Fetter Ch 4 - Ground-Water Flow
4.1 - Introduction
GW possesses energy in mechanical, thermal, chemical forms.
Flow of GW is in response to outside forces. Let's look at 3:
1. Gravity, pulling water downward
2. External pressure, both atmospheric and due to weight of overlying
water
3. Molecular attraction of water to solid surfaces
There are shear and normal forces acting as well….shear is tangential to
surface of solid, normal is perpendicular to it. These are external frictional
forces. Internal frictional force resisting flow by shear is "viscosity". Think
of molasses as high viscosity.
4.2 - Mechanical energy
here we consider kinetic energy, gravitational poten. energy, and fluid
pressure energy

kinetic energy is energy assoc with a moving mass:
Ek = mv2
2

potential energy can be visualized by first assuming that some work is
done to lift a mass of water to a certain elevation,
W = mgz
The water mass now has "acquired" potential energy equal to the amount
of work done to elevate it:
Ep = mgz

additional potential energy is present due to surrounding fluid pressure
P
Thus total energy per unit volume of fluid is
Etv = v2 + gv + P
More conversion of this equation gets units to energy per unit weight,
which becomes units of length (feet, meters).
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"sum of these 3 factors is total mechanical energy per unit weight" =
hydraulic head
4.3 Hydraulic Head
how do we measure head?
We use piezometers. So what's a piezometer? "small-diameter well with
a very short well screen or section of slotted pipe…..it is used to measure
the hydraulic head at a certain point in the aquifer".
By design, piezometer measures head at one point in the aquifer, rather
than averaging head over the total thickness.
The continued derivation of equations for head suggests that kinetic
energy can be ignored and removed from the equation. That leaves two
potential energy components, z and hp (Fig 4.2), defined as elevation
head and pressure head, respectively.
Hp = dist from screen to water level
Z = dist from screen to datum
Datum = Sea Level
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Htot = Hp + z
Let's do problem 2 on p.117….
4.4 Head in water of varying density
point is made that if you want to construct water table maps on a site
where water is variable density, then you need to convert the "point-water"
heads to freshwater heads in order to map the water table
let's not go there…pretty specialized application….
4.5 Force potential and hydraulic head
Force potential (total potential energy) is "driving impetus behind GW
flow".
Ф = gh (this potential energy is acceleration applied over a distance)
we essentially let g drop out, since gravity pull more or less constant
"everywhere on Earth" (not really true…)
so we are left with the driving impetus behind all GW flow being hydraulic
head,
h = z + hp
units of h start as energy per unit weight, but reduce to length, which is
easy to measure, especially relative to a datum such as sea level.
It is "total head", h which control movement of GW.
Interesting pictorial experiment in Fig 4.5, showing how total head controls
flow, not just pressure head or elevation head….both components are
critical.
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Fig 4.5A - note that water DOES flow uphill
Fig 4.5 B - flows downhill as expected
Fig 4.5 C - equal elevation heads, but pressure head higher at left (so total
head is higher at left)
Fig 4.5 D - equal pressure head, but elevation head higher at left (so total
head is higher at left)
Water flow is always from higher total head to lower total head, always,
and hydraulic head "always decreases in the direction of flow".
4.6 Darcy's Law
4.6.1 setting the stage for later 2 and 3D analysis…
4.6.2 Applicability
Laminar vs turbulent flow. Laminar in very slow, GW regimes. Turbulent in
higher velocity regimes, like rivers. Darcy only works in laminar flow.
4.7.2 - Specific discharge and average linear velocity
Once again, Q = v x A, where v is velocity and A is area through which
water flows
note that you can rewrite this with Darcy's Law in mind:
v = Q = -K dh = q (aka "little q")
A
dl
this is known as the "specific discharge"
I and many others use "Darcian velocity", but note that Fetter does NOT
like this terminology, because it is not TRUE velocity. He claims that this
is compared to flow through an open pipe, but in the ground water flow is
not through an open pipe. Yes, he has a point…..
he notes that in order to find the actual velocity, you must divide specific
discharge by the porosity of the material. Note that this will yield a value
considerably higher than the specific descharge, because you divide by
less than one…this means that the actual "seepage velocity" will be
considerably higher than the specific discharge (Darcy velocity)
OK this makes sense. For the same amount of water to travel through a
bunch of smaller openings as one large one, the velocity has to go up…
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4.7 - Equations of GW flow
OK hold on…the math looks bad, but you don't have to derive it….
4.7.1 - Confined aquifers
 "flow of fluids is governed by laws of physics"
 described by differential equations, where x,y,z, and time are all
independent variables
one rule for solution: no net change in mass of a unit volume of fluid…
another rule for solution - 1st Law of Thermodynamics - "conservation of
energy" - in a closed system, energy is neither created nor destroyed, but
it can change form
another rule for solution - 2nd Law of Thermodynamics - implies that
energy moves from higher energy, more useful form (like mech energy) to
a less useful form (often this form is heat)
using these rules and Darcy's law, main equations of GW flow are derived
we use a small cube, a control volume, to model the entire aquifer, and
make it parallel to an xyz coordinate system
flow can go through the control volume at any angle, like a vector, but we
can describe that vector on basis of the xyz coord system Fig 4.7
first portion of derivation is grouping terms to show the net accumulation of
mass in the control volume:
-
( qx +  qy +  qz) dx dy dz
x
y
z
assuming fully saturated porosity, the mass of the water is
density ,  (g ) x volume of porosity, n (cm3) = mass, M (g)
(cm3)
we can describe this change in mass with time
M =  ( n dx dy dz)
t
t
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this equation is tweaked and substituted, then set equal to the partial
diff equation above.
Transmissivity and storativity is also introduced, yielding an equation
for 2D flow with no vertical z component:
2 h
 x2
+
2 h
 y2
=
S h
T t
In steady state flow, there is no change in things over time….they stay
the same. With time not a variable to worry about, we can use the
LaPlace equation:
2 h
 x2
+
2 h
 y2
+
2 h
 z2
=0
this essentially says that the rate of change of head in the x,y, and z
directions is zero, which is true in steady state conditions.
The general equation for flow, 2 dimensions (the horizontal plane),
including consideration of leakage, becomes:
2 h
 x2
+
2 h
 y2
+e = S h
T
T t
this is a major equation used frequently in GW modelling.
4.7.2 Unconfined Aqs
backgrd: water comes out of water-table aquifers through vertical
drainage of pore space. So you see decline in actual surface of water
table around a pumping well - "cone of depression"
note that in the case of a confined aq, the potentiometric surface is the
thing that gets depressed, but the actual aqu stays saturated
potentiometric surface maps as lower near the well,
but the aq stays filled with water
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with an unconfined, water-table aq, the actual water table surface itself
gets drawn down
the general flow equation is known as the Boussinesq equation, which
is not solvable with calculus in this form
BUT, we make a simplifying assumption that the drawdown is very
small relative to saturated thickness (that is, it LOOKS saturated for the
most part), and we can re-write the equation in terms similar to the one for
confined aquifers:
2 h
 x2
+
2 h
 y2
=
Sy  h
Kb  t
where b is the average saturated thickness of the aquifer
this assumption makes the equation linear and solvable
4.8 - solution of flow equations
how do we solve these equations?
We use a mathematical model

governing flow equation

equations for hydraulic head at aquifer boundaries

equations describing "initial conditions" at start of modelling
if Aq is homogeneous and isotropic (same throughout and equal in all
directions), math model can be solved by "analytical" methods based on
integral calculus.
BUT, if aq does not meet these conditions (layered conditions, for
example) then a "numerical" solution is required. The concept here is that
the the partial diff equations are replaced by similar equations, but ones
that can be solved using basic arithmetic. "Numerical" solutions usually
require computers, addressed in Chap 13.
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4.9 - hydraulic head gradient
Definition - the "quantity we measure in the field, which represents
hydraulic head, is the depth to water in a piezometer"
Fetter goes through the mechanics of measuring head…we've done
this
Top of Casing (TOC) measuring point
Ground Level (GL) in feet above sea level
Water
Level
Water Table
Well screen (water entry)
Sea Level
Typical type of equation (all in feet, meters, whatever):
[(GL (dist above sea level)+ TOC (dist above GL)] - depth to water = Head
IMPORTANT twist: Head can change (increase or decrease) with depth in
an aquifer.
Fig 4.8 A shows no change in head with incr depth, but Fig 4.8 B shows
that head is higher at depth than it is shallow. This will cause lines of equal
potential (equal head) to be non-vertical, so that the path of a water
molecule would follow grad h,  to lines of equal potential.
Of course, to map things accurately, you need to have the aquifer covered
in 3D, with piezometers at different depths. Note that in block diagram of
Fig 4.8, the head values at top of block are those you'd record right at the
top of water table.
Grad h is the gradient of head, pointing in direction opposite of flow.
Fetter points out in Fig 4.8 that "equipotential lines at the top of the aquifer
represent the intersection of the equipotential surfaces with the water
table".
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Note that if grad h= zero, there will be no flow.
4.10 - GW flow relative to grad h
3 factors control GW flow:
 potential field orientation
 anisotropy of K
 permeability in relation to grad h
when aq is isotropic, K is same in any direction, so flow is parallel to grad
h
Fetter works through the tensor ellipse….let’s skip this.
4.11 - Flow Lines and Flow Nets
definitions:
flow line: "imaginary line that traces the path that a particle of ground
water would follow"
if aquifer is anisotropic, flow lines will not be completely perpendicular to
equipotential surface lines, whereas they would be in the isotropic aquifer.
La Place equation is solved graphically by constructing a flow net. Flow
net is defined as "network of equipotential lines and associated flow lines"
Bunch of criteria listed to be met, including boundary conditions.
Boundary conds are of 3 types…
 no-flow,
 constant-head,
 water-table.
No-flow is one where flow lines are parallel to it, so no flow crosses the
boundary
Constant-head is one where head values are constant all along the
boundary, which means it is an equipotential surface. This means flow
lines run into this boundary at 90o.
Water-table boundary is also present in unconf aqs.
Ideally, flow net should yield a pattern of squares or rectangles, because
of 90o angle of flow lines to the equipotentials
Let's start looking at what it will take to make a flow net..Fig 4-11
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In fig 4-11, note that no-flow boundaries are on top and bottom, high head
constant head boundary is on left, low head constant head boundary on
right. This means flow lines will move from high head on left to low head
on right.
Let's look at example flow tube problem on p.136.
4.12 - Refraction of Flow Lines - important to pros…perhaps not to this
class.
One interesting element of discussion:
Fig 4.14 - Like seismic and light refraction, flow path refracts away from
the orthogonal with higher velocity, refracts toward orthogonal with lower
velocity
4.13 - Steady Flow, confined aquifer
little easier to visualize than the unconfined example earlier:
Fig 4.16 shows elements. Note the units. q' here is in area units, like a
plate coming out the right side.
width
b
K
Note that multiplying width x K x b would yield a volume passing past a
point per unit time
Let's do the example problem.
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4.14 - Steady Flow in Unconfined Aquifer
Fig 4.17 shows the situation. Note that the slope of the water table
changes as you move down-gradient, while the slope of potentiometric
surface in a confined aq stays the same.
Dupuit assumptions are in place here:
 streamlines (flowlines) horizontal
 equipotentials vertical
Use the Dupuit equation to solve these
q' = K (h12 - h22)
L
note the units are in ft2 per sec, not feet per sec
Important to think about the units…visualize the plate of water coming out
of the aquifer
Use Fig 4.18 to put the equation to work
More manipulation leads to equations that can determine elevation of
water table between any two points (Eq 4.70, 4.71)
And discussion also of determining position of a water table divide.
Let's do a problem or two before leaving this chapter….
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