final review. ground water basics porosity head hydraulic conductivity transmissivity

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