Final Review
Ground Water Basics
•
•
•
•
Porosity
Head
Hydraulic Conductivity
Transmissivity
Porosity Basics
• Porosity n (or f)
n
• Volume of pores is
also the total volume
– the solids volume
Vpores
Vtotal
Vtotal  Vsolids
n
Vtotal
•
Porosity Basics
Vtotal  Vsolids
Can re-write that as:
n
Vtotal
• Then incorporate:
• Solid density: rs
= Msolids/Vsolids
• Bulk density: rb
= Msolids/Vtotal
• rb/rs = Vsolids/Vtotal
Vsolids
n  1
Vtotal
rb
n  1
rs
Porosity Basics
• Volumetric water
content (q)
– Equals porosity for
saturated system
Vwater
q
Vtotal
Ground Water Flow
•
•
•
•
•
•
•
Pressure and pressure head
Elevation head
Total head
Head gradient
Discharge
Darcy’s Law (hydraulic conductivity)
Kozeny-Carman Equation
Multiple Choice:
Water flows…?
• Uphill
• Downhill
• Something else
Pressure and Pressure Head
• Pressure relative to atmospheric, so P = 0
at water table
• P = rghp
– r density
– g gravity
– hp depth
Pressure Head
(increases with depth below surface)
Elevation
P = 0 (= Patm)
Head
Elevation Head
• Water wants to fall
• Potential energy
Head
Elevation
Elevation Head
(increases with height above datum)
Elevation datum
Total Head
• For our purposes:
• Total head = Pressure head + Elevation
head
• Water flows down a total head gradient
Elevation datum
Head
Total Head
(constant: hydrostatic equilibrium)
Elevation
P = 0 (= Patm)
Potential/Potential Diagrams
• Total potential = elevation potential +
pressure potential
• Pressure potential depends on depth
below a free surface
• Elevation potential depends on height
relative to a reference (slope is 1)
Head Gradient
• Change in head divided by distance in
porous medium over which head change
occurs
• dh/dx [unitless]
Discharge
• Q (volume per time)
Specific Discharge/Flux/Darcy
Velocity
• q (volume per time per unit area)
• L3 T-1 L-2 → L T-1
Darcy’s Law
• Q = -K dh/dx A
where K is the hydraulic
conductivity and A is the
cross-sectional flow area
1803 - 1858
www.ngwa.org/ ngwef/darcy.html
Darcy’s Law
• Q = K dh/dl A
• Specific discharge or Darcy ‘velocity’:
qx = -Kx ∂h/∂x
…
q = -K grad h
• Mean pore water velocity:
v = q/ne
Intrinsic Permeability
rw g
K k

L
T-1
L2
Kozeny-Carman Equation
3
2
m
n
d
k
2
1  n  180
Transmissivity
• T = Kb
Darcy’s Law
• Q = -K dh/dl A
• Q, q
• K, T
Mass Balance/Conservation Equation
 I   P  O   L   A
•
•
•
•
•
I = inputs
P = production
O = outputs
L = losses
A = accumulation
Derivation of 1-D Laplace Equation
qx|x
•
•
•
•
Dz
Inflows - Outflows = 0
(q|x - q|x+Dx)DyDz = 0
q|x – (q|x +Dx dq/dx) = 0
dq/dx = 0 (Continuity Equation)
h
q  K
x
(Constitutive equation)
h 

d K 
x 

0
dx
qx|x+Dx
Dx
Dy
h

0
2
x
2
General Analytical Solution of 1-D
Laplace Equation
h
0
2
x
2
h
 x 2 x   0x
2
h
A
x
h

x

A

x
 x 
h  Ax  B
Particular Analytical Solution of 1-D
Laplace Equation (BVP)
BCs:
- Derivative (constant flux): e.g., dh/dx|0 = 0.01
- Constant head: e.g., h|100 = 10 m
After 1st integration of Laplace Equation
we have:
h
A
x
Incorporate derivative, gives A.
After 2nd integration of Laplace Equation
we have:
h  Ax  B
Incorporate constant head, gives B.
Finite Difference Solution of 1-D
Laplace Equation
h/x|x+Dx/2
h|x
x
h|x+Dx
Estimate here
x +Dx
Need finite difference approximation for
2nd order derivative. Start with 1st order.
h x Dx  h x h x Dx  h x
h


x x Dx / 2 x  Dx  x
Dx
Look the other direction and estimate at x – Dx/2:
h
x

x  Dx / 2
h x  h x Dx
x   x  Dx 

h x  h x Dx
Dx
Finite Difference Solution of 1-D
Laplace Equation (ctd)
h|x-Dx
h/x|x-Dx/2
x -Dx Estimate here
h|x
h/x|x+Dx/2
x
Estimate here
h|x+Dx
x +Dx
2h/x2|x
Estimate here
Combine 1st order derivative approximations to get 2nd order derivative approximation.
 h

2
x
2
h
x

x  Dx / 2
Dx
h
x
h x  Dx  h x
x  Dx / 2

Set equal to zero and solve for h:
Dx
h x  h x Dx

Dx
Dx
hx 

h x  Dx  2h x  h x Dx
Dx 2
h x  Dx  h x  Dx
2
2-D Finite Difference Approximation
y +Dy
h|x-Dx,y
x -Dx
h|x,y+Dy
h|x,y
h|x+Dx,y
x, y
x +Dx
h|x,y-Dy
h x, y 
h x  Dx , y  h x Dx , y  h x , y  Dy  h x , y Dy
4
Matrix Notation/Solutions
4h2,2  h2,3  h1,2  h2,1  h3,2
 h2,2  4h2,3  h1,3  h2,4  h3,3
 4  1 h2, 2  h1, 2  h2,1  h3, 2 
  1 4  h   h  h  h 

  2,3   1,3 2, 4 3,3 
• Ax=b
• A-1b=x
Toth Problems
• Governing Equation
h h
 2 0
2
x
y
• Boundary Conditions
19
17
15
13
11
9
7
5
3
2
1
2
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
S11
10.09-10.1
10.08-10.09
10.07-10.08
10.06-10.07
10.05-10.06
10.04-10.05
10.03-10.04
10.02-10.03
10.01-10.02
10-10.01
Recognizing Boundary Conditions
S1
• Parallel:
S3
S5
– Constant Head
– Constant (non-zero) Flux
S7
S9
S11
• Perpendicular: No flow
• Other:
S13
S15
S17
S19
– Sloping constant head
S21
S23
S25
S29
13
10
7
4
1
S27
Internal ‘Boundary’ Conditions
• No flow
S11
S3
S5
S7
S9
S11
S13
S15
S17
S19
S21
S23
S25
S27
S29
4
7
10
– Wells
– Streams
– Lakes
13
• Constant head
S1
S3
S5
– Flow barriers
S7
S9
• Other
S11
S13
S15
S17
S19
15
13
11
9
7
5
3
1
S21
S23
Poisson Equation
R
• Add/remove water
from system so that q |
inflow and outflow are
different
• R can be recharge,
ET, well pumping, etc.
• R can be a function of
space and time
• Units of R: L T-1
xx
qx|x+Dx
b
Dx
Dy
Poisson Equation
R
(qx|x+Dx - qx|x)Dyb -RDxDy = 0
h
q  K
x

h
 K x

x  Dx
 h
 x

qx|x
qx|x+Dx
b
Dx
Dy
h 
K
Dyb  RDxDy

x x 
h 



x
R
x  Dx
x

Dx
T
 2h
R

2
x
T
Dupuit Assumption
• Flow is horizontal
• Gradient = slope of water table
• Equipotentials are vertical
Dupuit Assumption
(qx|x+Dx hx|x+Dx- qx|x hx|x)Dy - RDxDy = 0
h
q  K
x


h
h
hx Dx  K
hx  Dy  RDxDy
 K x
x x 
x  Dx

h
h
 2h
x
x
2
 h 2

 x
h 2 



x
R
x  Dx
x

2Dx
K
h
2R

2
x
K
2 2
Capture Zones
Water Balance and Model
Types
Water Balance
h
0
y x ,1000
• Given:
– Recharge rate
– Transmissivity
h
0
x 1000, y
h
0
x 0, y
• Find and compare:
– Inflow
– Outflow
h x ,0  0
Water Balance
• Given:
– Recharge rate
– Flux BC
– Transmissivity
• Find and compare:
– Inflow
– Outflow
Block-centered model
Effective
outflow
boundary
2Dy
Y
Only the area inside the
boundary (i.e. [(imax -1)Dx]
[(jmax -1)Dy] in general)
contributes water to what
is measured at the
effective outflow
boundary.
1Dy
0
0
1Dx
X
2Dx
In our case this was
23000  11000, as we
observed. For large imax
and jmax, subtracting 1
makes little difference.
Mesh-centered model
Effective
outflow
boundary
2Dy
An alternative is to use a
mesh-centered model.
Y
This will require an extra
row and column of nodes
and the constant heads
will not be exactly on the
boundary.
1Dy
0
0
1Dx
X
2Dx
Dupuit Assumption Water Balance
Effective
outflow
area
h1
(h1 + h2)/2
h2
Geostatistics
Basic definitions
• Variance:
n
1
2
var( K )   Ki  K mean 
n 1
  var( K )
2
• Standard Deviation:
  var( K )
Basic definitions
• Number of pairs
Basic definitions
• Number of pairs:
n ( n  1)
n pairs 
2
Basic definitions
• Lag (h)
– Separation distance
(and possibly
direction)
h
Basic definitions
• Variance:
n
1
2
var( K )   Ki  K mean 
n 1
h
• Variogram:
n(h)
1
2
K (x)  K (x  h)
 h  

2n ( h ) 1
The variogram
• Captures the intuitive notion that samples
taken close together are more likely to be
similar that sample taken far apart
Common Variogram Models
Common Variogram Models
Basic definitions
Kriging:
N
K x    wi Ki
1
N
w
i
1
1
BLUE
Kriging Estimates
Random Numbers; Pure Nugget
#
# One variable definition:
# to start the variogram modeling user interface.
#
data(K): 'rand.csv', x=1, y=2, v=3;
Unconditioned Simulation
•
•
•
•
Specify mean and neighborhood
Specify variogram
Simulation should honor variogram
.cmd file/mask map
# Unconditional Gaussian simulation on a mask
# (local neighborhoods, simple kriging)
# defines empty variable:
ncols
nrows
cellsize
xllcorner
yllcorner
0
data(dummy): dummy, sk_mean=100, max=20, min=10, force;
variogram(dummy): 10 Sph(10);
mask: 'gridascii.prn';
method: gs; # Gaussian simulation instead of kriging
predictions(dummy): 'gs.out';
60
40
1
0
0
0
0 ...
Unconditional Simulation
Simulated Field/Known Variogram
Conditional Gaussian Simulation
• Specify data
• Fit and specify variogram
• Simulation should honor variogram and be
responsive to values at ‘conditioning’
points
# Gaussian simulation, conditional upon data
# (local neighborhoods, simple kriging)
data(SC): 'SC_rand.csv', x=1, y=2, v=3,average,max=20, sk_mean=1400;
method: gs;
variogram(SC): 400000Nug(0)+3.5e+006 Gau(0.035);
#Gridded Output
mask: 'ga_SC.prn';
predictions(SC): 'SC_pred.prn';
Kriging
• Specify data
• Fit and specify variogram
• Simulation should honor variogram and
return exact values at sampling points
• Optimal estimate too far from sample data
is mean
#
# Kriging
# (local neighbourhoods, simple and ordinary kriging)
#
data(SC): 'SC_rand.csv', x=1, y=2, v=3,average,max=20, sk_mean=1400;
variogram(SC): 400000Nug(0)+3.5e+006 Gau(0.035);
#Gridded Output
mask: 'ga_SC.prn';
predictions(SC): 'SC_Krpred.prn';
Gaussian Simulation/Kriging
Kriging
500
400
300
200
100
0
Bin
5000
6000
7000
8000
9000
10000
More
6000
7000
8000
9000
10000
More
Histogram
5000
Bin
4000
3000
2000
1000
0
-1000
-2000
-3000
-4000
-5000
0
4000
3000
2000
1000
0
-1000
-2000
-3000
-4000
-5000
Frequency
Gaussian
Frequency
Gaussian Simulation/Kriging
Histogram
150
100
50
Transient Ground Water Flow
Transient Flow Equation
 I  O   A
DVw = DxDy S Dh
Vw
h
 DxDyS
t
t
(qx|x - qx|x+Dx)Dyb + (qy|y - qy|y+Dy)Dxb = SDxDyh/t

h
h
 K x  K x
x

 h

 x


h
h
 Dyb   K y  K y
x  Dx 
y


x  Dx
Dx

h   h

x x   y


y  Dy
Dy

h
 Dxb  SDxDy
t

y  Dy 
h 

y y 

S h
T t
 2 h  2 h S h


x 2 y 2 T t
Finite Difference
t-Dt
h|x, t-Dt
x -Dx
h/t|t-Dt/2
h Dh hx , y ,t  hx , y ,t Dt


t Dt
Dt
x +Dx
Estimate here
t
x
h|x, t
h x  Dx , y ,t Dt  2h x , y ,t Dt  h x Dx , y ,t Dt
Dx 2
hx , y ,t  hx , y ,t Dt

h x , y  Dy ,t Dt  2h x , y ,t Dt  h x , y Dy ,t Dt
Dy 2
S hx , y ,t  hx , y ,t Dt

T
Dt
TDt  h x  Dx , y ,t Dt  4h x , y ,t Dt  h x Dx , y ,t Dt  h x , y  Dy ,t Dt  h x , y Dy ,t Dt 



2
S 
Dx

CFL Condition
• The stability criterion (for 1-D) is:
T/S Dt/Dx2  ½
Quasi-3D Models
Leakance and head-dependent
boundaries
Assumptions:
• Flow is 2-D horizontal in ‘aquifer’ layers
• Flow is vertical in ‘confining’ layers
• There is a significant difference in
hydraulic conductivity between aquifers
and confining layers
• Aquifer layers are connected by leakage
across confining layers
Schematic
T2 (or K2)
b2 (or h2)
i=2
k2
d2
T1
b1
k1
d1
i=1
Pumped Aquifer Heads
T2 (or K2)
b2 (or h2)
i=2
k2
d2
T1
b1
k1
d1
i=1
Heads
h2
h2 - h1
T2 (or K2)
b2 (or h2)
i=2
h1
k2
d2
T1
b1
k1
d1
i=1
Flows
h2
h2 - h1
h1
T2 (or K2)
b2 (or h2)
i=2
qv
k2
d2
T1
b1
k1
d1
i=1
Leakance
Leakage coefficient, resistance (inverse)
• Leakance
k
d
• From below:
ki 1
qv  hi 1  hi 
d i 1
• From above:
ki 1
qv  hi 1  hi 
d i 1
Equations
• Fully 3-D
  h    h    h 
Kx   K y   Kz   0

x  x  y  y  z  z 
• Confined
  hi    hi 
ki 1
ki 1




T

T

h

h

h

h
 Ri  0
xi
yi
i 1
i
i 1
i




x  x  y  y 
d i 1
d i 1
• Unconfined
 
hi   
hi 
ki 1
K xi hi
  K yi hi
 hi 1  hi 
 Ri  0



x 
x  y 
y 
d i 1
Poisson Equation
h
R  qv

2
x
T
2
R  qv
h x  Dx  h x Dx  Dx
T

2
2
h x, y
Finite Elements
 f: basis functions
Finite Elements
 f: hat functions
Fracture/Conduit Flow
Basic Fluid Dynamics
Momentum
• p = mu
Viscosity
•
•
•
•
Resistance to flow; momentum diffusion
Low viscosity: Air
High viscosity: Honey
Kinematic viscosity:


r
Reynolds Number
• The Reynolds Number (Re) is a non-dimensional
number that reflects the balance between viscous and
inertial forces and hence relates to flow instability (i.e.,
the onset of turbulence)
• Re = v L/
• L is a characteristic length in the system
• Dominance of viscous force leads to laminar flow (low
velocity, high viscosity, confined fluid)
• Dominance of inertial force leads to turbulent flow (high
velocity, low viscosity, unconfined fluid)
Re << 1 (Stokes Flow)
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford
University Press, Oxford. 519 pp.
Separation
Eddies, Cylinder Wakes, Vortex
Streets
Re = 30
Re = 40
Re = 47
Re = 55
Re = 67
Re = 100
Re = 41
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford
University Press, Oxford. 519 pp.
Poiseuille Flow
y
z
u
x
a
L
Flow
Poiseuille Flow
• In a slit or pipe, the velocities at the walls are 0
(no-slip boundaries) and the velocity reaches its
maximum in the middle
• The velocity profile in a slit is parabolic and
given by:
2

G   a 
2
ux  
x



2    2 

• G can be due to gravitational
acceleration (G = rg in a vertical
slit) or the linear pressure gradient
(Pin – Pout)/L
u(x)
x=0
x = a/2
Poiseuille Flow
2


G  a 
2
ux  
x



2    2 

• Maximum
umax
G a

 
2  2 
2
u(x)
• Average
2
G 2
uaverage  umax 
a
3
12 
x=0
x = a/2
Kirchoff’s Current Law
• Kirchoff’s law states that the total current flowing into a
junction is equal to the total current leaving the junction.
I1
Gustav Kirchoff
was an 18th
century German
mathematician
I1 flows into the node
I2 flows out of the node
I3 flows out of the node
node
I2
I1 = I2 + I3
I3
• Ohm’s law relates the flow of current to the
electrical resistance and the voltage drop
• V = IR (or I = V/R) where:
– I = Current
– V = Voltage drop
– R = Resistance
• Ohm’s Law is analogous to Darcy’s law
• Poiseuille's law can related to Darcy’s law and
subsequently to Ohm's law for electrical circuits.
uave
1 DP 2

a
12  L
Q  uave A
1 DP 2
Q
aa
12  L
A = a *unit depth
• Cubic law:
a DP
Q
12 L
3
Q  Ki
3
a
K
12 
36 lu
Fracture Network
DP12
Q12
DP  DP12  DP23  DP34  DP45  DP56
L12
Q23 L
23
DP
DP23
Q34
DP34
L45
Q45
DP45
Q56
DP56
-216 lu -
Q12  Q34  Q56
Q23  Q45
Q12  2Q23
Matrix Form
DP23
DP12
K12  2
K 23  0
L
L
DP23
DP34
2
K 23 
K 34  0
L
L
DP34
DP45
K 34  2
K 45  0
L
L
DP45
DP56
2
K 45 
K 56  0
L
L
 K12
 0

 0

 0
 L12
 2 K 23
2 K 23
0
0
L23
DP  DP12  DP23  DP34  DP45  DP56
0
 K 34
K 34
0
L34
0
0
 2 K 45
2 K 45
L45
 DP12 
 L 
 12 
0   DP23   0 
0   L23   0 
  DP   
0   34    0 
 L34
 
 K56   DP   0 
 45 
L56   L  DP 
 DP45 
 56 
 L56 
Back Solution
• Have conductivities and, from the matrix
solution, the gradients
DP12
– Compute flows
Q
K12
L
• Also have end pressures
– Compute intermediate pressures from DPs
Hydrologic-Electric Analogy
Poiseuille's law corresponds to the Kirchoff/Ohm’s Law for electrical circuits,
where pressure drop Δp is replaced by voltage V and flow rate by current I
DaP  DP12  DP23  DP34  DP45  DP56
Vmax
I12
ΔP12
I23
 
DP a 2

2 rL 2
R
I34
ΔP34
I45
I45
I56
ΔP56
V
R
I23
ΔP23
ΔP45
I
Q = 0.11 lu3/ts
Q = 0.11 lu3/ts
Kirchoff
LBM
Re
0.66
1.0
1.8
4.1
7.2
43.0
1
K
Q (lu3/ts)
LBM
Kirchoff’s
0.11
0.11
0.14
0.14
0.18
0.19
0.27
0.28
0.36
0.37
0.87
0.92
Eddies
Serpa, CY, 2005, Unpublished MS Thesis, FIU
Bai, T., and Gross, M.R., 1999, J
Geophysical Res, 104, 1163-1177
Flow
3 mm
3.3 mm x 2.7 mm
Re = 9
Non-curving cross joint
4.0E-03
Non-linear
y = 0.29x + 0.00
R2 = 1.00
3.5E-03
2.5E-03
2.0E-03
1.5E-03
1.0E-03
5.0E-04
0.0E+00
0.0E+00
2.0E-03
4.0E-03
6.0E-03
8.0E-03
1.0E-02
1.2E-02
1.4E-02
HEAD GRADIENT
Non-curving cross joint
Poiseuille Law
Non-linear
0.295
0.290
HYDRAULIC CONDUCTIVITY (m/s)
FLUX (m/s)
3.0E-03
0.285
0.280
0.275
0.270
0.265
0.260
0.255
0.250
0.1
1.0
10.0
REYNOLDS NUMBER
100.0