# Geol_127_Lecture_11_Specific discharge_ equations of flow by keralaguest

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```									Geology 127                                                                        Lecture #11
Hydrogeology                                 Reading assignment: Fetter, pp. 122-132, 138- 146

- Last time: we talked about hydraulic head and refraction of flow lines
- Today: We will discuss specific discharge (Darcy flux), average linear flow velocity,
groundwater flow equations, and equations of flow in confined and unconfined aquifers

I) Darcy's law and moving water

A) Specific discharge and Darcy velocity

- Darcy’s law could be rearranged to predict ground water flow velocity, BUT it
does not account for all variables.
- The problem: in the next section we will find we need to add a correction to
account for actual particle path. Particles take a longer, more tortuous path
than Darcy assumed.
-If we used Darcy’s law to predict particle velocity:

- First: convert from Q to v:
Q = vA         OR: v = Q/A

- Substituting into Darcy's equation:
Q = -KA dH                    or   v = Q = - K dh
L                           A       L
- This gives the equation for SPECIFIC DISCHARGE (also called Darcy
Flux/ Darcy Velocity

- What is the problem here?

We now realize that calling this a “Darcy Velocity” isn’t correct, because it implies that
water is moving at this velocity. It isn’t!! Water moves faster than predicted by the
specific discharge equation when flowing through a porous medium.

- Specific discharge would predict accurate velocity for flow through a pipe, but
doesn’t account for the extra travel path length that water molecules
take in a porous medium
- The solution: To find the actual velocity of the moving water: we must include a
porosity term (see section C below)

B) Laminar and turbulent flow

- Another factor that must be considered
- Some of Darcy's assumptions begin to break down with faster-flowing water
- Darcy didn't account for turbulence (he assumed laminar flow)
- This approach gives an unrealistic view of each particle path
- Also gives an unrealistic view of particle velocity

See Figure 4.6 from Fetter, p. 123

- At many “natural” flow velocities: turbulence is created around grains
- You can calculate the flow velocity (based on temp, density, grain size,
viscosity) that occurs when Darcy’s law begins to break down: see p. 123-
124 in Fetter.
- Geologists refer to a Reynolds number (Re) as the boundary between
laminar and turbulent flow.
- In natural conditions (aquifers): experiments have shown the boundary between
laminar and turbulent flow occurs at Re ~10.
- This is MUCH lower than the boundary of Re = 500-2000 used in open flow
situations!
- The good news here: Water usually travels at slow enough velocity for
Darcy’s Law to be valid (true laminar flow)
- Most GW flow equations assume true laminar flow
- So: Darcy’s law produces an empirical result for K, but can’t accurately
predict flow velocity for a given particle

C) Seepage velocity (average linear flow velocity)

- Represents the average rate at which water moves between 2 points
- Includes the effective porosity:

Vx = Q = - K dh
ne A   ne dl

where           Vx = average linear flow velocity
ne = effective porosity: the volume of the void spaces through which water
or other fluids can travel in a rock or sediment divided by the total
volume of the rock or sediment. Units = dimensionless porosity;
this value is a decimal (NOT a %). Ex: 18% porosity = 0.18 ne
- Note: this equation does not account for diffusion or dispersion
- Therefore: cannot be used for calculations of contaminant plume velocity.

II) Equations of groundwater flow

A) Introduction

- Use law of conservation of mass, law of conservation of energy
- The theory: a fluid contained in a small area can neither gain nor lose energy
- Energy of the fluid is described by looking at fluid flux in x,y,z directions
- Note: Equations of flow are different for confined and unconfined aquifers
- Also: there are 2 basic aquifer conditions:
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- Some aquifers are steady state (flow conditions do not vary)
- Some aquifers are variable or transient (often from pumping)
- In class: I will give final flow equation, but won’t go through the derivation; you
can read this in the book.
- The basic approach: take a controlled look at the aquifer, describe changes in
head (h) in terms of changes in x,y,z, t
- Assumptions:
- Aquifer is homogeneous, isotropic
- All water is derived from the aquifer
- Solution uses partial differential equations: Solves Darcy=s Law for Qx, Qy, Qz

See Figure 4.7 from Fetter, p. 127

- Solve Darcy's law for Qx, Qy, Qz

B) Confined aquifer

- Remember: storativityconfined = b Ss

See Equation 4.42 from Fetter, p. 128 or simplified form: Equation 4.44

- Pick out components that we know:
- flux terms for x,y,z axis
- Compressibility term
- d/t term = velocity
- K term
- The result = total flow through the tube
- This is a steady state condition (no change in the Z dimension with time)
- The result: this equation gives total flow through the tube/layer/box (aquifer)
- Can be modified to account for leakage (book gives equation, but we
won’t discuss it in class)

C) Unconfined aquifers

- Are a different situation
- Water levels change with time: this produces a non-linear equation that cannot
be solved with Calculus (except in some specific instances!)
- Pumped water comes from specific yield
Remember: storativity = Sy + hSs
- Equation includes a Sy term

See overhead, equation 4.46, 4.47 (simplified version)
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- The equation can be made linear by assuming that drawdown is very small
compared to average aquifer thickness.

III) Solving flow problems
- Remember: flow is different in confined and unconfined aquifers

A) Steady flow in a confined aquifer
- Flow implies that the aquifer has a gradient (Remember: water flows downhill)
- Gradient of the aquifer is linear
See Figure 4.16 from Fetter, p. 139
- Note: the potentiometric surface is above the confining layer
- (Due to storativity, compressibility of the aquifer skeleton and water)
- We discussed this before: In a confined aquifer

S = b Ss
- and:
Ss = wg(+ n)

where:  = compressibility of the mineral skeleton
 = compressibility of water
w = density of water
- Now: how do you calculate the head for this confined aquifer between known
points?
- Formula:

h = h1 -    q1 x
Kb
where:
x = some intermediate distance between h1 and h2
b = aquifer thickness
h1 = height of the water table in highest well

- Hint: put all length terms in meters or feet

- There are two ways to find q1:

1) q1 = discharge per unit width (a variation on "big Q")
- calculate q1 by finding "Q" first, dividing "Q" by the total
width of the aquifer
- Width of aquifer must be given in problem
ex: 7000 ft wide gravel lense
- gives "discharge per unit area"
- units: L2/t (ex: m2/day)
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Or:    2) q1 = Kb dh
dl
- Now: the other possibility is that the aquifer is unconfined:

B) Steady flow in an unconfined aquifer

- This is different!!!
- The big difference:
The aquifer isn't always fully saturated,
Another consequence:
The gradient is not constant now:
Gradient increases in the direction of flow
- From a mathematical standpoint: this complicates the picture

See Figure 4.17 from Fetter, p. 141

- 2 problems:
- Equation for this sloping surface is much more complicated
- Also: recharge from the surface affects the gradient
- The solution: make some assumptions about boundary conditions, calculate
discharge through a section of the aquifer

See Figure 4.18 from Fetter, p. 142

- Assume: no flow in the "z" direction
- The equation for discharge through a face of this cube (Dupuit equation):

q1 = 1 K (h12 - h22)
2       L

where:           L = flow length
q1 = flow per unit width; units= L2/T (ft2/day, m2/day)

- The value of this equation: lets you calculate discharge per foot
of aquifer
- This determines discharge; what we really started out to calculate was
hydraulic head at some distance "x"
- With infiltration/evaporation included: Equation is fairly complex

See equation 4.70 from Fetter, p. 143
Write equation on board:
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- For our purposes: realize that it is possible to calculate hydraulic
head at a point that is distant from a known hydraulic head
- "w" term here: is a recharge factor (necessary in unconfined
aquifer)

2) Neglecting recharge:
- Equation is simpler (since w = 0)
See equation 4.71, p. 143
Write equation on board

- We just need to know:
x = distance of unknown point from origin

Draw picture on board

L = total distance between known wells
K = hydraulic conductivity
h1 = water level in first known well
h2 = water level in second known well
- Can also calculate the position of a groundwater divide, using some
different assumptions:

See Figure 4.19 from Fetter, p. 143
See eqtn. 4.73, p. 144

- "d" = distance from origin to water divide
- Be aware that it IS possible to make some of these predictions with
unconfined aquifers

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