final review. ground water basics porosity head hydraulic conductivity transmissivity
TRANSCRIPT
Final Review
Ground Water Basics
• Porosity
• Head
• Hydraulic Conductivity
• Transmissivity
Porosity Basics
• Porosity n (or )
• Volume of pores is also the total volume – the solids volume
total
pores
V
Vn
total
solidstotal
V
VVn
Porosity Basics
• Can re-write that as:
• Then incorporate:• Solid density: s
= Msolids/Vsolids
• Bulk density: b
= Msolids/Vtotal • bs = Vsolids/Vtotal
total
solidstotal
V
VVn
total
solids
V
Vn 1
s
bn
1
Porosity Basics
• Volumetric water content ()– Equals porosity for
saturated system total
water
V
V
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 = ghp
– density– g gravity
– hp depth
P = 0 (= Patm)
Pre
ssur
e H
ead
(incr
ease
s w
ith d
epth
bel
ow s
urfa
ce)
Pressure Head
Ele
vati
on
Head
Elevation Head
• Water wants to fall
• Potential energy
Ele
vatio
n H
ead
(incr
ease
s w
ith h
eigh
t ab
ove
datu
m)
Eleva
tion
Head
Ele
vati
on
Head
Elevation datum
Total Head
• For our purposes:
• Total head = Pressure head + Elevation head
• Water flows down a total head gradient
P = 0 (= Patm)
Tot
al H
ead
(con
stan
t: h
ydro
stat
ic e
quili
briu
m)
Pressure Head
Eleva
tion
Head
Ele
vati
on
Head
Elevation datum
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
www.ngwa.org/ ngwef/darcy.html
1803 - 1858
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
g
kK w
L T-1 L2
Kozeny-Carman Equation
1801
2
2
3md
n
nk
Transmissivity
• T = Kb
Darcy’s Law
• Q = -K dh/dl A
• Q, q
• K, T
Mass Balance/Conservation Equation
• I = inputs
• P = production
• O = outputs
• L = losses
• A = accumulation
ALOPI
Derivation of 1-D Laplace Equation
• Inflows - Outflows = 0
• (q|x - q|x+x)yz = 0
• q|x – (q|x +x dq/dx) = 0
• dq/dx = 0 (Continuity Equation)
x
hKq
x y
qx|x qx|x+xz
0
dxxh
Kd0
2
2
x
h(Constitutive equation)
General Analytical Solution of 1-D Laplace Equation
Ax
h
xAxx
h
0
2
2
x
h
xxx
h0
2
2
BAxh
Particular Analytical Solution of 1-D Laplace Equation (BVP)
Ax
h
BAxh
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:
Incorporate derivative, gives A.
After 2nd integration of Laplace Equation we have:
Incorporate constant head, gives B.
Finite Difference Solution of 1-D Laplace Equation
Need finite difference approximation for 2nd order derivative. Start with 1st order.
Look the other direction and estimate at x – x/2:
x
hh
xxx
hh
x
h xxxxxx
xx
2/
x
hh
xxx
hh
x
h xxxxxx
xx
2/
h|x h|x+x
x x +x
h/x|x+x/2
Estimate here
Finite Difference Solution of 1-D Laplace Equation (ctd)
Combine 1st order derivative approximations to get 2nd order derivative approximation.
h|x h|x+x
x x +x
h|x-x
x -x
h/x|x+x/2
Estimate here
h/x|x-x/2
Estimate here
2h/x2|x
Estimate here
22/2/
2
2 2
x
hhh
xx
hh
x
hh
x
x
h
x
h
x
h xxxxx
xxxxxx
xxxx
Set equal to zero and solve for h:
2xxxx
x
hhh
2-D Finite Difference Approximation
h|x,y h|x+x,y
x, y
y +y
h|x-x,y
x -x x +x
h|x,y-y
h|x,y+y
4,,,,
,
yyxyyxyxxyxx
yx
hhhhh
Matrix Notation/Solutions
• Ax=b
• A-1b=x
3,34,23,13,22,2
2,31,22,13,22,2
4
4
hhhhh
hhhhh
3,34,23,1
2,31,22,1
3,2
2,2
41
14
hhh
hhh
h
h
Toth Problems
• Governing Equation
• Boundary Conditions
1 3 5 7 9
11
13
15
17
19
S 1
S 2
S 3
S 4
S 5
S 6
S 7
S 8
S 9
S 10
S 11
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
10-10.0102
2
2
2
y
h
x
h
Recognizing Boundary Conditions
• Parallel: – Constant Head – Constant (non-zero) Flux
• Perpendicular: No flow
• Other: – Sloping constant head
Internal ‘Boundary’ Conditions
• Constant head – Wells– Streams– Lakes
• No flow– Flow barriers
• Other
Poisson Equation
• Add/remove water from system so that 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
x y
qx|x qx|x+xb
R
x y
qx|x qx|x+x
x yx yx y
qx|x qx|x+xb
R
Poisson Equation
x y
qx|x qx|x+xb
R
x y
qx|x qx|x+x
x yx yx y
qx|x qx|x+xb
R(qx|x+x - qx|x)yb -Rxy = 0
x
hKq
yxRybx
hK
x
hK
xxx
T
R
x
xh
xh
xxx
T
R
x
h
2
2
Dupuit Assumption
• Flow is horizontal• Gradient = slope of water table• Equipotentials are vertical
Dupuit Assumption
K
R
x
h 22
22
(qx|x+x hx|x+x- qx|x hx|x)y - Rxy = 0
x
hKq
yxRyhx
hKh
x
hK x
xxx
xx
K
R
x
xh
xh
xxx
2
22
x
hh
x
h
22
Capture Zones
Water Balance and Model Types
Water Balance
• Given: – Recharge rate – Transmissivity
• Find and compare:– Inflow– Outflow
0,1000
yx
h0
,0
yx
h
01000,
xy
h
00,
x
h
Water Balance
• Given: – Recharge rate – Flux BC– Transmissivity
• Find and compare:– Inflow– Outflow
X
0
2x1x
2y
1y
0
Y
Effective outflow boundary
Only the area inside the boundary (i.e. [(imax -1)x] [(jmax -1)y] in general) contributes water to what is measured at the effective outflow boundary.
In our case this was 23000 11000, as we observed. For large imax and jmax, subtracting 1 makes little difference.
Block-centered model
X
0
2x1x
2y
1y
0
Y
Effective outflow boundary
An alternative is to use a mesh-centered model.
This will require an extra row and column of nodes and the constant heads will not be exactly on the boundary.
Mesh-centered model
Dupuit Assumption Water Balance
h1
h2
Effective outflow area
(h1 + h2)/2
Geostatistics
Basic definitions
• Variance:
• Standard Deviation:
n
meani KKn
K1
21)var(
)var(2 K
)var(K
Basic definitions
• Number of pairs
Basic definitions
• Number of pairs:
2
)1(
nnnpairs
Basic definitions
• Lag (h)– Separation distance
(and possibly direction)
h
Basic definitions
• Variance:
• Variogram:
)(
1
2)()()(2
1 h
hxxh
hn
KKn
h n
meani KKn
K1
21)var(
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
iiKwK1
x
N
iw1
1BLUE
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:
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 krigingpredictions(dummy): 'gs.out';
ncols 60nrows 40cellsize 1xllcorner 0yllcorner 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 Outputmask: '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 Outputmask: 'ga_SC.prn';predictions(SC): 'SC_Krpred.prn';
Gaussian Simulation/Kriging
Gaussian Simulation/KrigingHistogram
0
50
100
150
-500
0
-400
0
-300
0
-200
0
-100
0 0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
0
Mor
e
Bin
Fre
qu
en
cyGaussian
Kriging
Histogram
0100200300400500
-500
0
-400
0
-300
0
-200
0
-100
0 0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1000
0
Mor
e
Bin
Fre
qu
ency
Transient Ground Water Flow
Transient Flow Equation
AOI
Vw = xy S h
t
hySx
t
Vw
(qx|x - qx|x+x)yb + (qy|y - qy|y+y)xb = Sxyh/t
t
hyxSxb
y
hK
y
hKyb
x
hK
x
hK
yyyxxx
t
h
T
S
y
y
h
y
h
x
x
h
x
h
yyyxxx
t
h
T
S
y
h
x
h
2
2
2
2
Finite Difference
x -x
h|x, tx
x +x
h/t|t-t/2 Estimate here
t-t
t
h|x, t-t
t
hh
t
h
t
h ttyxtyx
,,,,
t
hh
T
S
y
hhh
x
hhhttyxtyxttyyxttyxttyyxttyxxttyxttyxx
,,,,
2
,,,,,,
2
,,,,,,22
2
,,,,,,,,,,,,,,
4
x
hhhhh
S
tThh ttyyxttyyxttyxxttyxttyxx
ttyxtyx
CFL Condition
• The stability criterion (for 1-D) is:
T/S t/x2 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
i = 1
i = 2
d1
b1
d2
b2 (or h2)
k1
T1
k2
T2 (or K2)
Pumped Aquifer Heads
i = 1
i = 2
d1
b1
d2
b2 (or h2)
k1
T1
k2
T2 (or K2)
Heads
i = 1
i = 2
d1
b1
d2
b2 (or h2)
k1
T1
k2
T2 (or K2)
h1
h2
h2 - h1
Flows
i = 1
i = 2
d1
b1
d2
b2 (or h2)
k1
T1
k2
T2 (or K2)h1
h2 h2 - h1
qv
LeakanceLeakage coefficient, resistance (inverse)
• Leakance
• From below:
• From above:
d
k
1
11
i
iiiv d
khhq
1
11
i
iiiv d
khhq
Equations
• Fully 3-D
• Confined
• Unconfined
0
z
hK
zy
hK
yx
hK
x zyx
01
11
1
11
i
i
iii
i
iii
iyi
ixi R
d
khh
d
khh
y
hT
yx
hT
x
01
11
i
i
iii
iiyi
iixi R
d
khh
y
hhK
yx
hhK
x
Poisson Equation
2
2
,T
qRxhh
h
vxxxx
yx
T
qR
x
h v
2
2
Finite Elements
: basis functions
Finite Elements
: 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:
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 = 41Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford. 519 pp.
L
Flowuax
yz
Poiseuille 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:
x = 0 x = a/2
u(x)
• G can be due to gravitational acceleration (G = g in a vertical slit) or the linear pressure gradient (Pin – Pout)/L
2
2
22x
aGxu
Poiseuille Flow
• Maximum
• Average
x = 0 x = a/2
u(x)
2
2
22x
aGxu
2
max 22
aGu
2max 123
2a
Guuaverage
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.
II22 II33
node
II11 flows into the node
II22 flows out of the node
II33 flows out of the node II11 = = II22 + + II33
Gustav Kirchoff was an 18th century German mathematician
II11
• 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.
• Cubic law:
2
12
1a
L
Puave
AuQ ave
KiQ
aaL
PQ 2
12
1
L
PaQ
12
3
12
3aK
A = a *unit depth
Fracture Network
5645342312 PPPPPP
563412 QQQ
4523 QQ
2312 2QQ
L23
-216 lu -
L12
Q12
Q34
Q56
P
P12
P23
P34
Q23
Q45P45
P56
L45
36 lu
Matrix Form
02 2323
1212
K
L
PK
L
P
02 3434
2323
K
L
PK
L
P
02 4545
3434
K
L
PK
L
P
02 5656
4545
K
L
PK
L
P
P
L
PL
PL
PL
PL
P
LLLLL
KK
KK
KK
KK
0
0
0
0
2000
0200
0020
0002
56
56
45
45
34
34
23
23
12
12
5645342312
5645
4534
3423
2312
5645342312 PPPPPP
Back Solution
• Have conductivities and, from the matrix solution, the gradients– Compute flows
• Also have end pressures– Compute intermediate pressures from Ps
1212 K
L
PQ
a
Hydrologic-Electric AnalogyPoiseuille'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
I12
I23
I56
I45
ΔP12
ΔP23
ΔP34
ΔP45
ΔP56
I23
I45
R
VI 2max 22
aL
PV
KR
1
I34
0.66 0.11 0.111.0 0.14 0.141.8 0.18 0.194.1 0.27 0.287.2 0.36 0.3743.0 0.87 0.92
ReQ (lu3/ts)
Kirchoff’sLBM
Q = 0.11 lu3/ts Q = 0.11 lu3/ts
Kirchoff LBM
5645342312 PPPPPP
Eddies
Re = 93.3 mm x 2.7 mm
3 mm
2 m
m
Bai, T., and Gross, M.R., 1999, J Geophysical Res, 104, 1163-1177
Serpa, CY, 2005, Unpublished MS Thesis, FIU F
low
y = 0.29x + 0.00
R2 = 1.00
0.0E+00
5.0E-04
1.0E-03
1.5E-03
2.0E-03
2.5E-03
3.0E-03
3.5E-03
4.0E-03
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
FL
UX
(m
/s)
Non-linear
Non-curving cross joint
0.250
0.255
0.260
0.265
0.270
0.275
0.280
0.285
0.290
0.295
0.1 1.0 10.0 100.0
REYNOLDS NUMBER
HY
DR
AU
LIC
CO
ND
UC
TIV
ITY
(m
/s)
Poiseuille Law Non-linear
Non-curving cross joint