7/1994 26/8/1994

piezometric head (m)

measured computed piezometric

head (m)

Figure 5. Well No 178 - Measured and computed piezometric head Summary and conclusions

Α mathematical model was developed to predict the groundwater response to various withdrawal rates in the Pieria aquifer, Greece. The model is based on the numerical solution of the two-dimensional equations describing groundwater flow in unconfined and confined aquifers. The initiative for the development of the model was the construction by the Greek Ministry of Agriculture of a sprinkler irrigation system covering the Pieria plain. The model was calibrated by an automatic process using data collected from 1^st September 1992 to end of August 1994.

The calibrated model was also validated with data collected in the same period. Both calibration and validation process are considered very satisfactory. Therefore the model can constitute a very good tool for the ground water management of the Pieria aquifer.

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