CEL251 Hydrology

SUBSURFACE FLOW – II: Ground Water Flow

Introduction
Study of subsurface flow as ground water is important since about 30% of the world’s fresh
water resources exist in the form of groundwater. Moreover more than 99% of available
source of liquid fresh water lies underground. The main source of groundwater is
precipitation. A portion of rain falling on the earth’s surface infiltrates into ground travels
down and, when checked by impervious layer to travel further down, forms ground water.
The ground water reservoir consists of water held in voids within a geologic stratum.
Groundwater may flow into streams, rivers, marshes, lakes and oceans, or it may discharge in
the form of springs and flowing wells. Groundwater flows through permeable material, which
contains interconnected cracks or spaces that are both numerous enough and large enough to
allow water to move freely. In some permeable materials groundwater may travel several
meters in a day, in other places, it moves only a few centimeters in a very slowly through
relatively impermeable materials such as clay and shale. Thus, the residence time of the
groundwater i.e, the length of time water spends in the groundwater portion of the hydrologic
cycle, varies enormously. Groundwater is distinctive from surface water in the following
respects: (a) Groundwater exists in voids in the subsurface and its movement is very slow. (b)
Pore geometry, soil or rock fracture, surface tension, and flow resistance fundamentally affect
groundwater movement, both in saturated and unsaturated conditions. (c) Topographic and
geologic structures strictly govern groundwater flows. (d) Soil, stratum, rock mineral and
geothermal conditions exert a great influence on the chemical properties of groundwater. The
following Table lists the relative advantages and disadvantages of ground water over surface
water:
Groundwater Surface water
Advantages Disadvantages
1. Many large-capacity sites available 1. Few new sites available
2. Slight to no evaporation loss 2. High evaporation loss even in humid
3. Require little land area climate
4. Slight to no danger of catastrophic 3. Require large land area
structural failure 4. Ever-present danger of catastrophic
5. Uniform water temperature failure
6. High biological purity 5. Fluctuating water temperature
7. Safe from immediate radioactive fallout 6. Easily contaminated
8. Serve as conveyance systems-canals or 7. Easily contaminated by radioactive
pipelines across lands of others material
unnecessary 8. Water must be conveyed
9. May be used as a source of heat. 9. Can not be used for such purpose.

Disadvantages Advantages
1. Water must be pumped
2. Storage and conveyance use only 1. Water may be available by gravity flow
3. Water may be mineralized 2. Multiple use
4. Minor flood control value 3. Water generally of relatively low mineral
5. Limited flow at any point content
6. Power head usually not available 4. Maximum flood control value
7. Difficult and costly to investigate, 5. Large flows

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 evaluate, and manage 6. Power head available
8. Recharge opportunity usually dependent 7. Relatively easy to evaluate, investigate,
on surplus surface flows and manage
9. Recharge water may require expensive 8. Recharge dependent on annual
treatment precipitation
10. Continuous expensive maintenance of 9. No treatment required of recharge water
recharge areas or wells 10.Little maintenance required of facilities

Natural topographic and geologic systems control the occurrence of groundwater.

Thus groundwater has various types in flow systems based on the topographic and geologic
conditions. The water content in the geologic formations varies with depth below ground
surface. The subsurface occurrence of groundwater may be divided into zones of aeration and
saturation.
Zone of Aeration
The zone aeration consists of pores occupied partially by water and partially by air. In the
zone of aeration vadose water occurs. This general zone may be further subdivided into soil
water zone, the intermediate vadose zone, and the capillary zone. Water in the soil-water
zone exists at less than saturation except temporarily when excessive water reaches the
ground surface from rainfall or irrigation. Excessive or gravitational water drains through the
soil under the influence of gravity and some water is held in the soil by surface tension forces
as capillary water. The amount of water present in the soil water zone depends primarily on
the recent exposure of the soil to
moisture. The existing soil moisture is
extracted by the roots due to
evapotranspiration and soil dries out.
The zone extends from the ground
surface down through the major root
zone. Its thickness varies with soil type
and vegetation.
Available Water: Soils absorb and
retain water, which may be withdrawn
by plants during periods between
rainfalls or irrigations. This water
holding capacity is defined by the
available water, which is the range of plant available water, the moist end being the field
capacity and the dry end the wilting point. Field capacity can be defined as the amount of
water held in a soil after wetting and after subsequent drainage has become negligibly small.
The negligible drainage rate is often assumed after two days; however, different soils possess
varying drainage rates so that quantitative values may not be comparable. The wilting point
defines the water content of soils when plants growing in that soil are reduced to a permanent
wilted condition. Because factors such as soil type and volume and plant type and age
influence wilting point, this moisture content can also be variable.
Zone of Saturation
In the zone of saturation all pores are filled with water under hydrostatic pressure. The
saturated zone extends from the upper surface of saturation down to underlying impermeable
rock. In the absence of overlying impermeable strata, the water table or phreatic surface

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forms the upper surface of the zone of saturation. This is defined as the surface of
atmospheric pressure and appears as the level at which water stands in a well penetrating the
aquifer. Below the water table the porous medium is saturated and at greater pressure than
atmospheric. If the water table intersects the land surface the groundwater comes out to the
surface in the form of springs or seepage. A flat area having water table at or near the land
surface become water logged. Also the position of the water table relative to the water level
in a stream determines whether the stream contributes water to the groundwater storage or it
takes water from the groundwater storage. If the water level in the stream is below the water
table, the groundwater storage contributes water to the stream and such streams, which
receive groundwater flow, are called effluent streams. If the water table is below the bed of
the stream, the stream water percolates to the groundwater storage and such streams, which
contribute to the groundwater storage, are called influent streams. Above the water table
capillary forces can saturate the porous medium for a short distance in the capillary fringe,
however water is held here at less than atmospheric pressure. Above the capillary fringe the
porous medium is usually unsaturated except following rainfall when infiltration from the
land surface can produce saturated conditions temporarily.
In the zone of saturation, groundwater fills all the pores; hence the effective porosity
provides a direct measure of the water contained per unit volume. Groundwater storage
capacity and its movement depend upon the type of porous medium (geological formation).
There are four types of geological formations as follows:
Aquifer: An aquifer is a saturated formation of permeable material to yield significant
quantities of water to wells and springs. This implies an ability to store and to transmit water.
Thus an aquifer transmits water relatively easily due to its high permeability. Unconsolidated
deposits of sand and gravel form good aquifers. Aquifers are generally areally extensive and
may be overlain or underlain by a confining bed, which may be defined as a relatively
impermeable material stratigraphically adjacent to one or more aquifers.
Aquiclude: A saturated but relatively impermeable material that does not yield appreciable
quantities of water to wells though it may contain large amount of water due to its high
porosity; clay is an example.
Aquifuge: A relatively impermeable formation neither containing nor transmitting water
(neither porous nor permeable); solid granite belongs in this category.
Aquitard: A saturated but poorly permeable stratum that impedes groundwater movement and
does not yield water freely to wells, but that may transmit appreciable water to or from
adjacent aquifers and when sufficiently thick, may constitute an important groundwater
storage zone; sandy clay is an example.

Aquifers
Types of Aquifers
Most aquifers are of large areal extent and may be visualized as underground storage
reservoirs. Water enters a reservoir from natural or artificial recharge; it flows out under
action of gravity or is extracted by wells. Ordinarily, the annual volume of water removed or
replaced represents only a small fraction of the total storage capacity. Aquifers may be
classifies as unconfined or confined, depending on the presence or absence of a water table,
while a leaky aquifer represents a combination of the two types.
Unconfined Aquifer: An unconfined aquifer is one in which a water table varies in
undulating form and in slope, depending on areas of recharge and discharge, pumpage from
wells, and permeability. A well driven into an unconfined aquifer will indicate a water level

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corresponding to the water table level at that location. Rises and falls in the water table
correspond to change in the volume of water in storage within an aquifer. Only the saturated
zone of this aquifer is of importance in groundwater studies. Recharge of this type of aquifer
takes place through infiltration of precipitation from the ground surface.
Perched Aquifer: A special case of an unconfined aquifer involves perched water bodies.
This occurs wherever a groundwater body is separated from the main groundwater by a
relatively impermeable stratum of small areal extent and by the zone of aeration above the
main body of groundwater. Clay lenses in sedimentary deposits often have shallow perched

water bodies overlying them. Wells tapping these sources yield only temporarily or small
quantities of water.

Confined Aquifer: Confined aquifers (artesian or pressure aquifers) occur where

groundwater is confined under pressure greater than atmospheric by overlying relatively
impermeable strata. In a well penetrating such an aquifer the water level will rise above the
bottom of the confining bed. Water enters a confined aquifer in an area where the confining
bed rises to the surface; where the confining bed ends underground, the aquifer becomes
unconfined. A region supplying water to a confined aquifer is known as a recharge area;
water may also enter by leakage through a confining bed. Rises and falls of water in wells
penetrating confined aquifers result primarily from changes in pressure rather than changes in
storage volumes. The piezometric surface of a confined aquifer is an imaginary surface
coinciding with the hydrostatic pressure level of the water in the aquifer. Should the
piezometric surface lie above ground surface, a flowing well results. It should be noted that a
confined aquifer becomes an unconfined aquifer when the piezometric surface falls below the
bottom of the upper confining bed. Also, quite commonly an unconfined aquifer exists above
a confined aquifer.
Idealized Aquifer: For mathematical calculations of the storage and flow of ground water,
aquifers are frequently assumed to be homogeneous and isotropic. A homogeneous aquifer
possesses hydrologic properties that are everywhere identical. An isotropic aquifer is one

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with its properties independent of direction. Such idealized aquifers do not exist; however
good quantitative approximations can be obtained by these assumptions.
Aquifer Parameters
Storage Coefficient and Transmissivity are two important parameters used to describe the
behaviour of an aquifer. A portion of water can be removed from a saturated porous medium
by drainage or by pumping of a well; however, molecular and surface tension forces hold the
remainder of the water in place.
Specific Retention: The specific retention Sr of a soil or rock is the ratio of the volume of
water it will retain after saturation against the force of gravity to its own volume. Thus
S r = wr V
where wr is the volume occupied by retained water, and V is the bulk volume of the soil or
rock. (It should be noted that the terms field capacity and retained water refer to the same
water content but differ by the zone in which they occur.)
Specific Yield: The specific yield Sy of a soil or rock is the ratio of the volume of water that,
after saturation, can be drained by gravity to its own volume. Therefore
S y = wy V
where wy is the volume of water drained. Values of Sr and Sy can also be expressed as
percentages. Because wr and wy constitute the total water volume in a saturated material, it is
apparent that
η = (wr + w y ) / V = S r + S y
where η is porosity and all pores are interconnecting. Values of specific yield depend on
grain size, shape and distribution of pores, compaction of stratum, and time of drainage. It
should be noted that fine grained materials yield little water, whereas coarse grained materials
permit a substantial release of water- and hence serve as aquifers.
Storage Coefficient (S): Water recharged to or discharged from, an aquifer represents a
change in the storage volume within the aquifer. A storage coefficient (or storativity) is
defined as the volume of water that an aquifer releases from or takes into storage per unit
surface area per unit change in the component of head normal to that surface. The coefficient
is a dimensionless quantity involving a volume of water per volume of aquifer. For a vertical
column of unit area extending through a confined aquifer the storage coefficient S equals the
volume of water released from the aquifer when the piezometric surface declines a unit
distance. The storage coefficient for an unconfined aquifer corresponds to its specific yield.
Normally S varies directly with saturated aquifer thickness (S = 3 ×10-6b). Storage
coefficients can best be determined from pumping tests of wells.
Transmissivity (T): Transmissivity is defined as the rate at which water of prevailing
kinematic viscosity is transmitted through a unit width of aquifer under a unit hydraulic
gradient, thus
T = Kb
where b is the saturated thickness of the aquifer. Units of T is m2/s.

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Governing Equations for Flow in Saturated Porous Medium
Darcy's law
Flow in saturated porous medium is governed by Darcy's law. Darcy observed that the
volume per unit time passing through a porous medium is directly proportional to the area of
cross section A and the head difference between inlet & outlet (h1 – h2) and inversely
proportional to the length of the medium l therefore
Vol 1
= Q ∝ A (h1 − h2 )
t l
which in terms of discharge velocity v (m/s) is
Q ∂h
v= = −K = −K i
A ∂l
where constant of proportionality K = hydraulic conductivity (m/s); and i = ∂h/∂ l = hydraulic
gradient = rate of head loss per unit length of medium. The negative sign indicates that the
total head is decreasing in the direction of flow because of friction or resistance.
A medium has a unit hydraulic conductivity if it will transmit in unit time a unit
volume of groundwater at the prevailing kinematic viscosity through a cross section of unit
area measured at right angles to the direction of flow, under a unit hydraulic gradient. It has
units of velocity. The hydraulic conductivity of a soil or rock depends on a variety of physical
factors, including porosity, particle size and distribution, shape of particles, arrangement of
particles, and other factors. In general for unconsolidated porous media, k varies with square
of particle size; clayey materials exhibit low values of K, whereas sands and gravels display
high values. The hydraulic conductivity in saturated zones can be determined by a variety of
techniques, including calculation from formulas, laboratory methods, tracer tests, auger hole
tests, and pumping tests of wells.
The velocity v is referred to as the Darcy or discharge velocity because it assumes that
flow occurs through the entire cross section of the material without regard to solids and pores.
Actually the flow is limited to the pore space only so that the average seepage or interstitial
velocity (vs)
v Q
vs = =
η ηA
The pore spaces within the porous medium vary continuously with location. This means that
actual velocity is non-uniform, involving endless accelerations, decelerations, and changes in
direction. Darcy’s law describes a steady uniform flow of constant velocity in which net force
on any fluid element is zero. In applying Darcy’s law it is important to know the range of
validity within which it is applicable. Darcy’s law applies to laminar flow in porous media.
Reynold’s Number for flow through porous is
ρvd10
NR =
µ
where ρ is the fluid density, µ is the dynamic viscosity of the fluid and d10 is the effective
grain size. Darcy’s law is valid for NR < 1 and does not depart seriously up to NR = 10.
Fortunately, most natural underground flow occurs with NR < 1 so Darcy’s law is applicable.
Deviations from Darcy’s law can occur where steep hydraulic gradients exist, such as near

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pumped wells; also, turbulent flow can be found in rocks such as basalt and limestone that
contain large underground openings.

Governing Equations in Cartesian Coordinates

In deriving equation it is assumed that the bottom bed rock is horizontal and x and y axes are
in the horizontal plane coinciding with the bottom bed rock. Also the flow occurs throughout
the aquifer thickness so that flow is two dimensional in the vertical plane.
(a) Confined Aquifer: Applying mass balance for a control volume of size ∆x, ∆y in x-y
plane and of height b (equal to aquifer thickness b) i.e. Inflow – Outflow = Change in Storage

 ∂M x   ∂M y 
Mx −Mx + ∆x  + M y −  M y + ∆y  = ρS (h2 − h1 )∆x∆y
 ∂x   ∂y 
∂M x ∂M y
− ∆x − ∆y = ρS (h2 − h1 )∆x∆y
∂x ∂y
Since
∂h ∂h
M x = ρ b∆y v x ∆t ; M y = ρ b∆x v y ∆t ; u = −K x ; v = −K y
∂x ∂y
therefore assuming incompressible flow (ρ = mass density = constant)
∂  ∂h  ∂  ∂h 
 K x b  ρ∆x∆y∆t +  K y b  ρ∆x∆y∆t = ρS (h2 − h1 )∆x∆y
∂x  ∂x  ∂y  ∂y 

∂  ∂h  ∂  ∂h  ∂h
Further simplifying  Tx  +  T y  = S
∂x  ∂x  ∂y  ∂y  ∂t
This is the general partial differential equation for unsteady flow of groundwater in
the horizontal plane. If medium is isotropic then Tx = Ty = T and we have
∂ 2h ∂2h S ∂ h
+ =
∂ x2 ∂ y2 T ∂ t
If the flow is steady then
∂ 2h ∂2h
+ =0 ⇒ ∇2h = 0
∂ x2 ∂ y2
which is the Laplace Equation for potential flow.
(b) Unconfined Aquifer with Recharge: Applying mass balance for a control volume of
size ∆x, ∆y in x-y plane and of height h (equal to saturated thickness of aquifer = height of
water table from the bed rock)
 ∂M x   ∂M y 
M x − M x + ∆x  + M y −  M y + ∆y  + ρR∆x∆y∆t = ρS (h2 − h1 )∆x∆y
 ∂x   ∂y 
∂M x ∂M y
− ∆x − ∆y + ρR∆x∆y∆t = ρS (h2 − h1 )∆x∆y
∂x ∂y

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For an unconfined aquifer, direct analytical solution of the Laplace equation is not possible.
The difficulty arises from the fact that the water table in the 2-D case represents a flow line.
The shape of the water table determines the flow distribution, but at the same time the flow
distribution governs the water table shape. To obtain a solution Dupuit assumed (1) the
velocity of the flow to be proportional to the tangent of the hydraulic gradient instead of the
sine, and (2) the flow to be horizontal and uniform everywhere in a vertical section. These
assumptions, although permitting solution to be obtained, limit the application of the results.
With these assumptions
∂h ∂h
M x = ρ h∆y v x ∆t ; M y = ρ h∆ x v y ∆ t ; u = −K x ; v = −K y
∂x ∂y
For incompressible flow
∂  ∂h  ∂  ∂h 
 K x h  ρ∆x∆y∆t +  K y h  ρ∆x∆y∆t + ρR∆x∆y∆t = ρS (h2 − h1 )∆x∆y
∂x  ∂x  ∂y  ∂y 

∂  ∂h 2  ∂  ∂h 2  ∂h
 K x  +  K y  + R = S
∂x  2 ∂x  ∂y  2 ∂y  ∂t
If medium is isotropic then Kx = Ky = K and we have
∂ 2 h 2 ∂ 2 h 2 2 R 2S ∂ h
+ + =
∂ x2 ∂ y2 K K ∂t
This relation is known as Boussinesq Equation. If there is no recharge and flow is steady then
∂2h2 ∂2h2
+ = ∇2h2 = 0
∂x 2
∂y 2

which is the Laplace equation in h2 for unconfined flow.

Equations in Radial Coordinates

Wells form the most important mode of groundwater extraction from an aquifer. When a well
is pumped, water is removed from the aquifer surrounding the well, and the water table or
piezometric surface is lowered. The drawdown at a given point is the drop in the water table
or piezometric surface elevation from its previous static level. A drawdown curve shows the
variation of drawdown with distance from the well. In 3-D the drawdown curve describes a
conic shape known as the cone of depression. Also the outer limit of the cone of depression
defines the area of influence and its radial extent radius of influence of the well. At constant
rate of pumping, the drawdown curve develops gradually with time due to the withdrawal of
water from storage. Initially it is unsteady flow as the water table elevation at a given location
near the well changes with time. On prolonged pumping, an equilibrium state (steady state)
may be reached between the rate of pumping and the rate of inflow of groundwater from the
outer edges of the zone of influence. As soon as the pumping is stopped, the depleted storage
in the cone of depression is replenished by groundwater inflow into zone of influence called
recuperation or recovery stage, which is an unsteady phenomenon. Recuperation time
depends upon the aquifer characteristics. In confined aquifers the recovery takes place at a
very rapid rate.
To derive the radial flow equation the groundwater flow is assumed 2-D to a fully
penetrating well. In such case groundwater flow becomes axisymmetric about the well. For

8
axisymmetric groundwater flow to wells, radial coordinates are preferable because the
dimensional complexity is reduced by one degree e.g. 3D flow reduces to 2D and 2D flow
into 1D.
(a) Confined Aquifer: Applying mass balance for a control volume of ∆r thickness at
distance r
 ∂M r 
Mr − Mr + ∆r  = ρS (h2 − h1 )2πr∆r so
 ∂r 
∂M r
− = ρS (h2 − h1 )2π r
∂r
∂h
Using M r = ρ 2π rb v r ∆t ; vr = − K r and simplifying
∂r
1 ∂  ∂h  ∂h
 Krb r  = S
r ∂r  ∂r  ∂t
This is the general equation for groundwater flow in confined aquifer in radial coordinates in
the horizontal plane. If medium is homogeneous then Tr = Krb = T which is independent of r
and we have
∂ 2h 1 ∂ h S ∂ h
+ =
∂r2 r ∂r T ∂t
(b) Unconfined Aquifer with Recharge: Similar to the earlier cases
 ∂M r 
Mr − Mr + ∆r  + ρR 2πr∆r∆t = ρS (h2 − h1 )2πr∆r
 ∂r 
∂h
Using M r = ρ 2π rh v r ∆t ; vr = − K r and simplifying
∂r
1 ∂  ∂h  ∂h
 Krh r  + R = S
r ∂r  ∂r  ∂t
This is the general equation for groundwater flow in unconfined aquifer with recharge in
radial coordinates in the horizontal plane. If medium is homogeneous then Kr = K which is
independent of r and we have
∂ 2 h 2 1 ∂ h 2 2 R 2S ∂ h
+ + =
∂r2 r ∂r K K ∂t
For steady flow in homogeneous and isotropic unconfined aquifer without recharge
d  dh 2 
r =0
d r  d r 

Transformation of Medium
Transformation of Inhomogeneous into Homogeneous Medium
An equivalent transformed section is a fictitious section such that the seepage discharge is the
same as that for an actual anisotropic and/or inhomogeneous section of same dimensions

9
(thickness and cross-sectional area) under same head loss. Let there are n homogeneous and
isotropic layers of thicknesses d1, d2, d3, …, dn and hydraulic conductivities K1, K2, K3, …, Kn
respectively in a stratified (inhomogeneous) medium. The equivalent transformed section has
the hydraulic conductivities Kx along the stratification and Ky normal to the stratification and
dimensions Le = L and de = d1 + d2 + d3 + … + dn
(a) Flow Parallel to Stratification: First consider flow parallel to the stratification (x –
direction). Seepage discharges per unit width of the medium in each layer are
q1 = d1 K 1i1 ; q 2 = d 2 K 2 i 2 ; q 3 = d 3 K 3 i3 ; … q n = d n K n in

y qy

K1 d1

K2 d2 Kx

K3 d3 qx Ky
de = d1+ d2+…+dn

L Le

Kn dn
x
Stratified Medium Transformed Medium

For the equivalent transformed section, let discharge, hydraulic conductivity, and hydraulic
gradient in the x-direction are qx, Kx, and ix respectively, so that
q x = d e K x i x = (d1 + d 2 + d 3 + ... + d n ) K x i x
The conditions for this case are
∂h h2 − h1
i1 = = = i2 = i3 = ... = i n = i x and q x = q1 + q 2 + q3 + ... + q n
∂l L
Therefore
q x = (d 1 + d 2 + d 3 + ... + d n ) K x i x = d 1 K 1i1 + d 2 K 2 i 2 + d 3 K 3i3 + ... + d n K n i n

d1 K 1 + d 2 K 2 + d 3 K 3 + ... + d n K n n n
and hence Kx = = ∑ K jd j ∑d j
d1 + d 2 + d 3 + ... + d n j =1 j =1

(b) Flow Normal to Stratification: Now consider flow normal to the stratification (y –
direction). Seepage discharges per unit width of the medium in each layer are
q1 = LK 1i1 ; q 2 = LK 2 i 2 ; q 3 = LK 3i3 ; … q n = LK n i n
where hydraulic gradient in each layer is given by
∂h hL1 h hL 3 hLn
i1 = = ; i2 = L 2 ; i3 = ; in =
∂l d1 d2 d3 dn

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so hL1 = d1i1 ; hL 2 = d 2 i2 ; h L 3 = d 3 i3 ; hLn = d n in
For the equivalent transformed section, let discharge, hydraulic conductivity, and
hydraulic gradient in the y-direction are qy, Ky, and iy respectively, so that
∂h hL hL
q y = Le K y i y = LK y i y and iy = = =
∂l d e d1 + d 2 + d 3 + ... + d n
here the total head loss hL is the sum of head losses in each layer i.e.
hL = hL1 + hL 2 + hL 3 + ... + hLn
Therefore
( d1 + d 2 + d 3 + ... + d n )i y = d1i1 + d 2 i 2 + d 3 i3 + ... + d n i n

qy q1 q2 q3 qn
but iy = ; i1 = ; i2 = ; i3 = ; in =
LK y LK 1 LK 2 LK 3 LK n
and from the condition of continuity of flow in the y-direction
q y = q1 = q 2 = q 3 = ... = q n .
and hence
d 1 + d 2 + d 3 + ... + d n d1 d 2 d 3 d n n
= + + + ... + n ⇒ Ky = ∑d j ∑ (d j /Kj)
Ky K1 K 2 K 3 Kn j =1 j =1

(c) Kx > Ky: The hydraulic conductivity along the stratification is always greater than the
hydraulic conductivity normal to the stratification i.e. Kx > Ky or
n n n n

∑K
j =1
j dj ∑d
j =1
j > ∑d j
j =1
∑ (d
j =1
j /Kj)

To prove it consider a two layer porous medium

K 1 d1 + K 2 d 2 d + d2  
> 1 ⇒ (K1d1 + K 2 d 2 ) d1 +
d2
 > (d1 + d 2 )2
d1 + d 2 d d  K1 K2 
1
+ 2
K1 K 2
K 2 K1
+ >2 ⇒ K 12 + K 22 > 2 K 1 K 2 ⇒ ( K 1 − K 2 )2 >0
K1 K 2
which is always true so Kx > Ky. This can also be proved for three layered or more layered
soil medium.

Transformation of Anisotropic Medium into Isotropic Medium

If the porous medium is anisotropic then it can still be transformed into isotropic by
coordinate transformation as follows. The governing equation for steady flow is
∂  ∂h  ∂  ∂h 
 K x  +  K y =0
∂x ∂x ∂ y ∂ y 
If porous medium is homogeneous then Kx is independent of x and Ky is independent of y so

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 ∂2h ∂ 2h
Kx + K y =0
∂ x2 ∂ y2
rewriting
∂ 2h K y ∂ 2h ∂2h ∂2h
+ =0 ⇒ + =0
∂ x2 K x ∂ y2 (
∂ x2 ∂ y K / K
x y )
2

and letting Y = y K x / K y

∂ 2h ∂2h
+ =0 ⇒ ∇2h = 0
∂ x2 ∂Y 2
which is the Laplace equation in the transformed coordinates x and Y. Similar transformation
is possible in x – coordinate by rewriting
K x ∂2h ∂ 2h ∂ 2h ∂ 2h
+ =0 ⇒ + =0
K y ∂ x2 ∂ y2 (
∂ x Ky / Kx )
2
∂ y2

∂2h ∂2h
+ =0 ⇒ ∇2h = 0
∂ X 2 ∂ y2

where X = x K y / K x . Thus any stratified porous medium can be transformed into

equivalent homogeneous and isotropic medium first by converting into homogeneous by
knowing Kx and Ky and then into isotropic by transforming coordinate axes. The equivalent
hydraulic conductivity of the transformed section would be K e = K x K y .

Application of Governing Equations for Solving Groundwater Flow Problems

Steady Unidirectional Flow in Confined Aquifer
Let groundwater flow with a
velocity vx in the x-direction of a
confined aquifer of uniform
thickness. Then for steady flow
Laplace equation reduces into the
following ordinary differential
equation
d 2h
=0
d x2
which has its solution
h = C1 x + C 2
where C1 and C2 are constants of integration. Assuming h = h0 at x = 0 and h = h1 at x = L
(h0 − h1 ) dh (h − h1 )
h = h0 − x and v x = − K =K 0
L dx L
This states that the head decreases linearly with flow in the x-direction.

12
Steady Flow in a Varying Thickness Confined Aquifer
The governing equation for a steady ground-water flow in a homogeneous isotropic (wrt K)
confined aquifer of varying thickness in the x-direction becomes
∂  ∂h  d  dh 
 Tx =0 ⇒  bx =0
∂x  ∂x  dx  dx 
b0 − b1
where the thickness of aquifer at section x is bx = b0 − x
L

Impervious

h0
b0 h1
bx
x b1
x
L

Integrating this ordinary differential equation

dh C1 C1 L  b −b 
= ⇒h=− ln b0 − 0 1 x  + C 2
dx bx b0 − b1  L 
Boundary conditions h = h0 at x = 0 and h = h1 at x = L can be used to find C1 and C2 so
h0 − h1 b0 − b1 ln b0
C1 = − and C 2 = h0 − (h0 − h1 )
L ln(b0 / b1 ) ln(b0 / b1 )
Therefore
h0 − h1
h = h0 − {ln b0 − ln(b0 − (b0 − b1 ) x / L )}
ln(b0 / b1 )
h0 − h ln(b0 / bx ) ln b0 − ln bx
or = =
h0 − h1 ln(b0 / b1 ) ln b0 − ln b1
The discharge per unit width at any section x can be found by
dh h − h1 (b0 − b1 )
q = Ax .v x = bx .1.(−) K = − KC1 = K 0
dx L (ln b0 − ln b1 )
This shows that q is independent of x as it should be from the continuity of flow.

Steady Unidirectional Flow in Unconfined Aquifer

For steady unidirectional flow in an unconfined aquifer without recharge reduces to
d 2h2
=0
d x2

13
The integration of this ordinary
differential equation renders
h 2 = C1 x + C 2
To determine integration
constants, differentiate it
dh
2h = C1
dx
dh
Another expression of h is
dx
found in the flow rate per unit
width q at any vertical section
with Dupuit’s assumptions as
dh dh q
q = − Kh ⇒h =−
dx dx K
Therefore C1 = −2q / K . The constant C2 is determined by the condition that for x = 0, h = h0.
Therefore C 2 = h02 . The equation of water table is
2q
h2 = − x + h02
K
which indicates that the water table is parabolic in form. Boundary condition x = L, h = h1
may be used to determine q as

q=
K 2
2L
(
h0 − h12 )
In the direction of flow, the parabolic water table increases in slope and two Dupuit
assumptions 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. The
fact that the actual water table lies above the computed one is 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 for the same discharge. The above
solution fails to show the entry condition and the existence of a seepage face at the exit. The
slope of water table (which is streamline) at entry point (x = 0, h = h0) is
 dh  q
  =−
 dx  x = 0 Kh0
which has a finite value. Actually this must be zero as the streamline should be orthogonal to
the vertical upstream face which is equipotential line. Likewise 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 e.g. at the exit face with h1 = 0,
K 2
q= h0
2L
which is maximum and requires some area to pass through (exit area = 0 as h1 = 0). The
water table actually approaches the boundary tangentially above the water body surface and
forms a seepage face which provides the flow area.

14
 However for flat slopes, where the sine and tangent are nearly equal, Dupuit’s
solution closely predicts the water table position except near the outflow. The solution also,
accurately determines q or k for given boundary heads.

Steady Radial Flow to a Well in Confined Aquifer

We assume that the aquifer is homogeneous and isotropic, is of uniform thickness, and is of
infinite areal extent; that the well penetrates the entire aquifer, and is pumped at constant rate;
and that initially the piezometric surface is horizontal. The radial flow to a fully penetrating
well in a homogeneous and isotropic confined aquifer is everywhere horizontal; hence the
Dupuit assumptions apply without error. Using radial coordinates with the well as the origin,
for steady radial flow Q to a well. For steady radial flow in homogeneous and isotropic

confined aquifer the governing equation reduces to

d  dh 
r =0
d r  d r 
Integration leads to
dh
r = C1 and h = C1 ln r + C 2
dr
From Darcy law, the well discharge at any distance r equals
dh dh Q
− Q = Ar v r = 2π rb(−) K ⇒ r =
dr dr 2πbK
Therefore C1 = Q /(2πbK ) . The constant C2 is determined by one more condition at which h
is known at any radial distance r. This may be one observation well or the pumped well itself
or the radius of influence. Let for r = r1, h = h1; then C 2 = h1 − (Q 2πbK ) ln r1 . Therefore
equation of piezometric surface is

15
 Q Q Q
h= ln r + h1 − ln r1 ⇒ h = h1 + ln(r / r1 )
2πbK 2πbK 2πbK
which shows that h increases indefinitely with increasing r. Yet the maximum h is the initial
uniform head h0. Thus, from a theoretical aspect steady radial flow in an extensive aquifer
does not exist because the cone of depression must expand indefinitely with time. However,
from a practical standpoint, h approaches h0 with distance from the well, and the drawdown
varies with the logarithm of the distance from the well. From a practical standpoint, the
drawdown s ( = h0 – h) rather than the head h is measured. Because any two points define the
logarithmic drawdown curve, the method consists of measuring drawdowns in two
observation wells at different distances from a well pumped at a constant rate. In terms of
drawdowns
Q
s = s1 − ln(r / r1 )
2πbK
To determine Q the condition that for r = r2, h = h2 may be used as
h 2 − h1 s −s
Q = 2πbK = 2πbK 1 2
ln(r2 / r1 ) ln(r2 / r1 )
This solution is known as the Equilibrium or Thiem’s Equation. It enables the hydraulic
conductivity or the transmissivity of an aquifer to be determined from a pumped well. The
transmissivity is given by
Q Q
T = bK = ln(r2 / r1 ) = ln(r2 / r1 )
2π (h 2 −h1 ) 2π (s 1 − s 2 )

Steady Radial Flow to a Well in Unconfined Aquifer

An equation for steady radial flow to a well in unconfined aquifer also can be derived with
the help of the Dupuit assumptions. The well completely penetrates to the horizontal base and
a concentric boundary of constant head surrounds the well. Using radial coordinates with the
well as the origin, for steady radial flow in homogeneous and isotropic unconfined aquifer the

16
governing equation reduces to

d  dh 2 
r =0
d r  d r 
Integration leads to
dh 2
r = C1 and h 2 = C1 ln r + C 2
dr
From Darcy law, the well discharge at any distance r equals
dh dh Q dh 2 Q
− Q = Ar v r = 2π rh(−) K ⇒ hr = ⇒ r =
dr dr 2πK dr πK
Therefore C1 = Q /(πK ) . Similar to confined case, the constant C2 can be determined by one
more condition at which h is known at any radial distance r. Let for r = r1, h = h1; then
Q
C 2 = h12 − ln r1 . Therefore equation of free surface is
πK
Q Q Q
h2 = ln r + h12 − ln r1 ⇒ h 2 = h12 + ln(r / r1 )
πK πK πK
To determine Q the condition that for r = r2, h = h2 may be used as
h22 − h12 ( s − s )(2h0 − s 1 − s 2 )
Q = πK = πK 1 2
ln(r2 / r1 ) ln(r2 / r1 )
This equation fails to describe accurately the drawdown curve near the well because the large
vertical flow components contradict the Dupuit’s assumptions. In practice drawdowns should
be small in relation to the saturated thickness of the unconfined aquifer. The transmissivity
can be approximated
h1 + h2 Q Q
T=K = ln(r2 / r1 ) = ln(r2 / r1 )
2 2π (h 2 −h1 ) 2π (s 1 − s 2 )
Where drawdowns are appreciable the heads h1 and h2 can be replaced by h0 – s1 and h0 – s2
and the transmissivity for the full thickness becomes
Q
T = Kh0 = ln(r2 / r1 )
 s2   s2 
2π  s1 − 1  −  s 2 − 2 
 2h0   2h0 

17