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```CHAPTER 2
Principles of Groundwater Flow
Simon Mkhandi
Department of Civil Engineering. P.O.Box 35131, Dar es Salaam, Tanzania.
Akpofure E. Taigbenu
Department of Civil & Water Engineering, National University of Science &
Technology, P.O. Box AC 939. Bulawayo, Zimbabawe.
Brian K. Rawlins
Department of Hydrology, University of Zululand, P Bag X1001,
The following competencies can be achieved on completion of this chapter:
 Define some aquifer parameters and groundwater flow variables.

Explain storage mechanisms in confined and unconfined aquifer.

Derive the equations that govern flow in groundwater systems.

Provide solutions to simple flows in phreatic aquifers.

Highlight some features of surface water – groundwater interaction from
those solutions.
2.1 Groundwater flow
Flow in ground water is in general complex if the details of the flow within the
interconnected pores (interstices) are taken into consideration. Those details of
the flow at microscopic level through the irregular interstices with irregularly
shaped soil particles are overlooked by considering the flow on a macroscopic
scale.
To achieve this macroscopic perspective to the flow, a representative
elementary volume (REV) serves as a point in the continuum approach to the flow
analysis. The REV is a conceptual volume containing soil grains and pore space
within which the spatial and temporal distribution of flow variables (density,
velocity, temperature, pressure, etc) can be assessed in a continuum sense.
When this approach is not adopted, then a point could be in the solid phase when
it falls on a soil grain or in the liquid or gaseous phase when it falls in a pore space
(void).
Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
Considering the total flow Q normal to an area A (containing both soil grains and
pore spaces), then the flow velocity normal to the flow, referred to as the specific
discharge, is given by
q
Q
A
(2.1)
Taking the average volumetric porosity n (see definition in chapter 1) to be equal
to the average areal porosity, the actual area over which flow takes place is nA ,
the rest being occupied by the solid soil grains. The average velocity V through
connected interstices is given by
V
Q q

nA n
(2.2)
The actual available flow area within the pores is reduced by fluid particles that
are attached to the soil grains by capillary forces and ionic bonding to the soil
grains and as well as dead-end pores that may found within the porous matrix, so
that the effective flow area is now ne A , where ne is the effective porosity. The
expression for the average velocity that is given by eq. (2.2) is now modified to
V
Q
q

ne A ne
(2.3)
Darcy (1856) provided the relationship between the specific discharge q and the
gradient of the energy head that drives the flow. This relationship is now popularly
known as the Darcy law. Using an apparatus similar to that shown in Figure 2.1,
Darcy found that the specific discharge is proportional to the gradient of the
q g
 h h1  h2

L
L
(2.4)
where g denotes the hydraulic gradient and h  p /   z is the hydraulic head. It
is noted in eq. (2.4) that  h  h1  h2 represents the energy loss arising mainly
from frictional losses as the fluid meanders through the tortuous paths in the
interstices.
The kinetic energy or dynamic head component is usually not
considered in the energy loss calculations because it is considered negligible
compared to the pressure head p / 
and elevation or potential head z due
primarily to the sluggish nature of groundwater flow where velocities are generally
small. The constant of proportionality in eq. (2.4) is referred to as the hydraulic
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
conductivity K and it reflects the fluid transmitting capacity of the porous
medium. Introducing the hydraulic conductivity K , eq. (2.4) becomes
q  K
h1  h2
L
(2.5)
The way that Darcy law given by eq. (2.5) has been written indicates a flow that is
predominantly in one direction. Subsequently Darcy law is extended to flow in
higher dimensions of flow (2 and 3 dimensions).
Figure 2.1: Setup to demonstrate Darcy law
When the flow occurs in higher dimensions, in general three dimensions, then
Darcy law becomes
 qx 
  h /  x
 


q   q y    K h   K   h /  y 
q 
  h /  z
 z


(2.6)
The specific discharge is now a vector that has three components q x , q y and q z
in the x , y , and z directions. The minus sign in eq. (2.6) is intended to correctly
indicate that flow proceeds in the direction of decreasing hydraulic head. The
hydraulic conductivity K is now a tensor that, in general, has 9 elements. That is
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
K
 xx
K   K xy
K
 xz
K xz 

K yz 
Kzz 
K xy
K yy
K yz
(2.7)
It is a symmetric tensor that can be diagonalized into its principal directions x * ,
y * , and z * , so that K becomes
 K x* x*

K  0
 0
0
K y * y*
0
0 

0 
Kz*z* 
(2.8)
Further discussion on the hydraulic conductivity tensor is carried out on the basis
of its behaviour for different types of formations that can be encountered.
2.1.1 Hydraulic Conductivity
The hydraulic conductivity K is a parameter that occurs in most of the relations in
Darcy law, and it is sometime defined as the flux per unit area per unit hydraulic
gradient. It is a scalar with the dimension of L/T. It represents the capacity of the
formation to transmit water, and as such it depends on the porous matrix and as
well as the fluid properties. The factors of the porous matrix on which depends
the hydraulic conductivity include the soil grain size, shape, size distribution,
tortuosity, porosity, etc, while the fluid properties include density  and viscosity
 . These factors are expressed in the relationship for the hydraulic conductivity
that is given by
K
k g


kg
(2.19)

where    /  is the kinematic viscosity of the fluid, and k is the intrinsic
permeability that depends on the properties of the porous matrix. There are a
number of empirical formulae relating the intrinsic permeability and the properties
of the porous matrix (Bear, 1979). The hydraulic conductivity is determined in the
laboratory with permeameters of which are the constant-head permeameter that is
suited for noncohesive soils like sand, gravel and rocks, and the falling-head
permeameter that is generally suited for cohesive soils with low hydraulic
conductivity.
Typical values of hydraulic conductivity of various rocks are
presented in Table 2.1.
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
Table 2.1: Typical values of Hydraulic Conductivity (from Bouwer, 1978)
Geological classification
Unconsolidated materials:
Clay
Fine sand
Medium sand
Coarse sand
Gravel
Sand and gravel mixes
Clay, sand, gravel mixes (e.g. till)
K (m/d)
10-8 – 10-2
1–5
5 – 20
20 – 102
102 – 103
5 – 102
10-3 – 10-1
Rocks:
Sandstone
Carbonated rock with secondary porosity
Shale
Dense solid rock
Fractured or weathered rock (Core samples)
Volcanic rock
10-3 – 1
10-2 – 1
10-7
< 10-5
Almost 0 – 300
Almost 0 - 103
2.1.2 Range of Validity of Darcy law
Darcy law provides a linear relationship between the specific discharge q and the
hydraulic gradient g , and it is valid when the average flow is laminar in nature.
That is the case for most regional groundwater flow problems of practical interest
where the flow velocities are small so that inertia (driving force) is small compared
to the resisting force due to viscous action. The ratio of inertia force to viscous
force is accounted for by a dimensionless parameter known as the Reynolds
number Re . It is expressed as
Re 
qd
(2.9)

where d is a typical size of the solid grains. The typical size is sometimes taken
as the d10 size that can be obtained from a particle size analysis. It is the sieve
diameter that allows 10% by weight of the grains to pass through. Darcy law is
considered valid when the Reynolds number does not exceed 10, and in most
practical flows, this condition is satisfied, except in pockets of the flow region
(discharge areas) where it may not be satisfied.
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2.1.3 Isotropy and Anisotropy
Isotropy and anisotropy refer to the direction dependence of the hydraulic
conductivity. A formation is said to be isotropic if the hydraulic conductivity is the
same in all three directions x , y , and z . That is Kxx  K yy  Kzz . Conversely,
when Kxx  K yy  Kzz , then the formation is said to be anisotropic. Anisotropy
arise from depositional patterns, pressure from overburden soil, flow channels due
to chemical weathering, and structural fissures.
These factors give rise to
preferential flow directions in the formation.
2.1.4 Homogeneity and Heterogeneity
Geological formations are said to be homogeneous when the hydraulic
conductivity does not vary from one location to another. In that case, K xx , K yy
and K zz are not dependent on
x , y , or z .
A medium is said to be
heterogeneous or nonhomogeneous when the hydraulic conductivity varies with
location.
There are two types of heterogeneity.
The first type refers to
heterogeneity in which the variation in hydraulic conductivity can be represented
by some functional distribution. That is K xx  K xx ( x , y , z) , K yy  K yy ( x, y, z) and
Kzz  Kzz ( x , y , z) .
This type of heterogeneity occurs when the geological
processes that give rise to the formation produce a random distribution in
formation hydraulic conductivity. The second type of heterogeneity occurs when
there exist regions of homogeneous formations that are separated by sharp
discontinuities in hydraulic conductivity. When examining the flow in this type of
formation, the compatibility relations with respect to the hydraulic head and flux at
the interface of sharp discontinuities in hydraulic conductivity have to be
examined.
Type 2 heterogeneity formations may present it self as stratified
formations in a particular direction.
2.2 Mechanisms of storage in Aquifers
2.2.1 Confined Aquifers
Confined aquifers are known to be pressured aquifers because water in their
pores is under pressure from the confining strata and overburden soil. At any
typical location (Figure 2.2), the stresses will be borne by the soil grains (matrix of
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
the aquifer) and water in the pores. If we denote the total stress as  , then the
following relationship, first proposed by Terzaghi (1925), is given as (Lambe &
Whitman, 1979)
  p
(2.10)
where   is the stress borne by the soil grains and referred to as the intergranular or effective stress, and p the pore water pressure.
Both the
intergranular stress and pore water pressure act in a compensating manner when
there is no change in the total stress on the formation. That is the case when
there is withdrawal or addition of water into storage due to pumping activities from
wells located in the aquifer. In that case, the change in  is zero, that is d  0 ,
and equation (2.10) becomes
d    dp
(2.11)
Figure 2.2 Stresses at a plane in a soil.
When withdrawal is taking place, the reduced pore water pressure results in a
corresponding increase in the intergranular or effective stress, and vice-versa.
Because pressure changes can be quite significant, particularly in confined
aquifers, the compressibility of water and the soil matrix become quite important.
Under ordinary pressures within the range of unit atmosphere, water is considered
incompressible, but for larger values of pressures, water can no longer be treated
as incompressible.
When pore water pressure is reduced, water expands
(attempts to occupy a larger volume), and this results in expulsion of water from
the pores, thus enhancing depletion from storage.
At the same time, the
increased intergranular stress gives rise to the soil matrix collapsing (like a
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
sponge), resulting in reduced porosity that aids expulsion of water from the pores.
On the other hand, when the aquifer is being recharge through wells, increase in
storage is similarly enhanced by the compressibilities of the pore water and soil
matrix. In this case, the increased pore water pressure causes water in the pores
to contract, enhancing storage of water. At the same time, reduced intergranular
stress causes the soil matrix to expand, giving rise to marginal increase in porosity
that further enhances addition of water into storage. This elastic behaviour of
water and soil matrix of the confined formation are accounted for by the specific
storativity S0 which is defined as the amount of water released from (or added
to) storage in a unit volume of aquifer and per unit decline (or rise) in pressure (or
S0 
 Uw
U h
(2.12)
where  U w refers to change in volume of water in storage, U the volume of
aquifer, and  h is the change in piezometric head.
Another parameter,
commonly used for confined aquifers, is the storativity or storage coefficient S
which is defined as the amount of water released from (or added to) storage per
unit horizontal area of aquifer and per unit decline (or rise) in piezometric head.
S
 Uw
A h
(2.13)
When the flow is assumed to be essentially horizontal and the confined aquifer is
assumed to have uniform thickness b , then the relationship between the specific
storativity S0 and the storativity S is: S  S0b . If that is not the case, then both
aquifer parameters should not be related, as the specific storativity applies to a
three dimensional flow, while the storativity applies to two-dimensional horizontal
flow. The storativity ranges between 10-6 and 10-3, of which 40% is due to the
compressibility of water and the rest to the compressibility of the porous matrix.
2.2.2 Unconfined Aquifers
The mechanism for storage in unconfined aquifers is slightly different from that in
confined aquifers. For the unconfined aquifer, a water table serves as its upper
boundary.
(The water table is an approximate concept that, in our current
discussions, neglects the capillary fringe and unsaturated column of soil that have
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
influence to some extent on the flow in the saturated region.) The unconfined
nature of the aquifer and the pressures encountered preclude significant
compressibility effects in the water phase and that of the soil matrix. If there are
any elastic effects on storage, they occur at the onset of pumpage. During this
initial time, the unconfined aquifer behaves as a confined one with a
corresponding storativity S . As pumping continues, the water table is lowered
because of drainage of water from the pores at the water table due to pressure
differences at different points on the water table. (Drainage from the pores is
usually not complete because of some portion of water in the pores that will be
held to the soil grains by capillary and ionic forces that are usually much larger
than the drainage forces providing the flow.) This draining of the sediments is
accounted for by the specific yield S y which is the amount of water released
from (added to) storage per unit area of aquifer and per unit decline (rise) of water
table. The specific yield is related to the porosity of the aquifer by the relationship
S y  Sr  n
(2.14)
where S r is the specific retention, indicative of the amount of water retained in the
pores after drainage, and n is the porosity. Because of the above relationship,
the specific yield is also sometimes referred to as the effective porosity. Issues
of delayed yield of an unconfined are sometimes raised in the groundwater
literature, and they refer to the rate of fall of the water table in relation to how
quickly the pores are drained. When the water table falls quickly, the drainage of
the pores may lag behind, but when the water table falls more slowly, pores tend
to completely drain without any time lag. It is this phenomenon that raises the
issue of the time dependence of the specific yield, which largely depends on the
abstraction rate, hydraulic conductivity and porosity of the aquifer. Typical values
of the specific yield S y are in the range of 0.01 and 0.3, and are generally much
larger than the storativity. Figure 1.1 shows the general trend in the relationship
between S y and n for various soils.
2.3
Derivation of equations of Flow
Transient flow in groundwater systems can be described by equations that are
derived by combining Darcy law earlier discussed and the statement of mass
conservation or mass balance (continuity equation). This latter statement states
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
that mass inflow rate minus mass outflow rate equals rate of change of mass
storage. A suitable volume of the flow is defined so that fluid and solid grains are
allowed to enter and exit it while the demarcated volume remains fixed in space
and time. This suitable volume is referred to as the control volume (CV), and it
is chosen so that the information on the property or variable of the flow being
sought can readily be achieved. The CV can be of finite dimensions in which case
the flow property or variable cannot be described at each point or it can be of
infinitesimal dimension so that information on the property of the flow is obtained
at each point of the flow. When the CV is infinitesimal, the resultant equations are
in terms of differential equations that still have to be solved by mathematical tools.
In many instances, it may not be possible to solve these differential equations,
necessitating their being simplified further to become amenable to known solution
techniques.
Using the infinitesimal cube shown in Figure 2.3 with dimensions of  x ,  y , and
 z as CV, mass balance in, out and within it is examined. Across the face ABCD,
the mass inflow rate is given by
QABCD   q x  y z
(2.15)
where  is the fluid density. The mass outflow in the x direction across face
EFGH is obtained by taking the first term of the Taylor’s expansion of the quantity
QABCD . That is
QEFGH  (  q x 
 (  qx )
 x )  y z
x
(2.16)
In the x direction, the net mass flux is given by
QABCD  QEFGH  
 (  qx )
 x y z
x
(2.17)
Making similar calculations for the net mass flux in the other directions, the net
outflow through all the faces is given by
  (  q x )  (  q y )  (  qz ) 



 x y z
y
 z 
 x
2-10
(2.18)
Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
Figure 2.3: Infinitesimal control volume used to derive flow equations.
The rate of change of mass storage within the CV is given as
 (  n)
 x y z
t
(2.19)
On the basis of the statement of mass conservation earlier stated, eqs. (2.18) and
(2.19) can now be combined to yield
  (  q x )  (  q y )  (  qz ) 



y
 z 
 x
 (  n)
t

(2.20)
If the fluid density does not vary spatially but can vary temporally, then  is
treated as a constant with respect to the spatial coordinates x , y and z , and
taken out from the left hand side of eq. (2.20). That is
 q y  qz 
 q
 x 

y
 z 
 x

1  (  n)
 t
(2.21)
Note that eq. (2.21) can be written in short hand notation as
 q 
1  (  n)
 t
(2.22)
Applying Darcy law to the expression for the specific discharge vector, and
assuming that the principal coordinates coincide with the coordinates x , y and z ,
then eq. (2.22) becomes
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
  [K h] 
1  (  n)
 t
(2.23)
or, in long hand,
 
h  
h  
h
 Kxx  
 Kzz 
 K yy  
x 
x  y 
y  z 
z 

1  (  n)
 t
(2.23)
Eq. (2.23) is the governing equation for three dimensional groundwater flow in a
compressible porous formation. The storage term on the right hand side of the
equation can be related to the mechanism for storage change in confined and
unconfined aquifers discussed in section 2.2. That relationship is derived here for
a confined aquifer, and simply stated for an unconfined aquifer.
Expanding the storage term yields
1  (  n) 1   n
   p
 
n 
 t
  p
p  t
(2.24)
The term  /  p that indicates the change of the density of water to pressure
changes can be related to the compressibility of water. The compressibility of
water  is defined as the volumetric change of water with respect to increase in
pressure. That is

1  U w 1 

Uw p
 p
(2.25)
where U w is the volume of water in the pore spaces or void, and the minus sign in
eq. (2.25) indicates that the volume of water decreases with increase in pressure
and vice-versa. The term  n /  p can be related to the elastic behaviour of the
porous or soil or formation matrix. This elastic behaviour should not be confused
with that of the soil grains that are themselves essentially inelastic. Changes in
the intergranular or effective stress do not give rise to changes in the sizes of the
individual soil grains but the size of the porous matrix. The compressibility of the
porous matrix  is defined as
 
1  Ub
U b  '
(2.26)
where U b is the bulk volume of the soil matrix, and it is related to the volume of
the soil grains U s , which remains the same despite changes in pore pressure and
intergranular stress, as: U s  (1  n)U b  constant.
2-12
Noting the compensating
Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
character of changes in pore water pressure and changes in effective stress when
there is no change in the total stress on the formation, that is  p     ' , then the
compressibility of the porous matrix given by eq. (2.26) can be expressed as
 
1  [U s / (1  n)] 1  [U s / (1  n)]
1 n


Us
 '
Us
p
(1  n)  p
(2.27)
Now we are in a position to rewrite the storage term of eq. (2.24) in terms of the
elastic properties of water and the porous matrix that are given by eq. (2.25) and
(2.27). That is
1  (  n)
p
  (1  n)  n
 t
t


(2.28)
Recall the expression of the hydraulic head: h  z  p /  g .
The temporal
derivative of the pressure is expressed approximately as:  p /  t   g h /  t , so that
eq. (2.28) becomes
1  (  n)
h
  g[ (1  n)  n ]
 t
t
(2.29)
The factor premultiplying the temporal derivative of the hydraulic head in eq.
(2.29) is the specific storativity
S0
for the confined aquifer.
That is,
S0   g[ (1  n)  n ] so that eq. (2.29) becomes
1  (  n)
h
 S0
 t
t
(2.30)
The differential equation that governs three-dimensional flow in a confined aquifer
can now be expressed as
 
h  
h  
h
 Kxx  
 Kzz 
 K yy  
x 
x  y 
y  z 
z 

S0
h
t
(2.31)
When the aquifer is isotropic but heterogeneous, then K  Kxx  Kyy  Kzz and eq.
(2.31) becomes
  h   h   h
K  
K 
K  
x  x  y  y  z  z 

S0
h
t
(2.32)
Note that in this case, K is in general a function of x , y and z . For an isotropic
and homogeneous aquifer, the hydraulic conductivity is a constant and can be
taken out of the spatial derivatives. Eq. (2.31) becomes
 2h  2h  2h



 x2  y2  z2
S0  h
K t
(2.33)
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If the flow is steady and/or water and porous matrix is incompressible, the right
hand side of eq. (2.33) vanishes and the resultant governing equation becomes
 2h  2h  2h


 0
 x2  y2  z2
or
 2h  0
(2.34)
This is the well known Laplace equation. It governs steady groundwater flow in an
isotropic, homogeneous, incompressible confined formation.
For an unconfined aquifer where storage change is largely due to drainage of
water from the pores due to the lowering of the water table and minimally due to
elastic behaviour of water and the soil matrix, the storage term is negligibly small.
That is
1  (  n)
0
 t
(2.35)
For the unconfined aquifer, the differential equation that governs three
dimensional flow is given by
 
h  
h  
h
 Kxx  
 Kzz 
 K yy  
x 
x  y 
y  z 
z 

0
(2.36)
or
  (K h)  0
(2.36)
When the unconfined aquifer is isotropic and homogeneous, the flow equation
reduces to Laplace equation earlier presented.
It must be said here that although the unconfined aquifer is largely considered as
inelastic so that the mass conservation equation (2.22) is   q  0 and its
governing equation (2.36) in three dimensions appears simpler than its confined
flow counterpart (2.31), one must not loose sight of the fact that a water table or
phreatic surface serves as the upper boundary of the unconfined aquifer. The
location of the phreatic surface which is sought and its nonlinear movement
makes the solution to the unconfined flow problem more challenging than the
confined case. Without elaborating on the derivation of the equation that governs
the movement of the phreatic surface, the conditions that have to be satisfied on it
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
are simply stated. On the phreatic surface, the pressure is atmospheric ( p  0 ),
so that the hydraulic head is
h( x , y , z, t )  z or F ( x, y , z, t )  h( x, y, z, t )  z  0
(2.37)
where F ( x, y, z, t ) is the equation that described the phreatic surface.
The
movement of the surface is described by the relationship
Sy
F
 q  F  0
t
(2.38)
which is expanded to
2
   h  2  h  
 h
 h 
 h
Sy
  Kxx    K yy    Kzz        0
 x 
 t 
 z  
 y 
   z 
2
(2.39)
Observe how highly nonlinear the equation governing the movement of the
phreatic surface is. It is a great challenge to even numerical techniques in solving
the unconfined flow problem in three dimensions with this nonlinear equation on
the phreatic surface.
For an isotropic, homogeneous unconfined aquifer, eq.
(2.39) becomes
h
 h 
   h    h    h 
Sy
 K           0
t
 x 
   x    y    x 
2
2
2
(2.40)
It is noted that in all the flow equations so far derived, it has been assumed that
there are no sources or sinks arising from abstracting and recharge wells or
distributed recharge from accretion in the case of unconfined aquifers. When
those sources/sinks are present, an additional term has to be included in the flow
equation. To include such a term in eq. (2.31) yields
 
h  
h  
h
 Kxx  
 Kzz   Q 
 K yy  
x 
x  y 
y  z 
z 
S0
h
t
(2.41)
where Q is the rate of volume of water contributed by the source/sink by unit
volume of aquifer. It has a dimension of T 1 .
2.3.1 Horizontal Groundwater Flow
If the flow is considered in the confined or unconfined to be essentially horizontal,
then the piezometric head for the confined aquifer and water table elevation is
then dependent on x , y and t . In such a case, streamlines are horizontal. The
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
flow equations obtained earlier are simplified further. It is common to refer to the
approach that considers the flow to be horizontal as the hydraulic approach in
contrast to the hydrodynamic one that addresses the full three-dimensional
equations. The hydraulic approach is widely adopted because of the natural state
of most aquifers where the lateral dimensions are of many order of magnitude
larger than the vertical dimension so that variations in the flow in the vertical
dimension are negligible. For the unconfined aquifer, the hydraulic approach is
also commonly referred to as the Dupuit-Forchheimer approach or assumption
because of the observations of small slopes of the water table in unconfined
aquifers that were made by these two investigators. The hydraulic or essential
horizontal flow approach and Dupuit-Forchheimer assumptions are widely
adopted in solving regional groundwater flow problems largely because they
approximate most practical flow situations and the resultant equations are simpler
to solve. To obtain the differential equation for horizontal flow in aquifers, the
equation in three dimensions is integrated over the flow depth. For an anisotropic,
heterogeneous confined aquifer of uniform thickness b , the differential equation is
 
h  
h
 Kxxb  
 K yyb  
x 
x  y 
y 
S0b
h
t
or
  h   h
T
 
T
 
 x  xx  x   y  yy  y 
S
h
t
(2.42)
where Txx and Tyy are the transmissivities in the x and y directions, and S is the
storativity. If the medium is isotropic and homogeneous, then T  Txx  Tyy and eq.
(2.42) becomes
 2h  2h


 x2  y2
S h
T t
(2.43)
If the flow is steady and/or water and soil matrix are considered to be inelastic,
then the right hand side of eq. (2.43) vanishes. In that case, we have
 2h  2h

 0
 x2  y2
or
 2h  0
(2.44)
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This is again Laplace equation in two dimensions. It governs steady horizontal
flow in isotropic, homogeneous confined aquifers.
For horizontal flow in unconfined aquifers, integrating the differential equation in
three-dimensions over the flow depth from a horizontal impervious bottom to the
water table yields the equation
 
h  
h
 Kxx h  
 K yy h  
x 
x  y 
y 
Sy
h
t
(2.45)
Note that the nonlinear movement of the phreatic surface has been incorporated
in eq. (2.45) by the depth integration of eq. (2.36) over the flow thickness. If the
unconfined aquifer is isotropic and homogeneous, then the flow eq. (2.45)
becomes
  h   h
h  
h  
x  x  y  y 
Sy  h
K t
or
 2 h2  2 h2


 x2
 y2
2S y  h
K t
(2.46)
Eq. (2.46) is nonlinear in the water table distribution h , and it is commonly
referred to as the Boussinesq equation. When the flow is steady, then eq.
(2.46) becomes
 2 h2  2 h2

 0
 x2
 y2
or
 2 h2  0
(2.47)
Note that for steady horizontal unconfined flow, the Laplace equation is expressed
in terms of h 2 instead of h as in the confined flow case.
2.3.2 Physically Imposed Conditions
All the differential equations earlier derived for flow in aquifers do have unique
solutions only when appropriate conditions are applied on the boundaries of the
flow region and, for transient flows, as well as conditions at the initial time. The
conditions applied on the boundary are referred to as boundary conditions and
they reflect known relations for the hydraulic head and specific discharge when
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certain boundaries adjoin the aquifer of interest. It is with these known values for
the hydraulic head and/or discharge on the flow boundary that the distribution for
unknown quantities on the boundary and within the flow region can be calculated.
These conditions are discussed with respect to the horizontal approximation to
flow in aquifers.
This is a boundary on which the
aquifer is in hydraulic contact with a large water body like a river or lake. The
hydraulic head on the boundary equals the known water elevation in the river or
lake. This condition is stated as:
h  H1 ( x , y , t )
(2.48)
where H1 ( x, y, t ) is the water elevation in the river or lake.
Boundary of prescribed discharge:
Along such a boundary, the normal
discharge flux in or out of the aquifer is known. That is
Qn   KA
h
 f1 ( x, y, t )
n
(2.49)
where f 1 ( x, y, t ) is the known flux and A is the flow area. When the boundary in
impervious, f 1 ( x, y, t )  0 implying that  h /  n  0 .
There are other types of boundaries that can be encountered, but the above two
are the most common. The initial condition specifies the spatial distribution of
the water table an unconfined aquifer or the piezometric head for a confined
aquifer at the initial time.
It is this distribution that is used to calculate the
The solutions to the flow equations in three and two dimensions in confined and
unconfined aquifer can be achieved by analytical and numerical methods. The
approach based on the latter is the subject of another module on Groundwater
Modelling. Analytical methods are applied in this and subsequent chapters. They
can be applied only for simplified flow problems, and in certain situations the
above flow equations do not have to be solved directly but by prudent choice of a
CV and application of continuity and Darcy law, the analytical solutions can be
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obtained. Analytical solutions to a few unconfined flows are addressed in the
remainder of this chapter.
2.4
Flow in unconfined aquifers
2.4.1 Dupuit-Forchheimer assumptions for phreatic aquifers
The Darcy law coupled with the statement of mass conservation can be used to
solve simple systems of either lateral or vertical groundwater flow. It is common
though for systems to have both vertical and horizontal flow components. These
systems can be simplified so that the flow is in only one direction before Darcy law
is applied. Vertical flow components are often neglected where groundwater
moves primarily in a lateral direction and the flow is then considered to be purely
horizontal and also to be uniformly distributed with depth. These assumptions,
collectively called the Dupuit-Forchheimer assumption are used when solving
unconfined flow systems.
A simple illustration of the use of the Dupuit-Forchheimer assumption is outlined in
Figure 2.4 where a stream is situated in an alluvial floodplain where the water in
the stream is in direct hydraulic contact with the groundwater in the floodplain. A
horizontal impermeable layer occurs at a relatively small distance below the
stream bed. Water is flowing from the stream into the aquifer and the water table
falls away from the boundary. The flow from the stream will be vertical beneath
the stream and horizontal through the banks of the stream. To solve this situation,
the vertical flow components are ignored and all flow is considered to be
horizontal and the flow is also considered to be evenly distributed with depth (the
Dupuit-Forchheimer assumptions).
Applying Darcy’s law as used above to this situation between the water table at
the edge of the stream channel (site 1) and another location 300m away from the
stream channel using the water table surface as the streamline yields an estimate
of flow velocity of the water away from the channel.
V3
10.5  10
0.005m / d
300
The Dupuit-Forchheimer assumption indicates that all of the water below the
water table moves at this velocity so the calculation of the discharge rate from the
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stream is carried out using the average height of the water table between the two
sections (10.25m) as the flow thickness. If the width of the flow system is taken as
1m then the seepage from the stream per unit length of stream can be calculated
as 0.005  10.25  0.05125m3 / day . If both sides of the stream experience the same
conditions, the total seepage from the stream can be determined by doubling this
value to 01025
.
m3 / day .
Figure 2.4: Seepage from a stream in an unconfined aquifer with an
impermeable bottom at a relatively shallow depth
The Dupuit-Forchheimer theory loses accuracy in this case if the depth to the
impermeable layer increases since the vertical flow component gains increased
importance. Experimentation by Bower found that the theory gives reasonably
accurate seepage values if the distance of the impermeable layer below the
stream was not more than twice the height of water in the stream.
2.4.2 Steady unconfined flow between two reservoirs
The flow situation illustrated in Figure 2.5 shows flow through a homogeneous
isotropic phreatic aquifer situated between two surface water bodies (reservoirs)
of infinite extent. Flow proceeds from the higher reservoir towards the lower one.
Unlike a confined aquifer situation where the piezometric surface decreases
linearly between the reservoirs, in a phreatic system such as this, the difficulty
arises from the fact that the water table between the two reservoirs represents a
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flow line. The shape of the water table therefore determines the flow distribution
that at the same time governs the shape of the water table.
Figure 2.5: Steady flow in an unconfined aquifer between two reservoirs
To obtain a solution, Dupuit assumed that (1) the hydraulic gradient is equal to the
slope of the water table, and (2) the flow is horizontal and uniform everywhere in a
vertical section. These assumptions, although permitting a solution to be obtained,
limit the application of the results. For unidirectional flow in Figure 2.5, the
discharge per unit width Q at any vertical section can be given as:
Q   Kh
dh
K dh2

 constant
dx
2 dx
(2.50)
where h is the height of the water table above an impervious base and x is the
direction of flow. Integrating,
Qx  
K 2
h c
2
(2.51)
Applying the condition at x  0 where h  h0 , then the Dupuit equation becomes
Qx 
K 2
[h0  h 2 ]
2
(2.52)
which indicates that the water table has a parabolic profile. For flow between two
bodies of water of constant heads h0 and h L separated by a distance L as in
Figure 2.5, the water table slope at the upstream boundary of the aquifer is
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dh
Q

dx
Kh0
But the boundary h  h0 is an equipotential line because the fluid potential in a
water body is constant; consequently, the water table must be horizontal at this
section, which is inconsistent with eq. (2.53). In the direction of the flow, the
parabolic water table described by eq. (2.52) increases in slope. By so doing, the
two
Dupuit
assumptions,
previously
stated,
become
increasingly
poor
approximations to the actual flow; therefore, the actual water table deviates more
and more from the computed position in the direction of flow as indicated in Figure
2.5. The fact that the actual water table lies above the computed one can be
explained by the fact that the Dupuit flows are all assumed horizontal, whereas
the actual velocities of the same magnitude have a downward vertical component
so that a greater saturated thickness is required for the same discharge. At the
downstream boundary a discontinuity in flow forms because no consistent flow
pattern can connect a water table directly to a downstream free-water surface.
The water table actually approaches the boundary tangentially above the water
body surface and forms a seepage face.
Consequently, the water table does not follow the parabolic form but for shallow
slopes it closely predicts the water table position except near to the outflow point.
The above discrepancies indicate that the water table does not follow the
parabolic form of eq. (2.52), nevertheless, for flat slopes, where the sine and
tangent are nearly equal, it closely predicts the water table position except near
the outflow. The equation, however, accurately determines Q or K for given
2.4.3 Example of unconfined flow
An unconfined aquifer has a hydraulic conductivity of 1.8m/day and a porosity of
0.3. The aquifer is a bed of sand of uniform thickness of 35m. Two observation
wells A and B, separated by a distance of 200m, indicate water levels of 22.5m
and 24m below the land surface, respectively.
discharge per unit width, (b) the average pore velocity, and (c) the water table
elevation midway between the two wells.
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Solution
. m/ d
h0  35  22.5  12.5m , L  200m , hL  35  24  11m , K  18
Q
K (h02  hL 2 ) 18
. (12.52  112 )

 0157
. m3 / d / m width of aquifer.
2L
2  200
(b) The average pore velocity is
V
Q
nA
The flow area can be taken as the average flow area between sections A
. m2 / m width of aquifer. The average
and B. That is A  (12.5  11) / 2  1175
.
/ (1175
.  0.3)  0.0134m / d .
pore velocity is found to be V  0157
(c) The water table midway is found by making use of eq. (2.52). That is
hmidway 2  h02 
hmidway 
2Qx
2  0157
.
 100
 12.52 
 138.8m2
K
18
.
138.8m  118
.m
2.4.4 Steady unconfined flow between two reservoirs with uniform recharge
This particular unconfined flow is illustrated in Figure 2.6a. Examining continuity
of flow in the CV shown in Figure 2.6b of length dx , inflow = Q  Wdx and outflow
= Q  dQ . Under steady conditions, inflow equals outflow, and as such
dQ  Wdx
or
dQ
W
dx
(2.54)
By Darcy law: Q   Khdh / dx   K / 2dh2 / dx , so that eq. (2.54) now becomes
d 2h2
2W
2  
dx
K
(2.55)
Integrating eq. (2.55) twice yields
h2  
W 2
x  c1 x  c2
K
(2.56)
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where c1 and c2 are constants of integration. These constants are evaluated from
the conditions of the water table elevation at the two reservoirs, that is h  h0 at
x  0 and h  hL at x  L . Implementing these conditions yield
c2  h02 and c1 
hL 2  h0 2 W

L
L
K
(2.57)
Introducing these constants into eq. (2.56) yields
W
hL 2  h0 2
h  x( L  x) 
x  h02
K
L
2
(2.58)
(a)
(b)
Figure 2.6: (a) Steady unconfined flow with vertical recharge; (b) Control
volume used for analysis.
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A water divide could be encountered at which flow occurs from it in either direction
to the reservoirs and the height of the water table is a maximum. The location of
the divide is found from the fact that the flux is zero, that is dh / dx  0 at x  x D .
Differentiating the water table profile given by eq. (2.58) with respect to x yields
the expression for the location of the divide
L K (h0 2  hL 2 )
xD  
2 W
2L
(2.59)
The flow to the two rivers or reservoirs which represents the base flow is
estimated by evaluating the Darcy flux at x  0 and x  L . The Darcy flux is
Q 
K dh2
1
h 2  h02 
  W ( L  2 x )  L

2 dx
2
KL 
(2.60)
The base flow to the river at x  0 is
1
hL 2  h02 
Q0   WL 

2
KL 
(2.61)
while the base flow to the river at x  L is
1
hL 2  h02 
QL     WL 

2
KL 
(2.62)
When the water levels in the two reservoirs are the same with elevation h0 , the
above equations are simplified. The water table distribution of eq. (2.58) becomes
h2 
W
x ( L  x )  h0 2
K
(2.63)
The water table profile is known as the ellipse equation because the predicted
water table profile is a section of an ellipse. The water divide, at which the water
table elevation is highest, occurs midway between the reservoirs and the
maximum elevation of the water table is obtained by substituting x  L / 2 into eq.
(2.63) to obtain
hmax 2 
WL2
 h02
4K
(2.64)
Equations (2.63) and (2.64) are useful for analyzing flow toward subsurface
agricultural drains in humid regions where W is approximately constant (see
Figure 2.7). In particular, eq. (2.64) can be used to estimate the drain spacing L
required to ensure that hmax does not exceed a value that has been determined to
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be satisfactory for crop growth.
In this case, the rivers or reservoirs are
considered as part of a system of ridges and drains in a farm.
2.4.5 Example of unconfined flow with uniform recharge
A long canal was constructed running parallel to a river 1.5km away. Both fully
penetrate a sand aquifer with hydraulic conductivity of 1.2m/d. The area is subject
to a rainfall of 1.8m/year and evaporation of 1.3m/year. The elevation of water in
the river is 33m and in the canal it is 28m. Determine (a) the water divide (where
specific discharge is zero), (b) the maximum water table elevation, (c) the steady
discharge per kilometre into the canal, and (d) the steady discharge per kilometre
into the river
Solution
. m / d , W  18
.  13
.  0.5m / yr  137
.  103 m / d
h0  33m , L  1500m , hL  28m , K  12
(a) The water divide is given by eq. (2.59)
xD 
L K (h02  hL 2 )
12
.
(332  282 )

 750 
 661m
2 W
2L
0.00137 2  1500
(b) The maximum water table elevation is found from substituting x D  661m
for x in eq. (2.58)
hmax 2 
0.00137
282  332
 661  (1500  661) 
 661  332  1587.74m2
12
.
1500
hmax  1587.74  39.85m
(c) The steady discharge per kilometre into the canal is obtained from eq.
(2.62)
1
282  332 
. 
  2.055  0169
    0.00137  1500 
  103 m3 / d / km
   

2
12
.  1500 
2
Qcanal
Qcanal  111
.  103 m3 / d / km
(d) The steady discharge per kilometre into the canal is obtained from eq.
(2.61)
Qriver
1
282  332 
. 
  2.055  0169
    0.00137  1500 
  103 m3 / d / km
   

2
12
.  1500 
2
Qriver  0.943  103 m3 / d / km
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2.5
Flow through stratified phreatic aquifers
The previous solutions have all assumed that groundwater flow is in a
homogeneous and isotropic aquifer. In reality, such aquifers rarely exist and the
hydraulic conductivity is not the same in all directions. This is particularly so for
unconsolidated alluvial aquifers where anisotropy tends to be more common.
Variations in hydraulic conductivity are caused by the fact that where alluvial
material is deposited under water, the particles are rarely spherical and so when
the particles settle out they tend to come to rest with their flat sides down. This
tends to have the effect of leading to horizontal hydraulic conductivities being
greater than the vertical hydraulic conductivities. Another cause for anisotropy is
that alluvium typically consists of layers of different materials, each of which
possesses a unique value for hydraulic conductivity. If these layers are horizontal,
then any single layer with a relatively low hydraulic conductivity (e.g. clay) retards
vertical flow through the entire alluvial bed. Horizontal flow, however, can occur
easily through any layer having a relatively high hydraulic conductivity. In a typical
field condition with alluvial deposits it is therefore common to find a hydraulic
conductivity in the horizontal direction K x that is greater than the vertical hydraulic
conductivity K y . Ratios of Kx / K y usually fall in the range of 2 to 10 in alluvial
materials but values of up to 100 or more may occur where clay layers are
present. For consolidated geological materials, anisotropic conditions are
governed by the orientation of strata, fractures, joints, solution openings or other
structural conditions. These do not necessarily possess a horizontal alignment.
Figure 2.7: System of ridges and drains in a farm receiving uniform recharge
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
2.5.1 Flow in a Horizontally stratified Phreatic Aquifer
The case of flow in a horizontally stratified aquifer between two reservoirs at water
elevations h0 and hL is shown in Figure 2.8. The flow in the lower layer is
Q1   K1b
dh
dx
(2.65)
and that in the upper layer is given as
Q2   K2 (h  b)
dh
dx
(2.66)
The total flow per unit length of the aquifer at any section is: Q  Q1  Q2 
By carry the integration with respect to x and
implementing the condition that at x  0 h  h0 , yields
Qx  K1b(h0  h)  K2 [
h02  h2
 b(h0  h)]
2
(2.67)
Applying the condition that at x  L h  hL , then the Dupuit discharge is given as
Q  K1b
(h0  h) K2

[(h0  b) 2  (hL  b) 2 ]
L
2L
(2.68)
The expression for the discharge indicates that the steady discharge is as a
combination of confined flow in the lower stratum and unconfined flow in the upper
one.
Figure 2.8: Flow in a Horizontally stratified Phreatic Aquifer
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
2.5.2 Flow in a Vertically stratified Phreatic Aquifer
Using a vertically stratified unconfined formation with two strata that lies between
two rivers or reservoirs at water levels h0 and hL (Figure 2.9), the steady
discharge is estimated using Dupuit assumptions that the flow is essentially
In the first region 0  x  L1 , the expression for the water table is
horizontal.
similar to eq. (2.52). That is
2Qx
K1
h 2  h02 
(2.69)
Inserting the condition that at x  L1 , h  h1 , though still to be evaluated, into eq.
(2.69) yields
2QL1
K1
h12  h02 
(2.70)
For the second region L1  x  L , the expression for the water table is similarly
written
h 2  h12 
2Q( x  L1 )
K2
(2.71)
Inserting the condition that at x  L , h  hL into eq. (2.71) yields
hL 2  h12 
2QL2
K2
(2.72)
Substituting eq. (2.70) into (2.72) yields
 L
L 
hL 2  h02  2Q 1  2 
 K1 K2 
or
Q
h02  hL 2
 L
L 
2 1  2 
 K1 K2 
(2.73)
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
Figure 2.9: Flow in a Vertically stratified Phreatic Aquifer
The expression for the discharge is accurate, but not for the water table
distribution that are given by eqs. (2.69) and (2.71) due the failure of the Dupuit
approximation to account for the seepage face. The expression for the discharge
can be extended to the case when there are N strata as
h02  hL 2
Q
N L
2 i
i  1 Ki
2.6
(2.74)
Exercises
1. Water flows between a vertical sand column of 120cm length and 200cm 2
area.
The difference between water levels at the inflow and outflow
reservoirs is 30cm. If the porosity of the sand is 0.36 and its hydraulic
conductivity is 20cm/d, determine (a) the total discharge through the
column, (b) the specific discharge, (c) the average flow velocity, and (d) the
2. Define the storativity for confined and unconfined aquifers. A volume of
water of 40 × 106m3 has been pumped from a phreatic aquifer through
wells that are more or less, uniformly distributed over an area of 100km 2 of
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Waternet M.Sc in Integrated Water Resources Management: Introduction To Hydrogeology, Chapter 2
the aquifer. If the aquifer’s specific yield is 0.2, determine the average
drawdown of the water table over the area.
3. A long canal was constructed running parallel to a river 2.5km away. Both
fully penetrate a sand aquifer with hydraulic conductivity of 3.1m/d. The
area is subject to a rainfall of 2.6m/year and evaporation of 0.9m/year. The
elevation of water in the river is 45m and in the canal it is 40m. Determine
(a) the water divide (where specific discharge is zero), (b) the maximum
water table elevation, (c) the steady discharge per km length of canal into
the canal, and (d) the steady discharge per km length of river into the river.
The following standard groundwater textbooks have been consulted in the
creation of this chapter. They all cover in varying degrees of detail the material
covered in this chapter. Students are recommended to consult these for further
information and clarification of the course content. In addition, students are
encouraged to make use of the Internet where a wide variety of resources can be
found.
2.7
References
1. Bear, J (1979) Hydraulics of Groundwater, McGraw-Hill Book Company,
New York.
2. Bowen, R (1986)
Groundwater, Elsevier Applied Science Publishers,
London.
3. Bower, H (1978) Groundwater hydrology, McGraw-Hill, Tokyo.
4. Driscoll, F.G (1986)Groundwater and Wells, Johnson Screens, St Paul.
5. Fetter, C.W (1994) Applied hydrogeology, Prentice Hall, Englewood Cliffs.
6. Freeze, R. Allan and Cherry, J.C (1979) Groundwater, Prentice–Hall, Inc.,
New Jersey.
7. Todd, D.K (1980) Groundwater Hydrology, 2ed. John Wiley, New York.
8. Meinzer, O.E (1949) The Physics of the Earth IX, Hydrology, Dover
Publications, New York.
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