PRINCIPLES OF GROUNDWATER FLOW
Mechanical energy
- A moving body or fluid tends to remain in motion according to Newtonian
physics.
- This is because it has kinetic energy due to its motion
- the amount of kinetic energy the fluid contains is equal to ½ the product of
its mass and the square of the magnitude of its velocity
- water also has gravitational potential energy
Eg= mgz (mass x acceleration due to gravity x elevation)
- and water has potential energy owing to pressure
-Expressed as Pa or mb usually
-can be thought of as potential energy per unit volume of fluid is
Combining these three terms together we can arrive at a term defining the
total energy per unit mass of fluid:
this equation is also known as the Bernoulli Equation
If we divide all the terms on the right side of the Bernoulli equation by g
it will express all terms in units of energy per unit weight and will reduce the
three terms to units of length (e.g. m)
In this form the above equation is the total mechanical energy per unit weight
of fluid, which is also known as the hydraulic head
The total potential energy expressed above as Etm is also called the force
potential:
Φ=gh
Force potential differs from hydraulic head in that force potential is energy
per unit mass, while hydraulic head is energy per unit weight
Hydraulic head
Question: what is a piezometer?
Kinetic energy contributed by flowing groundwater is generally small enough to
be ignored, therefore hydraulic head becomes
since P=ρghp
where hp is the height of the water column caused by the pressure
h = Z + hp
Water density affect on head
Density corrections need to be considered if salinity varies considerably over
an area of consideration
For now we will focus only on variations in the vertical salinity concentration,
since variation in the horizontal is more complicated
Freshwater head - height of a column of freshwater in a well is just sufficient
to balance the pressure in the aquifer at that point
Point water head - the actual water level in a well or piezometer
In freshwater aquifer’s, heads are both freshwater and point water
In saline aquifers, where salinity varies in the aquifer, heads need to be
corrected to freshwater heads before flow calculations can proceed
z should be from the datum to the intake point of the well
The applicability of Darcy’s Law
In general Darcy Law holds:
1. For saturated and un-saturated flow
2. For steady state and transient flow
3. For flow in aquifers and flow in aquitards
4. For flow in homogeneous and heterogeneous systems
5. For flow in isotropic and anisotropic media
6. For flow in rocks and granular media
Darcy’s law is a linear law (a plot of discharge vs. hydraulic gradient should
be a straight line) and could be re-written as:
m=1, darcy’law
m…1, something else
This relationship is in question in at least two situations in flow through granular
materials:
1. Flow through very low permeability sediments under very low gradient’s
2.
For situation 1 it’s been suggested that there may be a lower threshold for flow
given low gradient
- hypothesized from lab studies but no conclusive explanation
- in most practical situations this can be ignored
And very high rates of flow Darcy’s law breaks down.
First need to define types of flow:
Laminar flow Turbulent flow – higher velocity/energy flows. As glossy increases so does
kinetic energy, and inertial forces dominate over viscous forces. Fluid particles
start behaving radically
Darcy’s Law is only applicable in laminar flows (where viscous forces
dominate).
Therefore we need a way to determine if flows are laminar or turbulent
Reynolds Number evaluates the condition of the flow:
R is dimensionless
ρ is fluid density
v is fluid velocity
d is the diameter of the passage way through which he fluid moves
μ is the viscosity
d is difficult to determine for a porous medium.
Often the average grain diameter, or some function of the square root of the
permeability is used
In general Darcy’s law is valid as long as the Reynolds Number based on
average grain diameter does not exceed some value between 1 and 10
Most R’s will be around or less than 1 or 2
Under most natural conditions groundwater velocity’s are sufficiently low that
Darcy’s law will apply
Flow rates that exceed the upper limit of Darcy’s law are common in:
1.
2. Dolomites
3.
4. Any formations with large openings
5. Areas with unnaturally steep hydraulic gradient’s (e.g. around pumping wells)
Specific Discharge
Velocity of water moving through the ground is different than velocity of water
flowing through an open pipe
If pipe is filled with sand the area through which water can flow is much
smaller
Velocity of groundwater flow (specific discharge) is an apparent velocity
representing the velocity at which water would move through the aquifer it
were an open conduct
To find the actual velocity of groundwater flow specific discharge is divided by
effective porosity
Question: what is effective porosity?
The result is the seepage velocity or average linear velocity
This equation does not account for dispersion.
Dispersion results from groundwater flowing through different pores at different
rates, and different flow paths have different lengths.
Specific Storage
Amount of water per unit volume of a saturated formation that is
stored or expelled from storage owing to compressibility of the
mineral skeleton and pore water per unit change in head.
Ss = Ve/V = is volume of water released from elastic storage per
unit cube of aquifer,
or
Specific Yield
Sy = specific yield is in an unconfined aquifer and related to water
table change
-ratio of the volume of water that drains from a sample under
gravity to the volume of sample: Vw/V
Also Sy = n –Sr
Recall from last week Specific Yield (Sy)
- volume of water that drains from a saturated rock by gravity to
volume of rock
- in an unconfined aquifer, S=Sy
Storativity
Volume of water that a permeable unit will absorb or expel from storage per
unit surface area per unit change in head.
In confined aquifers:
In unconfined aquifers: S = Sy+bSs
the second term (bSs) is v. small in unconfined aquifers and S
generally equals Sy
Groundwater Flow equations
Read Chapter 4 in Fetter for full equations and complete derivations
There are 5 general groundwater flow equations that can describe flow in an
aquifer
All are partial differential equations in which head is described in terms of
x,y,z (e.g. position in the flow field) and t (time)
All are based on
1.
- Can be no net change in mass of a fluid contained in a small volume of
the aquifer
2.
- Within any closed system there is a constant amount of energy
3. 2nd law of thermodynamics
- When energy changes form, it goes from more useful form (e.g. mechanical
energy) to a less useful form (e.g. heat)
Gradient of Hydraulic Head
Vector “grad h” defines the direction and magnitude of hydraulic head change
in an aquifer
where s is distance parallel to grad h and grad h has a direction that is
perpendicular to equipotential lines
- if potential is same everywhere (e.g. flat water table) grad h=0 since there is
no slope to h
- if grad h ≠ 0, flow will occur
Flow Direction and Grad h
For isotropic media flow will be in the opposite direction of grad h (i.e.
perpendicular to the equipotential lines)
For anisotropic media
- flow will not be parallel to grad h and will not cross equipotential ines at
right angles (unless by chance the anisotropy is lined up with grad h)
- method of calculating flow direction developed that assumes there will be
one plane where K does not vary with direction
-
Flow Lines and Nets
Construction of graphical representations of flow lines and flow nets can be
used to approximate flow
- flow line is an imaginary line that traces the path of an individual particle of
water as it move through the ground
- in isotropic media flow lines cross equipotential lines at right angles, but in
anisotropic media flow lines cross at same angle dictated by the tensor ellipse
calculations performed above
- flow net is a graphical representation of flowlines and equipontial lines for a
given aquifer
- very useful tools in isotropic media, but can also be used in anisotropic media
with modifications
for demonstration, our flow nets will be for:
1. a homegenous and isotropic aquifer,
2.
3. No change in potential over time
4. uncompressible aquifer and water
5.
6. All boundry conditions known
Boundary Conditions
- 3 types
no-flow boundary: boundary that groundwater can not pass
- adjacent flow line will be parallel
- equipotential lines intersect at right angles
constant-head boundary: head is constant every where on the boundary
- also an equipotential line
- flow lines will intersect at right angles
-
water-table boundary: only in unconfined aquifers
- not flow line or equipotential line,
- line of known head
- if recharge/discharge is occurring across water table, flowlines will be oblique
to it
-
Construction of Flow net (see also Fetter Ch. 4 or Freeze and Cherry Ch.5)
- goal of flow-net construction is a series of equipotential lines and crossing flow
lines that form a series of quasi squares
1.
2. Make scale drawing of boundaries (same scale on x and y)
3. Identify known equipotential and flow line conditions
4. draw trial flow lines, distance between flow lines should be the same at all
directions of the flow field
5. Draw equipotential lines from one end of field to other
- should cross perpendicular to flow lines
- follow boundary rules above
- space equipotential lines to form areas that are equidimensional
- resulting shapes should be as square as possible
6. Several re-iterations of 4 and 5 with pencil and paper will be necessary to get
desired product
- check that diagonals of “squares” are smooth curves that intersect each other
at right angles (do on a copy of drawing)
Use of Flow Nets for calculating flow
For simple flow systems with one recharge boundary and one discharge
boundary:
where q’ is discharge per unit width of aquifer
p is number of flow tubes bounded by adjacent pairs of flow lines
h is total head loss over length of flow lines
f is number of squares bounded by adjacent flow lines covering entire length
of flow
For more complex systems it is possible to find the discharge for each
streamtube:
and total flow can be summed
Refraction of Flow Lines
- direction of flow path changes when passing across unit boundary of
differing K
-variables:
Q1 is volume of water flowing in flow tube in stratum 1
Q2
a is width of stratum 1 flow tube
c is width of stratum 2 flow tube
dl1 is distance between equipotential lines in stratum 1
h1 is head loss between equipotential lines in stratum 1
Flow through each stream tube is found by modified Darcy equations:
dh1
Q1  K1a
dl1
and
Law of Continuity states that Q1 must = Q2
using the geometry of the refraction there is a line b which is the section of the
stratum boundary which falls between the flow tubes and creates two triangles
with sides dl1-a-b and dl2-c-b
-since head loss will be equal on both sides, can divide by h1 and work out the
triginometry of the triangles to find:
Interestingly, this tangent law allows water to take a more direct route through
low K media (i.e. get out of it as fast as possible) and a more indirect route
through high K media
- aquifers, flowlines tend to become almost horizontal
- aquitards, flowlines tend to be almost vertical
Rules for flow net construction in a heterogenous aquifer
1.
2. Equipotential lines must meet no flow boundaries at right angles
3. Equipotential lines parallel constant-head boundaries
4. Tangent law (above) must be satisfied at geologic boundaries
5. If the net is drawn so that squares exist in one portion of a geologic
formation, squares must exist throughout that formation and formations
with the same K. Rectangles will be formed in formations of differing K
Freeze and Cherry chapter 5 have good discussion for flow net
construction in different aquifers
Conductive-Paper analogs
- sheet of conductive paper cut in the shape of the aquifer cross-section in
question
- power supply sets up voltage differential across flow boundaries
- voltage meter then used to probe and map potential distribution across flow
field
- constant head boundaries constructed with highly conductive silver paint
- method limited to homogeneous, isotropic systems
- however capable of dealing with complex regional shapes and boundary
conditions
Resistance Networks
- uses the same principals, but instead of conductive paper, a network of
resistors is wired together
- adjusting the resistance in certain areas allows for simulation of
heteorogeneity and anisotropy