Mathematical Modeling of Bara Groundwater System Central Sudan

Groundwater has proved to be a major resource in Northern Kordofan State (NKS) in the development plans of water supply, irrigation as well as industrial sectors. Drinking water supply in the state relies on groundwater for more than 65% of the total consumption. In Bara Basin, the Cretaceous sediments extend underlying the Umm Ruwaba sediments, which are tapped by Abu Tenetin and Umm Sayala boreholes at the northern periphery of the Bara Basin (RRI, 1990). The main purpose of the modeling exercise was to assess the potential of the aquifer to satisfy the Bara well field needs. This designed model has predicted the possible drawdown for 25 years due to planned groundwater development in the project area. Other specific objectives were achieved such as: water balance calculation of the Bara basin (i.e. recharge from rainfall, leakage between upper, middle and lower aquifers, flow across the boundaries of the aquifers, water released from storage in the main aquifer due to heavy pumping), in addition to the prediction of the reactions of the hydraulic system due to groundwater withdrawal, and final determination of the safe yield of the basin.


Introduction
Bara and Umm Ruwaba Provinces are located in Northern Kordofan State. The two provinces are considered as the most important provinces as far as the natural resources are concerned [1]. The area lies within the Eastern Kordofan Groundwater basin. The Bara Basin lies in the lower right corner of the North Kordofan State between lat. 12° 30' and 14° 10' N, and long. 29° 50' and 31° 30' E, covering an area of around 58,250 km 2 . The model is covering an area of 136,330 m length and 62,080 width m ( Figure 1).
It extends over two climatic belts, corresponding to the two major E-W trending climatic zones in sub-Saharan region. The northern zone represents a semi-arid zone and the southern zone represents a poor Savannah one. The rainfall ranges and increases from 200 mm/y to 450 mm/y southward. The mean temperature is 27°C with 10°C and 46°C temperature extremes.
The area is generally an undulating plain of low relief with an average altitude ranging from 396 to 543 m above mean sea level but the ground surface slopes gently to the east and northeast towards the White Nile. The major drainage system in study area Khor Abu Habil, Wadi El Ghalla , Wadi El Milk , Wadi El Mirikh , Wadi El Muqaddam and Wadi El Hamra . Three major drainage basins can be identified as the White Nile, and the river Nile separated by a major water divide running E-W through Jubal Kordofan and Abu Zabad.
Evapotranspiration in the study area exceeds by far annual rainfalls resulting in average annual water deficits exceeding 1900 mm/year. During the rainy season when potential evapotranspiration diminishes to 10-20 mm/day and daily rainfall exceeds that amount, rainfall can generate runoff [2]. For Bara town the long term average annual rainfall  amounts are shown in figure 2.
There are no perennial streams in the region. Runoff from torrential rains forms a number of seasonal streams with irregular, short duration flows in July-August, lasting for few days or hours. Runoff is affected by rainfall intensity, topography, duration, amount, catchment areas, climatic conditions, geologic formation, and land use systems.
A mathematical model based on correlation of long term Umm Ruwaba rainfall and Abu Habil runoff, suggest with 80% probability, that the average annual flow is about 87.6 Mm 3 . Wadi Tagerger catchment area amounts to 3,800 km 2 ; discharge varies from 6 to 17 Mm 3 and average to 11.9 Mm 3 and that of Kajarr averages to 11.6 Mm 3 .

Geology and Hydrogeology
Stratgraphically, the area is composed of Mesozoic to Tertiary intercalations of continental origin varying in clastic and organic content related to the environment of deposition. The rifting phase leads to episodic variations in basin subsidence that influence the stratigraphic evolution of the study area. Two main lithostratigraphic units have been established in the Bara basin: Umm Ruwaba and Nubian formations. Rocks of Palaeozoic (Devonian-Carboniferous) to Precambrian age have been recognized in the drilled wells [3]. The groundwater occurs in the Cretaceous (Nubian sandstone) formations and the Tertiary (Umm Ruwaba) formation. The basement complex is divided into three members ranging from weather, fracture and crystalline rocks, also groundwater recognized in the drilled wells in weather/fracture basement rocks.
The general lithological log from the surface down to ~49 m depth consists of superficial deposits, fine to medium grained sand and clay, underlain by Umm Ruwaba formation down to 181 m depth, which are mainly clays, clayey sands and sand intercalated. Gravels and pebbles are frequent in sandy layer, while in clay calcareous nodules are abundant.
The main aquifer appears at depths below 181 m, composes of light gray to whitish sand ill sorted, gravelly and in many interbred, and in many intervals pebbly. It is intercalated with clayey interbred, and it is exceptionally clean, with high penetration rate from 265 to 338 ft. The base of the aquifer is at the depth of 480 ft. From 580 to 600 ft there are gray-green clays with intercalation of sand layers and it is not possible to separate an aquifer of good hydrological characteristics. The Nubian sediments extend underlying the Umm Ruwaba sediments of Bara-Umm Ruwaba basin. Nubian sediments are tapped by Abu Tenetin and Umm Sayala boreholes at the northern periphery of the Bara-Umm Ruwaba Basin.
Umm Ruwaba sediments are considered as the second important aquiferous zone, in NKS, filling the Bara-Umm Ruwaba basin and El Obeid trough [4]. Umm Ruwaba sediments are characterized by rapid facies change both lateral and vertical extends. Groundwater occurs in the sand lenses intercalated with mostly thick clay sediments and with medium hydraulic parameters with an average and less permeability of less than 3 m/D and transmissivity of 500 m 2 /D, and storativity of around 5%. The axis of the graben are filled with coarse and thick sands with high hydraulic parameters and a borehole yield of up to 100 m 3 / hour was reported from Bara and Umm Ruwaba town's wells. The water flows in general from the southwest to northeast direction.

Mathematical Modeling
The mathematical modeling is carried out for both stationary and transient groundwater flow. A model concept is first prepared based  on prior estimates of aquifer parameter is provided from the field measurements and hydraulic testing. Then a steady state model is calibrated to resemble an aquifer current state before introducing the planned development. Finally, the calibrated model is used to predict the consequences of 20 years well field abstractions. The hydraulic parameters and their mathematical relations are well studied by Chiles et al. [5], D`Agnese e al [6], and Todd et al. [7]. g. The flow system in the area is predominantly SE flow pattern. The piezometric levels described in figure 4 were used as starting hydraulic head for the steady flow model.

Model Description
• The Aquifer type is considered as confined deep aquifer governed by Poisson mathematical model as in eq.1 Franssen et al. [8], and unconfined upper aquifer governed by the flow equation (Eq.2) Fetter [9] below.
Boundaries described by three types of boundary conditions, namely: • 1st kind boundary condition (specified head) represented by the equation at the northern border • 3rd kind boundary condition (flow transfer/ reference hydraulic head) is assigned along the Nile and modeled by the • 4th kind boundary condition (single well type/ point source) is modeled by the equation

Results of Simulation Hydraulic parameters
Hydraulic parameter estimation is conditioned on head measurements. However, in the Bara case more than one parameter is unknown. The unknown parameters include the hydraulic properties T and S, the boundary conditions, the pumping rates, the areal recharge or vertical leakage entering or leaving the aquifer through boundary  aquitards. Therefore, the zonation method for parameter estimation is considered superior to other approaches due to limited and poor quality data. It enabled our interference and personal judgment. Zonation of hydraulic conductivity as shown in figure 5 below was however controlled by the few available test results, as well as by estimates from previous studies in the area. Another option for parameter optimization is the use of geo-statistical methods, which can hardly give better results with the number of data points available.

Steady state model
The steady state flow simulation is performed using the MODFLOW software package [10]. Figures 6 and 7 shows the simulated piezometric levels at steady state flow condition in both the upper and the lower aquifer with an overall scatter variance for the regional model within the range of the measured groundwater level variations. However, it is clear from the scatter plots that measured water levels at boreholes within the project boundaries fit the model results with a variance of about 0.5 meters.

Model predictions
Running the transient flow model for 20 years after the calibration year (2007), future groundwater level distribution due to continuous pumping at the current level of development is predicted. According to the head-time diagram in figure 10, Bara groundwater system will not reach a state of dynamic balance where recharge satisfies abstraction

Well field capture area
Flow lines are simulated to show the capture zone around the well field. In figure 11 flow lines indicate the instantaneous direction of flow throughout the aquifer at all time steps of the transient flow simulation. Figure 12 provides the transient flow field at the well field for 20 years stress period after 2007 with a time mark of 5 years.

Water budget
While limited information were available to have a prior estimate of the water budget within acceptable range of accuracy, the calibrated steady flow model has calculated a water budget assuming dynamic balance of In/outflows ( Table 1). The model boundary conditions figure  13 indicates the flow terms making the water budget.
According to the model water budget, inflow from the neighboring aquifer environments is estimated about 26,000 m 3 /day, and the safe yield of the model area would be within that estimated figure. The safe yield of the deep confined aquifer is linked to a dynamic water level above the top of the aquifer i.e. less than 181m.
Safe yield of the lower aquifer can be estimated from the presented model results. In confined aquifer conditions, safe yield is the volume abstracted without aquifer dewatering piezometric levels decline below the deep aquifer top (320 to 465 m NN). Thus, according to the model calculated drawdown, after 20 years under the current rate of abstraction the aquifer remains under confined conditions with an average dynamic water level of more than 25 m above the top of the deep aquifer. Table 2 below clearly confirms the above statement.

Conclusion
The scenario assessed by the study was continuous pumping at the current rate of abstraction from Bara well field and other wells tapping the basin. The current abstraction was estimated at 100 m 3 /d for the shallow aquifer and 300 m 3 /d from wells tapping the lower aquifer. There is no regional movement of groundwater in hard rock of transmissivity less than 100 m 2 /d. In addition to the slow movement of water, the groundwater gradient in hard rock aquifers is dominated by the topography with groundwater movement mainly towards the nearest valley.
A regional aquifer in the modelling terminology should not exceed 10 km of length in the case of Bara. This was based on the fact that slow groundwater flow (predominately horizontal with small gradient) in sandstone environments can hardly influence the local flow pattern beyond 10 km flow reach. As indicated in the report Umm Ruwaba formation starts at a depth of 49 m up to the depth of 181 m. The main aquifer extends vertically from a depth of 181 m to 480 m, with high penetration rate from 265 m to 338 m. According to the model water budget, inflow from the neighboring aquifer environments is estimated at ~ 26,000 m 3 /day.