Ground Water Essay – This is one of the best essays on ‘Ground Water’ especially written for school and college students.
Essay on Ground Water
Essay Contents:
- Essay on the Introduction to Ground Water
- Essay on the Subsurface Water and the Soil
- Essay on the Formation of Groundwater
- Essay on the Occurrence of Groundwater
- Essay on the Examples of Ground Water Flow
- Essay on the Water Table Contours and Regional Flow
- Essay on the Aquifer Properties and Ground Water Flow
- Essay on the Aquifers and Confining Layers
- Essay on the Ground Water Flow Equation under Steady State
- Essay on the Ground Water Flow Equations under Unsteady State
- Essay on the Two-Dimensional Flow in Aquifers
- Essay on the Steady One-Dimensional Flow in Aquifers
- Essay on the Classification of Groundwater Movement
Essay # 1. Introduction to Ground Water:
Most of the water that infiltrates into the soil travels down to recharge the vast ground water stored at a depth within the earth. In fact, the ground water reserve is actually a huge source of fresh water and is many times that of surface water. Such large water reserves remain mostly untapped though locally or regionally, the withdrawal may be high. Actually, as a result of excess withdrawal of ground water in many places of India (and also of the world), a number of problems have arisen. Unless the water resources engineer is aware of the consequent damages, this type of situation would lead to irreversible change in the quality and quantity of subsurface water which likely to affect our future generations.
In this article it is proposed to study how the water that infiltrates into the soil and the physics behind the phenomena. The study of subsurface movement of water from that of surface flow, has been separated because of the fact that the scale of movement of these two types of flows can vary by an order of magnitude 10 to more than 1000. This would be clear from Fig. 5.1 (a).
Essay # 2. Subsurface Water and the Soil:
Subsurface Water and the Soil – Rock Profile:
Figures 5.1 (b) and 5.1 (c) show two examples of underground soil-rock profiles and their relations with subsurface water that may exist both as confined and unconfined ground water reserves.
In fact, water is present in the pores of soils and fissures of rock up to a depth beyond which there is solid rock with no gaps which can store water. Although water is present in the pores of the soil and permeable rocks, there is difference between that stored above the water table and below it. The soil above the water table has only part of the voids filled up with water molecules whereas the soil below is completely saturated.
If we look more closely at the upper layers of the soil rock system, we find that it is only the change in moisture content that separates the unsaturated portion and the saturated portions of the soil. Figure 5.1 (d) shows a section through a soil – rock profile and corresponding graph showing the degree of saturation. Except the portion of the soil storing groundwater the remaining is unsaturated.
It may be noted that even in the driest climate, the degree of saturation in the unsaturated zone would not be zero as water clings to the soil particles by surface tension.
Some of the definitions related to subsurface water are as follows:
i. Soil Water:
The water stored in the upper layers of the soil from the ground surface up to the extent of roots of plants.
ii. Vadose Water:
That stored below in the region between soil water zone and the capillary fringe. It is a link between water infiltrating from the ground surface and moving down to the saturated layer of ground water.
iii. Capillary Water:
That which has risen from the saturated ground water region due to capillary action. Naturally, the pressure here would be less than atmospheric.
iv. Ground Water:
This is the water in the fully saturated zone. Pressure of water here would be more than atmospheric.
v. Water Table:
An imaginary surface within ground below which all the voids of the soil or permeable rock are completely filled with water. Below this imaginary surface, the pore water pressure is atmospheric. As one moves downwards from the water table, the pressure increases according to the hydrostatic law. Above the water table, the voids of soil/porous rock are only partially saturated with water clinging to the surface of the solids by surface tension. Hence, the pressure here is sub-atmosphere.
Essay # 3. Formation of Groundwater:
The level below which the soil is saturated with water is called groundwater table.
The groundwater is divided into two zones:
A. Unsaturated Zone, and
B. Saturated Zone.
These two zones are separated by water table where the pressure is atmospheric, as shown in Fig. 5.2.
A. Unsaturated Zone:
The zone above the water table is known as unsaturated zone. It is also known as vadose zone or aeration zone. In this zone, the pores in the soil may contain water or air, and the pressure is atmospheric.
This zone may be divided into three subzones:
(1) Soil-Water Zone,
(2) Intermediate Zone, and
(3) Capillary Zone.
(1) Soil Water Zone:
It lies close to the ground surface in the major root zone of vegetation. There may be loss of water from this zone by evapotranspiration. It may extend from a few metres to 15m, depending upon the nature of soil and vegetation.
(2) Intermediate Zone:
This subzone is between the soil-water zone and the capillary zone. Water from the soil zone may flow downwards due to gravity. The thickness of zone may vary from zero to several metres.
(3) Capillary Zone:
From the saturated zone, i.e. from the water may rise above due to capillarity, hence this zone is known as the capillary zone or the capillary fringe. The thickness may extend from a few centimetres to a few metres, depending on the porosity and structure of the soil. Water from this subzone may not move freely.
Water Pressure in Unsaturated Zone:
In literature, the term ‘ground water flow’ is used generally to describe the flow of water in the saturated portion of soil or fractured bedrock. No doubt it is important from the point of extraction of water from the zone using wells, etc.
But the unsaturated zone, too, is important because of the following reasons:
i. The water in the unsaturated zone is the source of moisture for vegetation (the soil water).
ii. This zone is the link between the surface and subsurface hydrologic processes as rain water infiltrates through this zone to recharge the ground water.
iii. Water evaporated or lost by transpiration from the unsaturated zone (mainly from the soil water zone) recharges the atmospheric moisture.
Further, the process of infiltration, quite important in hydrologic modeling catchment, is actually a phenomenon occurring in the unsaturated zone. Hence, knowledge about unsaturated zone water movement helps to understand infiltration better.
At the water table, the pressure head (conventionally denoted by Ψ) is zero (that is atmospheric), that in the unsaturated zone is (here Ψ is also called the moisture potential) and in the saturated zone, it is positive.
The hydraulic head at a point would, therefore, be defined as:
h = z + Ψ
Where, z is the elevation head, or the potential head due to gravity. According to the mechanics of flow, water moves from higher hydraulic head towards lower hydraulic head.
It may be noted that we may measure the negative pressure head within the unsaturated zone using a tensiometer. It consists of a porous ceramic cup connected by a water column to a manometer. The positive pressure head below water table can be determined using the hydrostatic pressure head formula ϒ D, where ϒ is the unit weight of water and D is the depth of water below water table.
Movement of Water in Unsaturated Zone:
The negative pressure head in the unsaturated zone of the soil can be metaphorically expressed as the soil being “thirsty”. All the pores of the soil here are not filled up. Hence, as soon as water is applied to the soil surface, it is “lapped up” by the soil matrix. Only if the water is applied in excess of the amount that it can “drink”, would water flow over the land surface as surface runoff.
This capacity of the soil in the unsaturated portion to absorb water actually depends on the volumetric water content, 9 expressed as:
θ = Vw/V
Where, Vw is the volume of water and V is the unit volume of soil or rock.
B. Saturated Zone:
Below the water table, all the pores in the soil are filled with water. Hence, it is known as the saturated zone. It is also known as the phreatic zone.
Water in this zone moves freely and may extend till the impermeable rock below. The pressure in this zone is more than atmospheric and increases as the depth increase.
Movement of Water in Saturated Zone:
The water that infiltrates through the unsaturated soil layers and move vertically ultimately reaches the saturated zone and raises the water table. Since it increases the quantity of in the saturated zone, it is also termed as ‘recharge’ of the ground water.
It may be observed from Fig. 5.9 that both before any infiltration took place, there existed a gradient of the water table which showed a small gradient towards the river. However, the rise of the water table after the recharge due to infiltrating water is not uniform and thus the gradient of the water table after recharge is more than that before recharge. This has a direct bearing on the amount of ground water flow, which is proportional to the gradient.
Based on actual observation or on mathematical analyses, we may draw lines of equal hydraulic head (the equipotentials) within the saturated zone, as shown in Figure 5.10. We may also draw the flow lines, which would be perpendicular to the equipotential lines. The flow lines, indicating the general direction of flow within the saturated soil zone is also drawn in the Fig. 5.10.
The rate of movement of the ground water, of course, varies with the material through which it is flowing since actually the flow is taking place through the voids which is different for different materials. The term hydraulic conductivity of a porous medium is used to indicate the ease with which water can flow through it.
It is defined as the discharge taking place through a flow tube (which may be thought of as a short pipe along a flow line) per unit area of the tube under the influence of unit hydraulic gradient (which is the difference of potential heads in unit distance along the flow line).
Hydraulic conductivity is generally denoted by ‘K’ and if the porous material is homogeneous, then K is also likely to be the same in any direction. However, in nature, the soil layers are often formed in layers resulting in the hydraulic conductivity varying between different directions.
Even porous bed rock, which is usually fractured rock, may not be fractured to the same extent in all directions. As a result, in many natural flows the flow is more in some preferential direction. This type of conducting media is referred to as being heterogeneous and the corresponding hydraulic conductivity is said to be anisotropic.
Essay # 4. Occurrence of Groundwater:
Groundwater occurs at various locations below the earth surface, depending on the formation of ground. The various formation are shown in Fig. 5.3.
i. Unconfined Aquifer:
An aquifer where the water table is the upper surface limit and extends below till the impermeable rock strata is called the unconfined aquifer, as shown in Fig. 5.4(a). It is also known as free aquifer phreatic aquifer, water table aquifer and non-artesian aquifer.
The water level in a well is an unconfined aquifer will be up to the water table.
ii. Confined Aquifer:
When an aquifer is sandwiched between two impermeable layers, it is known as a confined aquifer. It will not have water table, and the aquifer will be under pressure as shown in Fig. 5.4 (b).
If there is a well in this layer, the water in the well will rise upto piezometric head. If the piezometric head is above the ground, then the water from the well in this layer will flow over the ground. Then the well is called as free-flowing well or flowing well.
If the water level in a well in this layer is above the upper-confining layer level, but below the ground level, then such a well is called as artesian well.
Groundwater may move to the ground surface at a very small rate through faults, permeable material in joints, discontinuities, and so on. It is then called a spring.
iii. Perhed Aquifer:
An imperturbable sure-shaped stratum of a small aerial extent occurring in the zone of aeration may retain e hold some of water the called perched ground water as shown in Fig. 5.4 (c). It yields a limited quantity of water.
Formation of ground that contains water and may transmit water in usable quantity is known as aquifer. The aquifer may be unconfined or confined.
What happens to the water that got absorbed (that is infiltrated) at the surface of the unsaturated soil during application of water from above? It moves downward due to gravity through inter connected pores that are filled with water. With increasing water content, more pores fill, and the rate of downward movement of water increases.
A measure of the average rate of movement of water within soil (or permeable bed rock) is the hydraulic conductivity, indicated as ‘K’, and has the unit of velocity. Though it is more or less constant for a particular type of soil in the saturated zone, it is actually a function of the moisture content in the unsaturated portion of the soil.
As θ increases, so does K, and to be precise, it should correctly be written as K(θ), indicating K to be a function of θ. Figure 5.5 shows such a typical relation for an unsaturated soil.
Actually, the moisture potential (Ψ) is also a function of θ, as shown in Fig. 5.6.
The relationship of unsaturated hydraulic conductivity and volumetric water content is determined experimentally. A sample of soil placed in a container. The water content is kept constant and the rate at which water moves through the soil is measured. This is repeated for different values of θ (that is different saturation levels). It must be recommended that both K and Ψ very with θ and by its very nature, unsaturated flow involves many changes in volumetric moisture content as waves of infiltration pass.
The movement of a continuous stream of water infiltrating from the ground into the unsaturated soil may be typically seem to be as shown in Fig. 5.7.
If the source of the water is now cut off, then the distribution of water content with depth may look like as shown in Fig. 5.8.
Essay # 5. Examples of Ground Water Flow:
Although ground water flow is three-dimensional phenomenon, it is easier to analyse flows in two-dimension. Also, as far as interaction between surface water body and ground water is concerned, it is similar for lakes, river and any such body.
Here we qualitatively discuss the flow of ground water through a few examples which show the relative interaction between the flow and the geological properties of the porous medium. Here, the two-dimensional plane is assumed to be vertical.
1. Example of a Gaining Lake and River:
Fig. 5.11 shows an example of a lake perched on a hill that is receiving water from the adjacent hill masses. It also shows a river down in a valley, which is also receiving water.
2. Example of a Partially Loosing Lake, a Disconnected Loosing Lake, and a Gaining River:
Figure 5.12 illustrates this example modifies the situation of Example 1 slightly.
3. Example of Flow through a Heterogeneous Media, Case I:
This case (Fig. 5.13) illustrates the possible flow through a sub-soil material of low hydraulic conductivity sandwiched between materials of relatively higher hydraulic conductivities.
4. Example of Flow through a Heterogeneous Media, Case II:
This case (Fig. 14) is just opposite to that shown in example 3. Here, the flow is through a sub-soil material of high hydraulic conductivity sandwiched between materials of relatively low hydraulic conductivities.
Essay # 6. Water Table Contours and Regional Flow:
For a region, like a watershed, if we plot (in a horizontal plane) contours of equal hydraulic head of the ground water, then we can analyse the movement of ground water in a regional scale. Figure 5.15 illustrates the concept, assuming homogeneous porous media in the region for varying degrees of hydraulic conductivity (which is but natural for a real setting).
Essay # 7. Aquifer Properties and Ground Water Flow:
(a) Porosity:
Ground water is stored only within the pore spaces of soils or in the joints and fractures of rock which act as a aquifers. The porosity of an earth material is the percentage of the rock or soil that is void of material.
It is defined mathematically by the equation:
n = 100Vv/v
Where n is the porosity, expressed as percentage; K is the volume of void space in a unit volume of earth material; and V is the unit volume of earth material, including both voids and solid.
(b) Specific Yield:
While porosity is a measure of the water bearing capacity of the formation, all this water cannot be drained by gravity or by pumping from wells, as a portion of the water is held in the void spaces by molecular and surface tension forces, If gravity exerts a stress on a film of water surrounding a mineral grain (forming the soil), some of the film will pull away and drip downward.
The remaining film will be thinner, with a greater surface tension so that, eventually, the stress of gravity will be exactly balanced by the surface tension (Hygroscopic water is the moisture clinging to the soil particles because of surface tension). Considering the above phenomena, the Specific Yield (Sy) is the ratio of the volume of water that drains from a saturated soil or rock owing to the attraction of gravity to the total volume of the aquifer. If two samples are equivalent with regard to porosity, but the average grain size of one is much smaller than the other, the surface area of the finer sample will be larger. As a result, more water can be held as hygroscopic moisture by the finer grains.
The volume of water retained by molecular and surface tension forces, against the force of gravity, expressed as a percentage of the volume of the saturated sample of the aquifer, is called Specific Retention Sr, and corresponds to what is called the Field Capacity.
Hence, the following relation holds good:
n = Sy + Sr
(c) Specific Storage (ss):
Specific Storage (ss), also sometimes called the Elastic Storage Coefficient, is the amount of water per unit volume of a saturated formation that is stored or expelled from storage owing to compressibility of the mineral skeleton and the pore water per unit change in potentiometric head.
Specific storage is given by the expression:
Ss= ϒ(α + nβ)
Where, ϒ is the unit weight of water, α is the compressibility of the aquifer skeleton; n is the porosity; β is the compressibility of water.
Specific storage has the dimensions of length-1.
The storativity (S) of a confined aquifer is the product of the specific storage (Ss) and the aquifer thickness (b).
S = bSs
All of the water released is accounted for by the compressibility of the mineral skeleton and pore water. The water comes from the entire thickness of the aquifer.
In an unconfined aquifer, the level of saturation rises or falls with changes in the amount of water in storage. As water level falls, water drains out from the pore spaces. This storage or release due to the specific yield (Sv) of the aquifer. For an unconfined aquifer, therefore, the storativity is found by the formula.
S = Sy + hSs
Where h is the thickness of the saturated zone.
Since, the value of Sy is several orders of magnitude greater than hSs for an unconfined aquifer, the storativity is usually taken to be equal to the specific yield.
Essay # 8. Aquifers and Confining Layers:
It is natural to find the natural geologic formation of a region with varying degrees of hydraulic conductivities. The permeable materials have resulted usually due to weathering, fracturing and solution effects from the parent bed rock.
Hence, the physical size of the soil grains or the pre sizes of fractured rock affect the movement of ground water flow to a great degree.
Based on these, certain terms that have been used frequently in studying hydrogeology are:
1. Aquifer:
This is a geologic unit that can store and transmit water at rates fast enough to supply reasonable amount to wells.
2. Confining Layers:
This is a geologic unit having very little hydraulic conductivity.
Confining layers are further subdivided as follows:
i. Aquifuge:
An absolutely impermeable layer that will not transmit any water.
ii. Aquitard:
A layer of low permeability that can store ground water and also transmit slowly from one aquifer to another. Also termed as “leaky aquifer”.
iii. Aquiclude:
A unit of low permeability, but is located so that it forms an upper or lower boundary to a ground water flow system.
Aquifers which occur below land surface extending up to a depth are known as unconfined. Some aquifers are located much below the land surface, overlain by a confining layer. Such aquifers are called confined or artesian aquifers. In these aquifers, the water is under pressure and there is no free water surface like the water table of unconfined aquifer.
Till now, qualitative assessment of subsurface water whether in the unsaturated or in the saturated ground was made. Movement of water stored in the saturated soil or fractured bed rock, also called aquifer, was seen to depend upon the hydraulic gradient. In this article, we derive the mathematical description of saturated ground water flow and its exact and approximate relations to the hydraulic gradient.
Essay # 9. Ground Water Flow Equation under Steady State:
Continuity Equation and Darcy’s Law under Steady State Conditions:
Consider the flow of ground water taking place within a small cube (of lengths Δx, Δy and Δz respectively the direction of the three areas which may also be called the elementary control volume) of a saturated aquifer as shown in Fig. 5.16.
It is assumed that the density of water (ρ) does not change in space along the three directions which implies that water is considered incompressible. The velocity components in the x, y and z directions have been denoted as Vx, Vy and Vz respectively.
Since water has been considered incompressible, the total incoming water in the cuboidal volume should be equal to that going out.
Defining inflows and outflows as:
Inflows:
Outflows:
This is continuity equation for flow. But this water flow is due to a difference in potentiometric head per unit length in the direction of flow. A relation between the velocity and potentiometric gradient was first suggested by Henry Darcy, a French Engineer, in the mid nineteenth century.
He found experimentally (see Fig. 5.17 below) that the discharge ‘Q’ passing through a tube of cross sectional area ‘A’ filled with a porous material is proportional to the difference of the hydraulic head ‘h’ between the two end points and inversely proportional to the flow length ‘L’.
It may be noted that the total energy (also called head, h) at any point in the ground water flow per unit weight is given as:
Where (dh/dl) is known as hydraulic gradient.
The coefficient ‘K’ has dimensions of LIT, or velocity, and this is termed as the hydraulic conductivity.
Thus the velocity of fluid flow would be:
It may be noted that this velocity is not quite the same as the velocity of water flowing through an open pipe. In an open pipe, the entire cross section of the pipe conveys water. On the other hand, if the pipe is filed with a porous material, say sand, then the water can only flow through the pores of the sand particles.
Hence, the velocity obtained by the above expression is only an apparent velocity, with the actual velocity of the fluid particles through the voids of the porous material is many time more.
But for our analysis of substituting the expression for velocity in the three directions x, y and z in the continuity relation, and considering each velocity term to be proportional to the hydraulic gradient in the corresponding direction, one obtains the following relation:
Here, the hydraulic conductivities in the three directions (Kx, Ky and Kz) have been assumed to be different as for a general anisotropic medium.
Considering isotropic medium with a constant hydraulic conductivity in all directions, the continuity equation simplifies to the following expression:
In the above equation, it is assumed that the hydraulic head is not changing with time, that is, a steady state is prevailing.
If now it is assumed that the potentiometric head changes with time at the location of the control volume, then there would be a corresponding change in the porosity of the aquifer even if the fluid density is assumed to be unchanged.
Essay # 10. Ground Water Flow Equations under Unsteady State:
For an unsteady case, the rate of mass flow in the elementary control volume is given by:
This is caused by a change in the hydraulic head with time plus the porosity of the media increasing accommodating more water.
Denoting porosity by the term ‘n’, a change in mass ‘M’ of water contained with respect to time is given by:
Considering no lateral strain, that is, no change in the plan area ΔxΔy of the control volume, the above expression may be written as:
Where, the density of water (ρ) is assumed to change with time. Its relation to a change in volume of the water Vw, contained within the void is given as:
The negative sign indicates that a reduction in volume would mean an increase in the density from the corresponding original values.
The compressibility of water, β, is defined as:
Where ‘dp’ is the change in the hydraulic head ‘p’ Thus,
The compressibility of the soil matrix, α, is defined as the inverse of Es, the elasticity of the soil matrix.
Hence,
Where σz is the stress in the grains of the soil matrix.
Now, the pressure of the fluid in the voids, p, and the stress on the solid particles, σz, must combine to support the total mass lying vertically above the elementary volume.
Thus,
p + σp = constant
Also since the potentiometric head ‘h’ given by:
Where Z is the elevation of the cube considered above a datum.
We may therefore rewrite the above as:
First term for the change in mass ‘M’ of the water contained in the elementary volume, is:
This may be written as equal to:
Considering the compressibility of the soil grains to be nominal compared to that of the water or the change in the porosity, we may assume dVs to be equal to zero.
Which may substituted in second term of the expression for change in mass, M, of the elementary volume, changing it to:
Thus, the equation for change of mass, M, of the elementary cubic volume becomes:
Combining earlier equation with the continuity expression for mass within the volume, gives the following relation:
Assuming isotropic media, that is, KX = KY = KZ = K and applying Darcy’s law for the velocities in the three directions, the above equation simplifies to:
Now, since the potentiometric (or hydraulic) head h is given as:
The flow equation can be expressed as:
The above equation is the general expression for the flow in three-dimensions for an isotropic homogeneous porous medium. The expression was derived on the basis of an elementary control volume which may be a part of an unconfined or a confined aquifer.
Essay # 11. Two-Dimensional Flow in Aquifers:
Under many situations, the water table variation (for unconfined flow) in areal extent is not much, which means that there the ground water flow does not have much of a vertical velocity component. Hence, a two-dimensional flow situation may be approximated for these cases.
On the other hand, where there is a large variation in the water table under certain situation, a three dimensional velocity field would be the correct representation as there would be significant component of flow in the vertical direction apart from that in the horizontal directions. This difference is shown in the illustrations given in Fig. 5.19.
In case of two-dimensional flow, the equation flow for both unconfined and confined aquifers may be written as:
There is one point to be noted for unconfined aquifers for hydraulic head (or water table) variations with time. It is that the change in the saturated thickness of the aquifer with time also changes the transmissivity, T, which is a product of hydraulic conductivity K and the saturated thickness h.
The general form of the flow equation for two-dimensional unconfined flow is known as the Boussinesq equation and is given as:
Where Sy is the specific yield.
If the drawdown in the unconfined aquifer is very small compared to the saturated thickness, the variable thickness of the saturated zone, h, can be replaced by an average thickness, b, which is assumed to be constant over the aquifer.
For confined aquifer under an unsteady condition though the potentiometric surface declines, the saturated thickness of the aquifer remains constant with time and is equal to an average value ‘b’. Solving the ground water flow equations for flow in aquifers require the help of numerical techniques, except for very simple cases.
Two-Dimensional Seepage Flow:
In the last section, examples of two-dimensional flow were given for aquifers, considering the flow to be occurring, in general, in a horizontal plane. Another example of two-dimensional flow would that be when the flow can be approximated to be taking place in the vertical plane. Such situations might occur as for the seepage taking place below a dam as shown in Fig. 5.20.
Under steady state conditions, the general equation of flow, considering an isotropic porous medium would be:
However, solving the above would require advanced analytical methods or numerical techniques.
Essay # 12. Steady One-Dimensional Flow in Aquifers:
Some simplified cases of ground water flow, usually in the vertical plane, can be approximated by one-dimensional equation which can then be solved analytically.
We consider the confined and unconfined aquifers separately, in the following sections:
A. Confined Aquifers:
If there is a steady movement of ground water in a confined aquifer, there will be a gradient or slope to the potentiometric surface of the aquifer. The gradient, again, would be decreasing in the direction of flow. For flow of this type, Darcy’s law may be used directly.
i. Aquifer with Constant Thickness:
This situation may be shown as in Figure 5.21.
Assuming unit thickness in the direction perpendicular to the plane of the paper, the flow rate ‘q’ (per unit width) would be expressed for an aquifer of thickness ‘b’
q = b * 1 * v
According to Darcy’s law, the velocity ‘v’ is given by:
Where h, the potentiometric head, is measured above a convenient datum. Note that the actual value of ‘h’ is not required, but only its gradient δh/δx in the direction of flow, x, is what matters.
Here K is the hydraulic conductivity:
Hence,
The partial derivative of ‘h’ with respect to ‘x’ may be written as normal derivative since we are assuming no variation of ‘h’ in the direction normal to the paper.
Thus,
For steady uniform flow, q should not vary with time, t, or spatial coordinate, x. hence:
Since the width, b, and hydraulic conductivity, K, of the aquifer are assumed to be constants, the above equation simplifies to:
Selecting the origin of coordinate x at the location of well A (as shown in Fig. 5.21), and having a hydraulic head, hA and also assuming a hydraulic head of well B, located at a distance L from well A in the x-direction equal to hB, we have:
ii. Aquifer with Variable Thickness:
Consider a situation of one-dimensional flow in a confined aquifer whose thickness, b, varies in the direction of flow, x, in a linear fashion as shown in Fig. 5.22.
The unit discharge, q, is now given as:
Where K is the hydraulic conductivity and dh/dx is the gradient of the potentiometric surface in the direction of flow, x.
For steady uniform flow, we have,
A solution of the above differential equation may be found out which may be substituted for known values of the potentiometric heads hA and hB in the two observation wells A and B respectively in order to find out the constants of integration.
B. Unconfined Aquifers:
In an unconfined aquifer, the saturated flow thickness, h is the same as the hydraulic head at any location, as seen from Figure 5.23.
Considering no recharge of water from top, the flow takes place in the direction of fall of the hydraulic head, h, which is a function of the coordinate, x taken in the flow direction. The flow velocity, v, would be lesser at location A and higher at B since the saturated flow thickness decreases. Hence v is also a function of x and increases in the direction of flow.
Since, v, according to Darcy’s law is shown to be:
The gradient of potentiometric surface, dh/dx, would (in proportion to the velocities) be smaller at location ‘a’ and steeper at location ‘b’. Hence the gradient of water table in unconfined flow is not constant, it increases in the direction of flow.
This problem was solved by J. Dupuit, a French hydraulician, and published in 1863 and his assumptions for a flow in an unconfined aquifer is used to approximate the flow situation called Dupuit flow.
The assumptions made by Dupuit are:
i. The hydraulic gradient is equal to the slope of the water table, and
ii. For small water table gradients, the flow-lines are horizontal and the equipotential lines are vertical.
The second assumption is illustrated in Fig. 5.24.
Solutions based on the Dupuit’s assumptions have proved to be very useful in many practical purposes. However, the Dupuit assumption do not allow for a seepage face above an outflow side.
An analytical solution to the flow would be obtained by using the Darcy equation to express the velocity, v, at any point, x, with a corresponding hydraulic gradient dh/dx as:
Considering the origin of the coordinate x at location A where the hydraulic head is hA and knowing the hydraulic head hB at a location B, situated at a distance L from A, we may integrate the above differential equation as:
Rearrangement of above terms leads to, what is known as the Dupuit equation:
An example of the application of the above equation may be for the ground water flow in a strip of land located between two water bodies with different water surface elevations, as shown in Fig. 5.25.
The equation for the water table, also called the phreatic surface may be derived from above equation as follows:
In case of recharge due to a constant infiltration of water from above the water table rises to a many as shown in Fig. 5.26:
There is a difference with the earlier cases, as the flow per unit width, q, would be increasing in the direction of flow due to addition of water from above. The flow may be analysed by considering a small portion of flow domain as shown in Fig. 5.27.
Considering the infiltration of water from above at a rate i per unit length in the direction of ground water flow, the change in unit discharge dq is seen to be:
For no recharge due to infiltration, I = 0 and the expression for qx is then seen to become independent of x, hence constant, which is expected.
Essay # 13. Classification of Groundwater Movement:
The theory of groundwater movement into a well is commonly known as well hydraulics. The movement of groundwater is in accordance with the Darcy’s law, coupled with the hydraulic principles.
When the water table is higher than the bed of the stream, groundwater will flow to the stream. Such a stream is known as effluent strewn. On the other hand, when the water table is lower than the bed of the stream, water will flow from. The stream to the groundwater. Such a stream is called influent stream. Both types are shown in Fig. 5.31.
In both the cases, the flow will depend on the soil characteristics. Similar is the case of a well. When water flows into a well, the well is known as discharging well; when water flows from the well to the adjacent ground, it is called recharging well.