Week 3
beginning October 12th
Section 3
Use of Darcy’s Law to predict and calculate groundwater flow
1.3.1. Introduction
In 1856, a French hydraulic engineer, Henry Darcy carried out a series of experiments
with water flowing through sand under different heads of water. He determined that
the flow rate through porous media is proportional to the head loss and inversely
proportional to the length of the flow path. Darcy’s Law provides the basis for ground
water flow calculations.
He showed that the velocity of water flowing through a porous medium is equal to the
hydraulic gradient times a constant that he called permeability. The value of water
permeability for a porous medium is called hydraulic conductivity. Discharge is
calculated by measuring the velocity through the cross-sectional area of the sample.
The parameters of the Law are expressed in Figure 1.3.1.
Figure 1.3.1. Diagramatic representation of Darcy’s experiment showing that the
velocity (v) of water flowing through a porous medium is equal to the hydraulic
gradient (h/l), times a constant (k) [=permeability]. The value of permeability for a
porous medium varies according to hydraulic conductivity. As the amount of flow
(Q) is determined by the velocity (v) and the cross-sectional area of the sample (A),
therefore Darcy’s Law may be used to calculate discharge.
Source: Brassington (1990:49).
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1.3.2. Application of Darcy’s Law.
The driving force of groundwater flow is the hydraulic head, that is, difference in
level of the piezometric surface /water table, from recharge to discharge. The flow of
the water through the saturated zone of an aquifer may be represented by the
equation:
Q
=
Groundwater
discharge (m3/day)
A
x
cross-sectional
area through
which flow takes
place (m2)
K
hydraulic
conductivity(m/d)
x
h/l
hydraulic
gradient
This may be expressed:
V
=
Ki
Velocity of flow through aquifer = coefficient of permeability x hydraulic gradient
i
=
Δh
L
Hydraulic gradient
=
head loss in length of flow path L
Q
Vol. rate of flow
=
=
AV
=
cross sectional area
of aquifer
[width x thickness (wb)]
[discharge or yield]
Aki
A = wb
T
=
Kb
Coefficient of transmissibility of aquifer = coefficient of permeability x thickness
Therefore Q
= wbki
discharge or yield = coefficient of perm. x cross sectional area x hydraulic gradient
Therefore Q
=
Tiw
Discharge/Yield = coefficient of transmissibility x hydraulic gradient x width of
aquifer
In aquifers containing large diameter solution openings (e.g. in the karstic Mountain
Aquifer of the West Bank) flow is no longer laminar due to high gradients and
exhibits non-linear flow between the velocity and hydraulic gradient. This will also
be the case with some of the coarser gravels present in the Gaza Coastal Aquifer.
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The transmissibility is the flow capacity of an aquifer per unit width under unit
hydraulic gradient and is equal to the product of permeability times the saturated
thickness of the aquifer.
In a confined aquifer, T = Kb and is independent of the piezometric surface.
In a phreatic aquifer, T = KH, where H is the saturated thickness.
As the water table drops, H decreases and the transmissibility is reduced.
Thus the transmissibility of an unconfined aquifer depends upon the depth of the
groundwater table.
Source: Raghunath, H.M.(1985) Hydrology: Principles, Analysis and Design, Wiley Eastern Ltd., New Delhi.
Darcy’s Law provides a value for the hydraulic conductivity and therefore may be
used to assess the overall flow through an aquifer (Figure 1.3.2). It is also possible to
use Darcy’s Law to calculate the length of time it will take water to flow through the
aquifer. Using the data available from Figure 1.3.2 and the equations used earlier:
5 x 60
V=kh/l =
=0.015m/d
20,000
This means that it would take over 3,560 years for water to travel from the recharge
area to emerge from the springs in the discharge zone.
Figure 1.3.2. Regional flow (Q) through a sandstone aquifer can be calculated using
Darcy’s Law. The sandstone has an average thickness of 200 m and is 10 km wide.
The distance from the recharge area to the discharge springs is 20 km, and the head
difference averages 60 m. The hydraulic conductivity is 5 m/day. With these data,
Darcy’s Law may be used to determine the following:
source: Brassington (1990)
Q = kAh/l = 5 x (200 x 10,000 x 60/20,000
= 30,000m3/day
If the specific yield is 15%, then the total volume of usable aquifer storage in will be:
200 x 10,000 x 20,000 x 0.15 = 6Mm3
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1.3.3. Groundwater flow calculations in the Eastern Basin , Mountain Aquifer.
The water level along the Hebron and Ramallah anticlinal axes is at about 450m
above mean sea level (amsl). The hydraulic gradient dips eastwards to the Jordan
Valley and Dead Sea, that is to 300m to 400m below mean sea level (bmsl). Thus
there is a head of 750m to 850m over a lateral distance of 25-30kms. If this gradient
is uniform, this represents a very steep flow gradient.
In fact the west to east gradient has been shown by Guttman (2000) and other
investigators not to be uniform. These hydrologists have identified relatively
moderate flow gradients from studies based on readings taken from the production
wells in the Herodion water well field and further north in the Jericho area. The
gradient is steeper between the Jerusalem Number 5 well and the Azzariya Number 1
well. It may therefore be surmised that low transmissivity zones are created by
transmissivity barriers that influence groundwater flow.
The groundwater flow bypasses the low transmissivity zones as may be seen in the
flow direction from the Herodion field to Shedema, which is initially northeast and
subsequently east. There are also structural controls of groundwater flow. There is,
for example, a difference of more than 100m in the water table between Ein Samia
Well No. 3, on the up-throw side of a fault and Ein Samia Well No. 4 and Kokhav
Hashahar No. 1 on the down-throw side of the fault. There is therefore a probable
hydrological barrier either side of this fault. Groundwater flow must be parallel to the
fault up to the point where the throw becomes more moderate and allows hydraulic
contact between the two sides of the fault (Guttman, 2000:9).
Evidence from the Auja wells indicates that the base of the lower aquifer is at or
above the water table for the upper phreatic aquifer. This is a structural control
resulting from the proximity to the Ein Samia fault system and the Wadi Far’ah
monocline. This creates a hydrological barrier or watershed/divide. Comparison of
the water table in the Auja wells reveals that it is higher than in the Fasa’el and Gitit
wells further north or the Jericho wells further south. Thus, groundwater to the north
of the Auja wells flows towards the Jordan River whereas south of the Auja wells
groundwater flows to the Dead Sea springs Figure 1.3.3.
1.3.4. Spring discharge
Groundwater from the Eastern Basin of the Mountain Aquifer discharges naturally
into the Jordan River and Dead Sea. Darcy’s Law expresses the link between spring
discharge and head:
Q = A(h-h0)
Where
Q = Spring discharge in million m3/y
A = Spring constant in m2/yr
h = elevation of water level in metres
h0 = threshold elevation of spring in metres
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Figure 1.3.3. Direction and velocity of groundwater flow in the Eastern Basin.
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1.3.4. Spring discharge
Groundwater from the Eastern Basin of the Mountain Aquifer discharges naturally
into the Jordan River and Dead Sea. Darcy’s Law expresses the link between spring
discharge and head:
Q = A(h-h0)
Where
Q = Spring discharge in million m3/y
A = Spring constant in m2/yr
h = elevation of water level in metres
h0 = threshold elevation of spring in metres
Figure 1.3.4. shows the gradient of the surface of saturated rock in the eastern aquifer.
Under the recharge area in the Hebron Mountains, infiltrating water reaches this
water table at about 450m elevation. The average gradient dips eastwards to the Dead
Sea, where the springs discharge at more than 400m bmsl.
1.3.5. Transmissivity
Guttman used his model (1995 and 2000) to calculate that transmissivity ranges from
200 to 400 m2/day in the natural replenishment zone along the Ramallah and Hebron
axes. In the Jerusalem area it is about 50m2/day. In the Jericho area transmissivity is
much higher, ranging from 1,000 to 5,000m2/day. In contrast, the hydrologic barriers
reduce transmissivity to 4 to 15m2/day.
1.3.6. Water balance
A steady-state water balance is demonstrated the Guttman model (1995 and 2000) in
which inflow or recharge of the aquifer is 118.61 Mm3/yr. Spring discharge is
summed as follows:
Mm3/yr
To Jordan River and associated springs
Feshkha
Kana
Samar
Kedem
Yesha
Mazor
Total
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25.77
53.30
16.22
18.32
0.21
0.69
4.11
118.61
Figure 1.3.4. Groundwater table contours in the Eastern Basin of the Mountain
Aquifer.
Source: Guttman (2000).
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