INTRODUCTION
Groundwater management involves water extraction, its quality for irrigation and the optimal use for the welfare of agrarian community. Such considerations are confined not only to geological and hydrological but also to social aspects. In general, efficient water management includes cost effective groundwater pumping while maintaining the quantity and quality of water for agriculture and drinking. It is an admitted fact that periodic measurement of water level in irrigation wells is important in a particular area or district to determine the quantity of total water extraction and the aquifer flow capacity. This type of information is helpful for determining groundwater depletion, operational cost involved, pumping schedule, design of irrigation system and water management programs for optimizing the use of underground water aquifer and develop water use program for an aquifer. The water sedimentary basin is defined as a natural underground reservoir. However, pumping water from a sedimentary basin at certain sites may affect the quantity of water available to other sites within the basin if good governance and efficient water management program are not followed.
Many people believe that water well should be continuous and permanent productive
without interruption. But if water pumping is not properly scheduled and managed
it may lead to partially or complete depletion of the aquifer (Powers
et al., 1966; AlSomayien, 1989). The necessary
measurements include well discharge, groundwater level (static and dynamic)
and frequency of pumping water. In the absence of balance between water availability
and the consumption, more water is expected to be pumped from the aquifer which
needs monitoring of the unwanted wells. In fact, the water withdrawal at the
beginning of any agricultural project begins with a small number of productive
wells. But with the passage of time, new wells are drilled to meet the growing
demand of water for agriculture expansion, provision of drinking water to houses,
industrial uses and landscape development. Therefore, new rules and regulations
have to be formulated for drilling new wells to avoid unnecessary depletion
of aquifer. Also, the water extraction should not exceed the recharge phenomenon.
Because excess pumping of water may lead to aquifer depletion resulting in dryness
of the aquifer (AlSomayien, 1989). It is also important
to determine a balance between the water consumption and the rate of groundwater
pumping in an arid country like Saudi Arabia (AlSagabi,
1978). This could be attributed to unnecessary use of groundwater by pumping
from wells installed illegally in the agricultural areas of the Kingdom. It
is very difficult to use mathematical models of groundwater management under
wetland conditions and directly in dry areas. Therefore, safety and security
of an aquifer is very important for sustainability of an arid region of a country.
It demands to develop an applicable groundwater management program in an arid
zone with little recharge due to arid climate and low rainfall.
Fischer et al. (2011) observed that groundwater
levels in the capital Hanoi decreased dramatically. In order to manage, they
described the “state of the art” and the development of sustainable
solutions to maintain and increase the declined groundwater levels in Hanoi.
Kenabatho and Montshiwa (2006) stated that water is
an essential resource affecting many aspects of development and the natural
environment. They concluded that with the current fragmented, uncoordinated
institutional and legal arrangements in water resources management, there is
an urgent need to adopt integrated water demand management as envisaged in the
overall concept of Integrated Water Resources Management (IWRM). Segosebe
and Parida (2006) stated that water is one of the most important elements
essential not only to attain food and health security, but also for the economic
development of a country. They examined the various strategies in a semiarid
country like Botswana to manage the growing demand for water. These strategies
encompass the use of tariffs, water reuse/recycling and water restrictions.
Other attempts encourage water conservation through rainwater harvesting and
implementation of technological innovations with exploration of nonconventional
sources. Khadim et al. (2013) developed a mathematical
formula to assess the rate of tidal sedimentation due to Tidal River Management
(TRM) in parts of Khulna and Jessore districts. The study found the evidences
of considerable advancements in regional livelihood i.e., flood resistance,
cultivated lands, cultivable area, cropping intensities and food security due
to Integrated Water Resources Management (IWRM) approaches.
In order to regulate proper groundwater management, some initial and essential steps such as preliminary survey and geological field surveys are required in the study area. The changes in the groundwater level can be attributed either to the local aquifer flow itself or directly through the influence of pumping wells in the area. The configuration of sandstone aquifer in the NorthCenter are tangent to the rocks of Arabian shield of Saudi Arabia. Also, the support program defines the maximum permissible allowance as the amount of water that can be pumped from a well without affecting the aquifer and the well conditions. Therefore, water pumpage must be limited to the quantity of flow rate which should be less than or equal to the ideal flow. The current problem in the study area (Hail Agriculture Development Company, HADCO) is that wells are dug in advance and spacing between the wells is either horizontal or vertical direction.
Therefore, the main objective of this study is to calculate maximum permissible allowance, discharge and depletion in groundwater level. Also, to clarify the optimal quantity of pumping water using field data from the sandstone aquifer in the Saq formation southwest of Hail province, Saudi Arabia.
MATHEMATICAL MODEL FOR ADMINISTRATIVE PROGRAM
The Maximum Permissible Flow rate (MPF) is defined as the quantity of water
that can be extracted from a well without affecting the aquifer and the productive
well. So, pumping of water from a well should not exceed the optimal flow rate
in order to avoid excess costs, depletion of water level and the interaction
between the radius of influence of two adjacent wells. One of the major problems
in the study area (Hail Agriculture Development Co., HADCO) is that wells are
dug in advance and the distance between wells is either horizontal or vertical
in direction. Maximum Permissible Flow rate (MPF) is an unstable value depending
on the value of dry zone, space or distance between wells and physiochemical
properties of subsurface layer of each well. Therefore, the concept of maximum
water withdrawal should consider that the well is in good condition and is not
damaged from continuous use. Also, there is no effect from interaction between
the radius of influence and the cone of depression between adjacent wells. It
is also clear that the concept of radius of flow velocity of underground water
increases near the center of the well. This increase in flow velocity allows
the movement of colloidal particles from the aquifer around the well casing
thus increasing the pressure loss due to blockage of pump filter resulting in
high drawdown of water in the well (Ali et al., 1997).
Furthermore, mathematically the groundwater velocity can be expressed at any
point by the movement of groundwater zone through each section of the porous
material (solid part and pores) which is equal to the value of specific flow
rate (well pumping flow rate, Q) or called Darcy velocity to total crosssectional
area (A) depending on the flow direction as follows:
Where q is the actual flow velocity of groundwater to the crosssectional area
and is measured by the space of the porous material of aquifers. Therefore,
the actual velocity of groundwater is much more than velocity of Darcy (Nonner,
2003).
The relationship between the actual velocity (V_{a}) and velocity of Darcy (V) is as follows:
where, A_{cap} is the sum of cross sectional area of the capillary tubes. The porosity (n) is (A_{cap}/A).
Equation 2 can also be written in another form, after dividing the drainage capacity on porosity (n), as follows:
where, A_{cap} is the crosssectional area close to the well in the
aquifer. From physical point of view, the velocity near the center of the well
depends on the distribution of particle density of aquifer and the hydraulic
conductivity than other hydrological factors. All these factors can be obtained
by estimation using the empirical correlation. This correlation is based on
the observation from a numbers of wells. Driscoll (1986)
observed a strong relationship between the hydraulic conductivity (K) and the
depletion of ground water. This relationship can also be expressed empirically
by using approach velocity (V_{a}) equation with the introduction of
safety factor according to Huisman (1972). This equation
can be written as follows:
Equation 3 gives accurate measurement of hydraulic conductivity.
This can be verified by pumping test or by the measurement of specific flow
rate of each well. However, the other alternative approach is Logan
(1964) method that was used in the present study for calculating the conductivity
coefficient. After that, this coefficient was converted to hydraulic conductivity
by dividing the conductivity coefficient by the thickness of aquifer. Equation
3 can be rewritten to determine the Maximum Permissible Flow rate (MPF)
as follows:
where, r_{wi} is the radius of irrigation well, K_{i} is the hydraulic conductivity, b is the thickness of the ground water reservoir and n is the number of wells in the study area.
Besides, implementation of the second phase of water management program requires interaction between the drawdown and rate of pumping water from a well at a critical level of 5%. In fact, if water is pumped from two or more adjacent wells from the same aquifer layer, it shows interaction between the drawdown and flow rate of well resulting in more drawdown in groundwater. This interaction between radius of influence among the wells will cause high water level depletion in many wells. As the aquifer in the study area is of closed type, so the flux equation in the steady state for any well can be written as follows:
where, S_{wi} is the level of drawdown in irrigation well, R_{i}
is the effective well Radius, T_{i} is conductivity coefficient, the
value of the conductivity coefficient can be expressed by T_{i}. Equation
5 is the vertical distribution in wells without receiving any recharge from
surface sources AlNaeem (1999) conducted studies that
used the effective well Radius as follows:
By substituting Eq. 6 in Eq. 5 we obtained:
After rearranging Eq. 7, the value of the maximum permissible drawdown in wells can be obtained as follows:
All the variables in Eq. 8 are known except the value of
water drawdown (S_{wi}) in the wells. Equation 8 can
be used to calculate the maximum permissible drawdown for each well. The main
advantage of the last method is that it does not consider the distance between
the wells. By this way it can be converted to a series of effective well radius
(R_{wi}) by using Eq. 6. Also, it is possible to determine
the distances between the adjacent wells from the well map. Assuming that D_{ab}
indicate the distance between two wells close to each other at location a and
b and there are many unknown wells (n). Which means that there are many wells
with different distances and interactions with each other. It can be calculated
from n (n1)/2. Each distance must be equal or more than the summation of the
effective radius of two close (adjacent) wells. This close distance can be expressed
as follows:
Which represents the algebraic distance in the administrative program. At the critical level Eq. 9 can be rewritten as follows:
where, α is the critical level. It is clear that if a critical level is equal to zero (this means that there is no interaction between the wells), Eq. 10 is converted to Eq. 9 after obtaining the effective radius of well. It has to be satisfied for each two wells using Eq. 10. If the algebraic distance in Eq. 10 is unsatisfied, the value of α will be equal to 0.05. In this case, the value of effective radius of a well has to be reduced until the same value of algebra distance is obtained in Eq. 10 and this value is satisfied. Therefore, the value of maximum permissible drawdown in well water can be calculated as follows:
By this method, all the adjacent wells were investigated to obtain final values of water drawdown and the radius of wells. Hence, the permissible groundwater flow velocity can be calculated as follows:
According to the method of Theis (1935), the values
of aquifer properties such as hydraulic conductivity (K), water layer storage
coefficient (S) and permissibility factor (T) can be estimated. The value of
(T) can be obtained from water pumping experiments. Theis
(1935) method can be derived from the following equation:
Equation 13 can be written in terms of radial coordinator
as follows:
In 1935, Thiees created a solution to the differential Eq. 14 for unstable flow in two directions based on the symmetry between the groundwater flow and thermal conductivity as follows:
where, the value of u is:
Whereas, s = (h_{o}h) is the level of drawdown (meter) at any point
which is the distance from the observation well to the pumping well at a steady
flow rate, Q represents the well discharge in m^{3} per unit time (L^{3}
t^{1}), r is the radius of influence which is the distance between
the pumping well to the observation well. T is the starting time from the beginning
of pumping.
MATERIALS AND METHODS
The development and implementation of the mathematical model program for the management of groundwater in Saq sand aquifer was carried in the southeast part of the Hail northern region of Saudi Arabia (Hail Agriculture Development Co., HADCO). The study area is located between longitudes 42°39'43°E and width of 27°16'27°23' N. It is about 120 km from the province of Hail. The study area covers an area of approximately 350 km^{2} and is rectangular in shape (Fig. 1).
The land topography of Hail is 9801000 m above sea level. In this study, the wells were distributed in the study area into a small area as horizontal lines and were named A, B, C, D, E, F, G, H, J, K, L and M and the vertical lines from 112 represented the names of wells by the interaction of the horizontal line with the vertical lines, such as (A1, A2, ...etc). as shown in Fig. 2.
A total of 193 wells were selected from the study area. The location of wells
and distribution in the study area is rectangular in shape (Fig.
2). The average horizontal distance between the wells is 1641 m while the
average vertical distance is 821 m. The average depth of wells is 509 m in the
selected study area. The groundwater levels ranged from 6171 m in 1982, 6869
m in 1983, 144168 m in 1997/1998 and 168 and l86 m in 20112012 below the ground
surface.

Fig. 1: 
Location of the study area in Hail region, Saudi Arabia 

Fig. 3: 
Geological layer series of Saq sand aquifer in the Arabian
shield of Saudi Arabia (Powers et al., 1966) 
The total No. of wells used in the study were 117 representing about two third
of the total number of wells in the whole area.
The study area consist of a sedimentary layer of quaternary age and covers most of the area. These deposits consist of silt, sand stones and gravel layers with a thickness of 215 m. It is known that sandstone layers of Saq aquifer cover large area of the region. The Saq sand layer composed of the coastal lower part of the Tabuk layer with a thickness of 800 m and mostly consists of sandstone. The bottom layer of Tabuk aquifer is separated from Saq sand aquifer by Alhanadr clay which is impermeable with approximate thickness of 20 m as given in Fig. 3.
The thickness of Saq sand aquifer in the study area is identical. The aquifer
is horizontal and its eastward slope is about 0.01. The selected location of
wells in Saq aquifer is from a well in location A1 to the well in location G9.
The total number of wells in the study area were 68 and mostly used for agricultural
purposes. The groundwater level was measured through the observation wells.
The pumping experiments were carried to a constant flow rate and recovery of
well located in G5 and G7 locations where the water level drawdown was monitored
with time. Then, Theis (1935) method was applied to determine
the aquifer properties namely hydraulic conductivity (k), coefficient storage(s)
and permissible coefficient (T) using Groundwater For Windows (GWW) computer
program. The results of pumping tests were analyzed to estimate the values of
different properties of aquifer. The values of different aquifer properties
are presented in Table 1. Besides, Surfer program was used
to draw contour maps for various aquifer properties.
Data analysis: Data were analyzed by ANOVA and regression techniques
for treatment evaluation at 5% level of significance according to SAS
(2001).
RESULTS AND DISCUSSION
Mean values of permeability (T, M^{2}/tor) was 1862.5 and 3475.9 for
wells located in G5 and G7, respectively (Table 1). Mean hydraulic
conductivity (K, m day^{1}) was 5.39 and 4.05 for wells in G5 and G7
locations, respectively in the study area. Whereas, the storage factor (S) was
0.0025 and 0.0029 for wells in G5 and G7 locations, respectively. These values
were determined by well pumping tests in the study area. The maximum permissible
velocity was calculated using Eq. 12 and ranged between 2.17.7
m day^{1}.
Data in Table 2 show the mean maximum permissible flow velocity
(MMPFV) for different locations in the study area. The values of MMPFV came
to 5.06, 4.65, 3.48, 4.70, 4.56, 3.92, 5.23, 4.38 and 5.06 m day^{1}
for J, H, G, F, E, D, C, B and A fields, respectively in study area. The results
indicated the aquifer homogeneity with respect to water storage characteristics
as all these values are very close and the difference among these seems to be
insignificant.
Table 1: 
Different properties of subsurface reservoir calculated by
Theis (1935) method 

Table 2: 
Mean maximum permissible velocities in the study area (m^{2}
day^{1}) 

Table 3: 
Mean values of actual flow rate, maximum permissible flow
rate and residual flow rate for the study sites in the study area (m^{3}
sec^{1}) 

While looking on the values of velocities, it infers that losses through aquifer
cracks is bare minimum according to Driscoll (1986).
It is also expected that under these aquifer conditions, the rate of scaling
and corrosion of well casing is minimum.
A contour map was prepared for the values of maximum permissible velocities for the wells in the study area (Fig. 4). The contour map shows that local variation in the maximum permissible velocity ranges between 2.4x10^{5} to 8.89x10^{5} m sec^{1}. Based on the distance between contour lines, values of the maximum velocities for a particular area in Saq sand aquifer were used for all the agricultural related activities especially the irrigation in the study area. The actual flow rate of wells ranged between 0.0480.049 m^{3} sec^{1}. While, the values of Maximum Permissible Flow rate (MPF) calculated by using Eq. 7 ranged between 0.0070.017 m^{3} sec^{1}.
Table 3 shows the mean values of actual flow rate, maximum permissible flow rate and residual flow rate of wells in the study area. The ranges of different flow rates (m^{3} sec^{1}) were 0.0410.085, 0.0070.017 and 0.0480.098 for residual flow rate, mean maximum permissible flow rate and mean actual flow rate, respectively for all the investigated wells in different study sites. These values were calculated to determine the quantity of groundwater available and to know the change in the volume of this water with time. In other words, on permanent basis, the quantity of water available and the actual characteristics of the subsurface Saq aquifer tied with the possible side effects during pumping.
Data in Fig. 5, 6 and 7
represent the contour maps of distributions of maximum permissible flow rate,
actual flow rate and the residual flow rate, respectively.

Fig. 4: 
Contour map showing the maximum permissible velocity in the
study area (m day^{1}) 

Fig. 5: 
Contour map shows the maximum permissible flow rate of wells
in the study area (10^{2} m^{3} sec^{1}) 

Fig. 6: 
Contour map shows the actual flow rate of wells in the study
area (10^{2} m^{3} sec^{1}) 
The values of actual flow rates shown in the contour map were estimated based
on the actual field data. It was noticed that the values of actual flow rate
measured in the field were higher than the calculated values of Maximum Permissible
Flow rate (MPF).
This point is clear from the range of actual flow rate and the maximum permissible
flow rate when compared with the average flow rate in the study area as given
in Table 3. It can be concluded that actual flow rates of
wells in the study area are higher than the maximum permissible flow rate by
about 90%. Also, the actual flow rates for two wells is more than the optimal
production of wells in the study area. The residual flow rate is the difference
between the actual flow rate and the maximum permissible flow rate. From the
study results, it was found that the residual flow rates of wells tend to change
from one location to another location as shown in Fig. 5 and
Table 3.

Fig. 7: 
Counter map shows the residual flow rate of wells in the
study area (10^{2} m^{3} sec^{1}) 
Table 4: 
Values of mean maximum permissible drawdown of some wells
in the study area 

The results in Figures and Tables explain the values of the residual flow
rates with the maximum quantity of groundwater and justify the maximum permissible
flow rate. In fact, unnecessary consumption of groundwater can be seen from
Table 3 and is likely to affect the aquifer water storage
in future. Consequently, this will increase the slope of the pizometeric water
level in future thus making it impossible for the hydraulic conveyance of the
aquifer to recharge the aquifer flow rate in the area. Similar views were reported
by Qahman et al. (2005). Who reported that water
problems occur when excessive pumping at certain individual wells lowers the
potentiometric surface locally and causes upconing of the interface between
fresh water and saline water. They researchers investigated the optimal and
sustainable extraction of groundwater from a coastal aquifer under the threat
of seawater intrusion. The physical model is based on the densitydependent
advectivedispersive solute transport model.
The study of the contour maps (Fig. 5, 6
and 7) show that if the counter line is high, this means that
the actual flow rate is high. Therefore, it is easy from the counter maps to
determine the high and low flow rates of wells in the study area which is verified
from the distance between different counter lines.
Lastly data in Fig. 7 show the simulation of counter maps for actual flow rate and the maximum permissible flow rate of wells in the study area. The intersection of contour lines for the two maps at any point represents the residual flow rate. The study of a similar map indicates that if the values of the counter lines are positive, then the actual flow rate is more than the maximum permissible flow rate of wells in an aquifer.
The above classification for the actual flow rate, the optimum flow rate and the residual flow rate gives an indication of good administrative process for drawing the counter map of groundwater pizometric level during the irrigation interval in the irrigated agriculture project in the study area. In general, the water consumption is less than recharge in the study area. It is an admitted fact that when water is pumped from a well, the water stored is consumed around the well resulting in decreasing the piezometric water level. Above all, data of groundwater piezometric level such as vertical flow and horizontal flow is required for calculating the capacity of an aquifer. Because, this data is an important part of input data in developing the mathematical model in the computer. The mathematical model, so developed, will be used to predict the effect of the recent pumping flow rates on longterm basis and to draw optimal plan of future groundwater development.
The maximum permissible water drawdown was calculated from Eq.
10 and presented in Table 4. Table 4
shows that the mean maximum permissible water drawdown was 4.65 m which is the
drawdown required to produce the desired flow rate partially for possible determining
the natural hydraulics of the aquifer. Also, it will further help to develop
and design productive wells. Besides, if the drawdown in the well is greater
than the maximum permissible drawdown, then disturbance occurs in the flow rate
of wells. The tendency of specific capacity of a well is gradual when the increase
in the flow rate is above the Maximum Permissible Flow rate (MPF). Moreover,
the intersection occurs between the effective radius of counters. Data in Fig.
5 shows the counter map of maximum permissible flow rate. The data in Fig.
5 can help to observe local variation and to estimate the maximum permissible
drawdown in well water at any desired point in the study area. The study findings
agree with those of Qahman et al. (2009) applied
two multiobjective management models in a coastal aquifer to maximize the total
volume of water pumped, minimizing the salt concentration of the pumped water
and controlling the drawdown limits on a part of the aquifer with 9 existing
pumping wells located at various depths. The study showed that the optimum pumping
rate is in the range of 2634% of the total natural replenishment and the proposed
technique is a powerful tool for solving this type of management problems. Soni
and Pujari (2010) analyzed the hydrochemical data of groundwater samples
of three different limestone mine sites which are in close proximity and covers
a tract along the Gujarat coast of Indian peninsula. They found sea water intrusion
in the coastal aquifer in the study area and recommended measures for sustainable
use of groundwater by the mining companies and other stake holders.
CONCLUSION
A mathematical model was developed consisting of easy and simple steps to formulate a procedure for aquifer management in dry regions. Since the aquifers represent the national wealth of Saudi Arabia, therefore this program was designed to manage the water wells located at irregular distances. The variables for the mathematical computer program include flow velocity and the maximum permissible drawdown. The developed program is useful to satisfy the principal variables such as hydraulic conductivity of aquifer related to each well location, well diameter, distance between wells and the aquifer thickness. The effective administrative program was executed without using local continuity equation. The application of this computer program requires actual field data including the properties and composition of Saq aquifer formation. Overall, the variables studied were very effective for the establishment of aquifer administrative program for efficient utilization in agriculture.
RECOMMENDATIONS
In order to apply a simple and effective aquifer management program, the following points need consideration:
• 
Monitoring of aquifer water level by using a simple and sophisticated
administrative rules as suggested in the study 
• 
The interaction between the radius of influence of wells must be minimized
and should be less than 5% for introducing aquifer administrative program 
• 
To avoid any problems in the pump filters and the stones around the well
casing, the aquifer velocity must be less than the maximum permissible flow
rates 
• 
It is required to modify the actual flow rate and should be equal to the
maximum permissible flow rate as suggested in this study 
• 
Additional requirements such as quantity of water pumped and its actual
use must to be compared or matched for better utilization of aquifer 