Darcy’s Law: The percolation of water through soil was first studied by Darcy who demonstrated experimentally that for laminar flow conditions in a saturated soil,

Statement: The flow rate through porous media is proportional to the head loss , total cross-sectional area of soil mass perpendicular to the direction of flow and inversely proportional to the length of the flow path, is known as Darcy’s law.

Or

The rate of flow or the discharge per unit time is proportional to the hydraulic gradient, and it could be expressed as follows:

$$q=kiA$$ $$v=\frac{Q}{A}=ki$$

Where Q = Rate of flow i = Hydraulic gradient k = Darcy’s coefficient of permeability A = Total cross-sectional area of soil mass perpendicular to the direction of
Flow v = Flow velocity

Darcy’s law is valid only for laminar flow. Because of very small pore dimensions in fine grained soils, a laminar flow should exist, but in course grained soils turbulent flow may be expected under certain conditions. I

Aquifer and It’s Types

1. A permeable stratum or a geological formation of permeable material, which is capable of yielding appreciable quantities of ground water under gravity, is known as an aquifer.
2. The term ‘appreciable quantities’ is relative, depending upon the availability of ground water. In the regions, where ground water is available with great difficulty, even fine-grained materials containing very less quantities of water may be classified as principal aquifers.
3. When an aquifer is overlain by a confined bed of impervious material, then this confined bed of overburden is called an aquiclude.
4. The amount of water yielded by a well excavated through an aquifer, depends on may factors; some of which, such as well diameter, are inherent thickness of the aquifer are most important.
5. Aquifer vary in depth, lateral extent and thickness; but in general, all aquifer falls into one of the two categories, i.e.
   • Unconfined or Non-artesian aquifer
   • Confined or artesian

1. Unconfined Aquifer or Non-Artesian Aquifer:

• The top most water bearing stratum having no confined impermeable over burden (i.e., aquiclude) lying over it, is known as an unconfined aquifer or non-artesian aquifer.
• The ordinary gravity wells of 2 to 5m diameter, which are constructed to tap water from the top most water bearing strata, i.e., from the unconfined aquifer, are known as unconfined or non-artesian well. The water levels in these wells will be equal to the level of water table. Such wells are, therefore, also known as gravity wells or water table wells.

2. Confined Aquifer or Artesian Aquifer:

• When an aquifer is confined on its upper and under surface, by impervious rock formations (i.e., aquicludes) and is also broadly inclined so as to expose the aquifer somewhere to the catchment area at a higher level for the creation of sufficient hydraulic head, it is called a confined aquifer or an artesian well.
• A well excavated through such an aquifer, yields water often flows out automatically, under the hydrostatic pressure, and may thus, even rise or gush out of surface for a reasonable height.
• However, where the ground profile is high, the water may remain well below the ground level. The former type of artesian wells, where water gushes out automatically, are called flowing wells.
• The level to which water will rise in an artesian well is determined by the highest point on the aquifer, from where it is fed from the rains falling in the catchment. However, the water will not rise to this full height, because the friction of the water moving through the aquifer, uses up some of the energy.
• The question whether it will be flowing artesian well or a non-flowing artesian well depends upon the topography of the area, and is not the inherent property of the artesian aquifer.
• In fact, if the pressure surface lies above the ground surface, the well will be a flowing artesian well, whereas, if the pressure surface is below the ground surface, the well will be artesian but non-flowing and will require a pump to bring water to the surface, as shown figure 37. Such non-flowing artesian wells are sometimes called sub-artesian wells.

3. Perched Aquifer:

• Perched aquifer is a special case which is sometimes found to occur within an unconfined aquifer
• Sometimes a lens or localised patch of an impervious stratum can occur inside an unconfined aquifer in such a way that it retains a water table above the general water table as shown below.
• Such a water table retained around the impervious material is known as perched water table, while the supported body of the saturated material is known as perched aquifer.

Difference between Aquifuge, Aquitard and Aquiclude,

1. Aquifuge: It is that geological formation, which is, neither porous nor permeable; and hence it neither contains nor yields ground water. Granite is an example of aquifuge.
2. Aquitard: It is that geological formation, which does not yield water freely to wells due to its lesser permeability, although seepage is possible through it. The yield from such a formation is, thus, insignificant. Sand clay is an example of aquitard.
3. Aquiclude: It is highly porous, containing large quantities of water, but essentially impervious, as not to yield water. A clay layer is an example of aquiclude.

Steady Radial Flow to a Well: Dupuit’s Theory

1. When a well is penetrated into an extensive homogeneous aquifer, the water table remains horizontal in the well. When the well is pumped, water is removed from the aquifer and the water table or piezometric surface, depending upon the type of the aquifer, is lowered resulting in a circular depression in the water table or the piezometric surface.
2. This depression is called the cone of depression or the drawdown curve. At any point away from the well, the drawdown is the vertical distance by which the water table or piezometric surface is lowered.

3. The analysis of such radius flow towards a well was originally proposed by Duouit in 1863 and later modified by Thiem.

   • Unconfined Aquifer: Figure below shows a well penetrating an unconfined or free aquifer to its full depth

Let r = Radius of the well. H = thickness of the aquifer, measured from the impermeable layer to the initial level of water table. s = drawdown at the well h = Depth of water in the well measured above impermeable layer.

Considering the origin of co-ordinates at a point O at the centre of the well at its bottom, let the co-ordinates of any point P on the drawdown curve be (x, y).

• Assumptions and limitations of Dupuit’s Theory:
• The velocity of flow is proportional to the tangent of the hydraulic gradient instead of sine.
• The flow is horizontal and uniform everywhere in the vertical section.
• Aquifer is homogeneous, isotropic and of infinite aerial extent.
• The well penetrates and receives water from entire thickness of the aquifer.
• The coefficient of transmissibility is constant at all places and at all times.
• Natural ground water regime affecting an aquifer remainsconstant with time.

• Flow is laminar and Darcy’s law is applicable.

  • Confined Aquifer

Aquifer Test

There are two tests

1. Constant Level Pumping Test

• In this test, a pump with suitable pumping arrangement is used. The water level is depressed by an amount h (say) as the depression head.
• The head of the pump is so adjusted that whatever water enters under this depression head is pumped out and a constant water level is maintained in the well
• The amount of water pumped out is measured with the help of a V-notch or any other arrangement, in a given amount of time for which the pump speed was regulated to a constant value.
• The quantity pumped in one hour gives the yield of the well per hour.

The formula for discharge in cumecs from an open well with impervious may be written as: Q = A X v Q = A X C X h Where Q = Discharge in cubic metres per sec A = Cross-sectional area of flow into the well at its base in m2 v = Mean velocity of water percolating into the well, in m/sec h = Depression head in m C = Percolation intensity coefficient

2. Recuperation Test

1. Through the constant level pumping test gives an accurate value of safe yield of an open well, it is sometimes very difficult to regulate the pump in such a way that constant level is maintained in the well.
2. In such a circumstance, a recuperation test is resorted to. In the recuperation test, water is depressed to any level below the normal and the pumping is stopped. The time taken for the water to recuperate to the normal level is noted.

Let aa =static water level in the well, before the pumping started.

bb = Water level in the well when the pumping stopped

$h_1$= depression head in the well when the pumping stopped

cc = water level in the well at a time T after the pumping stopped

$h_2$ = depression head in the well at a time T after the pumping stopped

h = depression head in the well at a time t after the pumping stopped

dh = decrease in depression head in a time dt

t,T = time in hours.

In time dt after this, the head recuperates by a value dh meters.

Volume of water entering the well, when the head recuperates by dh is

dV = A dh …(1)

Where A = cross-sectional area of well at its bottom

Again, if Q is the rate of discharge in the well at the time t, under the depression head h, the volume of water entering the well in them t hours is given by

dV = Q dt

But Q is proportional to h

Or Q = Kh ….(2)

dV = K h dt …..(3)

where K is a constant depending upon the soil at the base of the well through which water enters.

Equating (1) and (3), we get K h dt = -A dh

The minus sign indicates that h decreases as time t increases. Integrating the above between the limits: t = 0 when h = $h_1$ ; t = T when h = $h_2$ .

$$\text{We get, }\frac{K}{A}\int_0^T dt=\int_{h1}^{h2}\frac{dh}{h}$$ $$\text{Or }\frac{K}{A}\int_0^T dt=-\int_{h2}^{h1}\frac{dh}{h}$$ $$\text{From which, }\frac{K}{A}T=[\log_eh]_{h2}^{h1}$$ $$\frac{K}{A}=\frac{1}{T}\log_e\frac{h1}{h2}=\frac{2.303}{T}\log_e\frac{h1}{h2}$$

Thus, knowing the value ofh1, h2 and T from recuperation test, the quality K/A can be calculated. K/A is known as the specific yield or specific capacity of an open well.

Knowing the value of K/A by observation, the discharge from a well under a constant depression head H can be calculated as under:

$$Q=KH$$ $$Q=\lgroup \frac{K}{A}\rgroup A.H.$$ $$Q=\frac{2.303}{T}\log_{10}\frac{h1}{h2}AH\ m^3/hour$$