Brief Review of Vector Calculus - New Mexico Tech Earth ... · ERTH403/HYD503 Lecture 6 Hydrology...

ERTH403/HYD503 Lecture 6

Hydrology Program, New Mexico Tech,

Prof. J. Wilson, Fall 2006 1

Darcy’s Law in 3D

• Today

– Vector Calculus

– Darcy’s Law in 3D hKq !"=

Brief Review of Vector Calculus

• A scalar has only a magnitude

• A vector is characterized by both direction

and magnitude.• Vectors are represented by :

– boldface (in books, papers & reports), or

– Characters topped by a bar or arrow, or by an underline

(eg, handwritten on paper or the board):

– e.g.,

h!"",,,, e.g, vqg

sSShn ,,,,,, e.g., !"µ

)(

),,(

.or ,, velocity seepage

zyx

v

zyx

z

y

x

zyx

vvv

v

v

v

vvv

vvv

++=

!!!

"

#

$$$

%

&

==

=r





where a, b, c are all scalarszcybxaf ++=

z

x

y

(unit vector in the Z direction)

(unit vector in the Y direction)

(unit vector in the X direction)x

y

z

We’ll top a character with a bar, or use boldface, to indicate a vector.

x

y

z

xy

z

xa

yb

zcf

Project vector onto each axisf

Orthogonal projection onto each axes.

zcybxaf ++=





(written )

To find the magnitude (length) of a vector:

f f

222cba ++ zcybxaf ++= where

The magnitude of vector is

!!!

"

#

$$$

%

&

=++=

mc

mb

ma

zmcymbxmafm

Multiplication by a scalar only affects the vector length.

To multiply a vector by a scalar:

A vector multiplied by a scalar results in a vector.

=f


Multiplying vectors by each other:

For groundwater flow, the dot or inner product is

the most commonly used product.

Given two vectors:

The dot product is a scalar:

zgygxggzyx

++=

zkykxkk zyx ++=

g

k

kg !

g k g

k k

( dot ) is the product of the component of

in the direction of with magnitude

The result is a scalar.





This is the component of in the direction of .

It equals .

zzyyxx kgkgkgkgkg ++==! )cos("

)cos(!g

g k g

k k

( dot ) is the product of the component of

in the direction of with magnitude


g

k!

g k

= sum (products of components)


Direction of movement

F1

100% of force

contributed

F2Force contributed is

F2 cos (45°)45°

F3

No contribution

Reminder from freshman Physics: pushing a car

Force in the x-direction moves the car.

Find and sum the x-direction forces:

y

x





Special Cases:

Orthogonal Vectors: dot product is zero

Parallel Vectors: dot product is the product of vector magnitudes:

g

k

kg kgkgkg ==! )0cos( o

00)90cos( o===! kgkgkg


Gradient operator:

The gradient operator is a way of doing differentiation with vectors.

It is a vector operator.

Gives the rate of change of a scalar field

in the direction of the greatest rate of change.

!( ) : pronounced “del”

The ( ) indicates the vector

del is operating on a scalar

In this case it is !z because del is

operating on elevation z.

x

y

z

x1

y1

z1

!z

!( )





zz

yy

xx !

!+

!

!+

!

!="

) () () () (

zyxzyxh ++= 2),,( 2

zyxxzyxh ++=! 22),,(

Example:


The driving force for groundwater flow is -!h, where h is head,

because !h points in the increasing direction.

Define the hydraulic gradient vector hJ !"=

Gradient operator:

Hydraulic gradient:

For a one-dimensional system (a Darcy column)

dx

dhx

x

xhz

z

xhy

y

xhx

x

xhxh =

!

!=

!

!+

!

!+

!

!="

)()()()()(


h varies only in x-direction, ie h(x)

Gradient of h :

If conductivity and area is uniform within the column (and no

sources/sinks), then dxdh is linear, so

l

h

x

h

dx

dh

!

!=

!

!=




f!"

Divergence:

The divergence is obtained by taking the dot product of the gradient

operator and another vector, eg., specific discharge.


Divergence operator:

It’s basically taking the partial derivative of each component of a

vector and then summing the result.

is the “divergence of vector f ”

)(!"


( )z

c

y

b

x

azcybxaz

zy

yx

xf

!

!+

!

!+

!

!=++"#

$

%&'

(

!

!+

!

!+

!

!=")

) () () (

[ ]) (!"! is the “divergence of the gradient”.

[ ]2

2

2

2

2

2 ) () () () () () () (

zyxzzyyxx !

!+

!

!+

!

!=

!

!

!

!+

!

!

!

!+

!

!

!

!="#"


It is basically taking the second partial derivative.

Example. Given then

Written as and called the Laplacian Operator.( )2!

),,( zyxhh =

2

2

2

2

2

22 ),,(),,(),,(

z

zyxh

y

zyxh

x

zyxhh

!

!+

!

!+

!

!="





The divergence shows the presence of sources/sinks

and other internal forcings.

If thenspace of functiona or constant a 2==! "h

02=! hIf then

Laplace’s equation

Poisson’s equation

More on Tensors

!!!

"

#

$$$

%

&

=

zzzyzx

yzyyyx

xzxyxx

KKK

KKK

KKK

K

Kij is the entry from the ith row, jth column

Kij gives flux in ith -direction for a unit gradient applied in the jth-direction

is a symmetric tensor (Kij = Kji)K




Extrapolation of Darcy’s Law to 3D

hKq !"=

for anisotropic, heterogeneous porous media

j

ijix

hKq

h

!

!"=

#"= Kqor

!Einsteinian notaton

!

q =

qx x

qy y

qz z

"

#

$ $ $

%

&

' ' '

= (

Kxx Kxy Kxz

Kyx Kyy Kyz

Kzx Kzy Kzz

"

#

$ $ $

%

&

' ' '

)h

)xx

)h

)yy

)h

)zz

"

#

$ $ $ $ $ $

%

&

' ' ' ' ' '

= (

(Kxx

)h

)x+ Kxy

)h

)y+ Kxz

)h

)z) x

(Kyx

)h

)x+ Kyy

)h

)y+ Kyz

)h

)z) y

(Kzx

)h

)x+ Kzy

)h

)y+ Kzz

)h

)z)

qZ

1 2 4 4 4 4 3 4 4 4 4

z

"

#

$ $ $ $ $ $ $ $ $

%

&

' ' ' ' ' ' ' ' '


hKq !"=

i.e.,





)(

)(

)(

z

hK

y

hK

x

hKq

z

hK

y

hK

x

hKq

z

hK

y

hK

x

hKq

zzzyzxz

yzyyyxy

xzxyxxx

!

!+

!

!+

!

!"=

!

!+

!

!+

!

!"=

!

!+

!

!+

!

!"=

hKq !"=

Kmax

Kmin

If our coordinate axes for our problem are aligned differently than the principal directions

of K, we must use a full tensor for K

x

z

y

Darcy’s Law in 3D




Instead, we can rotate coordinate axes so they are parallel to the “axes” of

the conductivity ellipsoid

x

z

y

!!!

"

#

$$$

%

&

=

!!!

"

#

$$$

%

&

=

zz

yy

xx

zzzyzx

yzyyyx

xzxyxx

K

K

K

KKK

KKK

KKK

K

00

00

00

)( zz

hKy

y

hKx

x

hKhKq zzyyxx

!

!+

!

!+

!

!"=#"=

Then

(all off-diagonal

components

equal zero)

this implies off-diagonal terms in are zero,

and that

i.e., flow is strictly in the direction of gradient

K

Kxx = Kyy = Kzz = K

Suppose K is isotropic:

Darcy’s Law in 3D

hKzz

hy

y

hxx

hKq !"=

#

#+

#

#+

#

#"= )(

If we consider flow in only one direction, say x:

x

hKq!

!"=




• Next time

– Conservation of Mass

– Continuity

– Field Equations

Darcy’s Law in 3D

• Review

– Vector Calculus

– Darcy’s Law in 3D

)( hKqt

hSs !!="!=

#

#

Groundwater Balance• Aquifer wide:

Change in storage = Recharge

– Pumping – GW Discharge

– ETGW ± Underflow = Forcings

• What about the water balance withinan individual Control Volume (CV)

somewhere (any arbitrary CV)within the aquifer?

Pumping

Pumping

flow




Continuity Equation

• Assumptions– 1. Medium is not deformable but undergoes elastic compression

in the vertical direction– Terzaghi assumption

– 2. Fluid is only slightly compressible– density is nearly constant

From 1 and 2 all compressibility effects are described by specificstorage, Ss, which is the only form of storage

– 3. Darcy’s law applies

• Statement of Mass Conservation (or Continuity)– Rate of change of storage =

Volumetric flux in – volumetric flux out

– No other sources/sinks or forcings

Volumetric flux into the left x face =

Volumetric flux into the back y face =

Volumetric flux into the top z face =

Total volumetric flux in =

dydzqAq xxinx =,

dxdzqAq yyiny =,

dydxqAq zzinz =,

dydxqdxdzqdydzq zyx ++

Continuity EquationVolumetric flux into the CV:

“x = co

nstant”

face

Note positive down in this derivation




Volumetric flux out of the right x face = dydzx

qq xx )(

!

!+

Continuity EquationVolumetric flux out of the CV:

At “x+

dx = co

nstant”

face

!

qx,out AX = qxAx +"qx

"xdxAx

= qxdydz +"qx

"xdxdydz

Where does this come from?

If qx,out ! qx,in , then qx must have changed over dx.

The "qx/ " x grasps this change.

Mathematical justification: Taylor Series Approximation

Volumetric flux out of the right x face =

Volumetric flux out of the front y face =

Volumetric flux out of the bottom z face =

Total volumetric flux out =

dydzdxx

qq xx )(

!

!+

dxdzdyy

qq

y

y )(!

!+

dydxdzz

qq ZZ )(

!

!+

Continuity EquationVolumetric flux out of the CV:

At “x+

dx = co

nstant”

face

dxdydzz

qqdxdzdy

y

qqdydzdx

x

qq z

z

y

yx

x )()()(!

!++

!

!++

!

!+




Continuity Equation

])()()[( dxdydzz

qqdxdzdy

y

qqdydzdx

x

qq z

z

y

yx

x!

!++

!

!++

!

!+"

dxdydzq!"#=

Net volumetric flux into the CV:

dydxqdxdzqdydzq zyx ++

Net volumetric flux into the CV =

volumetric flux in – volumetric flux out =

dxdydzz

q

y

q

x

q

dzdxdyz

qdydxdz

y

qdxdydz

x

q

zyx

zyx

)(

][

!

!+

!

!+

!

!"=

!

!+

!

!+

!

!"=

Rate of change in storage =t

Vw

!

!

hVSVTsw!=!

Rate of change in storage = dx dy dzt

h S

t

h VS sTs

!

!=

!

!

Continuity EquationRate of change of storage in the CV:

But where VT = dx dy dz




Continuity Equation

• Statement of Mass Conservation (or Continuity)

Rate of change of storage =


)(z

q

y

q

x

q

y

hS zyxs

!

!+

!

!+

!

!"=

!

!

dx dy dzt

h Ss!

!dxdydzq!"#=dxdydz

z

q

y

q

x

q zyx )(!

!+

!

!+

!

!"=

0 or =!"+#

#!"$=

#

#q

t

hSq

t

hS ss

Continuity Equation

• Statement of Mass Conservation (or Continuity)

Rate of change of storage =


!

Ss"h

"t= # $ % q

!

q = "K#h

!

"# $ q = "# $ "K#h( ) = K#2h

If hydraulic conductivity is isotropic:

!

" Ss

#h

#t= K$

2h

!

" Ss

#h

#t= K$

2h




t

hShKS

!

!="

2

t

hbShKbS

!

!="

2

t

hShT!

!="

2

t

h

T

Sh

!

!="

2

Another homology to heat conduction or solute diffusion

a “diffusion equation”

T

San upscaled hydraulic diffusivity

Vertically Integrated Aquifer Equation

Aquifer equation,

for homogeneous K:

Multiply by thickness, bto “upscale”

Define T and S

Rewrite as

h(x,y,z,t)

h(x,y,t)

• Next time

– Boundary Conditions

– Initial Conditions

– Proper Mathematical

Statement

`

Darcy’s Law in 3D

• Review

– Conservation of Mass

– Continuity

– Field Equations)( hKq

t

hSs !!="!=

#

#

Brief Review of Vector Calculus - New Mexico Tech Earth ... · ERTH403/HYD503 Lecture 6 Hydrology...

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Transcript of Brief Review of Vector Calculus - New Mexico Tech Earth ... · ERTH403/HYD503 Lecture 6 Hydrology...