Principles of groundwater flow

Objectives: On completion of the lesson, the student shall be learn
• The description of steady state of ground water flow in the form
of Laplace equation derived from continuity equation and
Darcy’s law for ground water movement.
• The quantitative description of unsteady state ground water
flow.
• The definition of the terms Specific Yield and Specific Storage
and their relationship with Storativity of a confined aquifer.
• The expressions for ground water flow in unconfined and
confined aquifers, written in terms of Transmissivity.
• Expression for two – dimensional flow in unconfined and
confined aquifers; Boussinesq equation.
• Expression for two – dimensional seepage flow below dams.
• Analytical solution of steady one dimensional flow in simple
cases of confined and unconfined aquifers.

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.

•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 νx, νy, νz
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:
outflows: inflows:
In X-direction: ρ [νx + (∂vx/∂ x) Δx Δx] (Δy.Δz) In x-direction: ρ νx (Δy.Δx)
In Y-direction: ρ [νx + (∂vy/∂ y) Δy] (Δx.Δz) In y-direction: ρ νy (Δx.Δz)
In Z-direction: ρ [νx + (∂vz/∂ z) Δz] (Δy.Δx) In z-direction: ρ νz (Δx.Δy)

The net mass flow per unit time through the
cube works out to:
•This is continuity equation for flow.
•is due to a difference in potentiometric head per unit length in the direction
of flow

Henry Darcy
•French Engineer
•He found experimentally (see figure 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’.

Where
•Z is the elevation of the point above a chosen datum;
•γ/p is the pressure head, and
•v^2/2g is the velocity head
# Since the ground water flow velocities are usually very small, v^2/2g is
neglected and h = Z+γ/p is termed as the potentiometric head (or piezometric
head in some texts)

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,
equation (2) 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.
Porosity: It is ratio of volume of voids to the total volume of the soil and is generally
expressed as percentage.

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 Δx.Δy
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,
That is …

The compressibility of the soil matrix, α, is defined as the inverse of ES, the
elasticity of the soil matrix.
The compressibilWhere 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+σz= constant (18) that is… dσz = -dp

Thus,
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,
This may be written, based on the derivations shown earlier, as equal to
Also the volume of soil grains, VS, is given as
Thus,

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. Hence,
or

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

Assuming isotropic media, that is, KX=K=YKZ=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 h

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. The next section looks into the
simplifications for each type of aquifer.

Ground water flow expressions for ground water flow unconfined and
confined aquifers
Unsteady flow takes place in an unconfined and confined
aquifer would be either due to:
• Change in hydraulic head (for unconfined aquifer) or
potentiometric head (for confined aquifer) with time.
• And, or compressibility of the mineral grains of the soil
matrix forming the aquifer
• And, or compressibility of the water stored in the voids
within the soil matrix

We may visually express the above conditions as shown in Figure, assuming
an increase in the hydraulic (or potentiometric head) and a compression of
soil matrix and pore water to accommodate more water

Since storability S of a confined aquifer was defined as
The flow equation for a confined aquifer would simplify to the
following:

Defining the transmissivity T of a confined aquifer as a product of the
hydraulic conductivity K and the saturated thickness of the aquifer, b,
which is:
The flow equation further reduces to the following for a confined aquifer

For unconfined aquifers, the storability S is given by the following expression
Where Sy is the specific yield and Ss is the specific storage is equal to
# Usually, Ss is much smaller in magnitude than Sy and may be neglected.
Hence S under water table conditions for all practical purposes may be taken
equal to Sy.

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 Figure.

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 Figure .

Under steady state conditions, the general equation of
flow, considering an isotropic porous medium would be

However, solving the above Equation for would
require advanced analytical methods or
numerical techniques. More about seepage flow
would be discussed in the later session.

Steady one dimensional flow in aquifers
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.

Aquifer with constant thickness

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’
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 h∂/∂x in
the direction of flow, x, is what matters. Here is 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 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:
Which may be analytically solved as

Selecting the origin of coordinate x at the location of well A (as shown in
Figure 6), 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 and having a
hydraulic head hB, we have:
hA = C1.0+C2 and
hB = C1.L+C2
Giving
Thus the analytical solution for the hydraulic head ‘h’ becomes:

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

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 flow, we have,

Which may be simplified, denoting dx/db as b′
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.

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

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:
•The hydraulic gradient is equal to the slope of the water
table, and
•For small water table gradients, the flow-lines are horizontal
and the equipotential lines are vertical.

The second assumption is illustrated

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 dx/dh, as

Thus, the unit discharge, q, is calculated to be
Considering the origin of the coordinate x at location A where the
hydraulic head us 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:

Which, on integration, leads to
Rearrangement of above terms leads to, what is known as the Dupuit equation:Rearrangement of above terms leads to, what is known as the Dupuit equation:Rearrangement of above terms leads to, what is known as the Dupuit equation:
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

The equation for the water table, also called the phreatic surface may
be derived from Equation (61) as follows:
In case of recharge due to a constant infiltration of water from
above the water table rises to a many as shown

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 analyzed by
considering a small portion of flow domain as shown

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
or
From Darcy’s law,

Substituting the expression for dx/dq, we have,
or
The solution for this equation is of the form

If, now, the boundary condition is applied as
At x = 0, h = h1, and
At x = L, h = h2
The equation for the water table would be
And,
Where q0 is the unit discharge at the left boundary, x = 0, and may be found out to be

Which gives an expression for unit discharge qx at any
point x from the origin as
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.

References
Raghunath, H M (2002) Ground Water (Second
Edition), New Age International Pvt. Ltd
Principles of Ground Water Flow
Version 2 CE IIT, Kharagpur

Principles of groundwater flow

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Principles of groundwater flow