gwii 1 basics

7/31/2019 GWII 1 Basics

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Wolfgang KinzelbachInstitute of Environmental Engineering

ETH Zrich

Groundwater modeling:Basics and equations

Groundwater II


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Where are groundwater modelsrequired?

For integrated interpretation of data For improved understanding of the functioning of

aquifers For the determination of aquifer parameters For prediction

For design of measures For risk analysis

For planning of sustainable aquifer management


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Examples Installation of new well fields

Sustainability? Environmental impacts? Water quality?

Well head protection zones

Size and shape of catchment, travel times Design of remediation measures

Feasibility, optimization of costs

Risk analysis of repositories (e.g. for radioactivewaste)


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Why models? Relative inaccessibility of object of interest

Scarce data in geosciences required conceptualmodel for analysis

Complexity of nature requires simplification Connection between quantities of interest and

measurable quantities

Slowness of processes requires predictive capability Big effort in interpretation justified by high cost of

data


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Types of groundwater models Flow models (saturated flow, homogeneous fluid)

Modeled quantity: Piezometric head h

Derived quantity: Darcy-velocity v Solute transport models (hydrodynamically

inactive, dissolved substances) Modeled quantity: concentration c Derived quantity: mass flux j

Unsaturated flow and multiphase flow Coupled models

density, soil mechanics, temperature complex chemistry


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Basic equations Starting from first principles:

Conservation of mass (or volume if density is constant) Conservation of dissolved mass (solute)

General approach based on the following quantities: Extensive quantity (volume, dissolved mass) Intensive quantity = Extensive quantity per unit volume

(porosity, concentration) Flux j (of volume, diss. mass) (Quantity per unit area and time)

Source-sink distribution (volume per geometric volume andtime, diss. mass per water volume and time)


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General principle: in 1D (for simplicity)

x

x

Flux in Flux out

Storage is change inextensive quantity

Gain/loss from volumesources/sinks

Conservation law in words:

Cross-sectional area A

Volume V = A x

x+ x

Time interval [t, t+ t]

( ) _in out Flux Flux A t source density V t Storage + =


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General principle in 1D

)()())()(( t t t t V t A x x j x j +=++

t t t t

x x x j x j

+=+

+ )()()()(

Division by txA yields:

In the limit t, x to 0:

t x

j

=+


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General principle in 3D

t z j

y j

x j z y x

=+

t j =+

G

or


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Water balance: in 1D

x

x

Storage can be seen as changein intensive quantity

Gain/loss by volumesources/sinks

Conservation equation for water volume

x+ x

Time interval [t, t+ t]

)(v x x j in =G

)(v x x x jout +=G

)()())(v)(v( xx t V t t V t V wt A x x x water water +=++

Density assumed constant!

V=A x


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Water balance: in 1D continued

t

h

dh

dnw

x =+

xv

xh

K and S dhdn

x == v0

t

n

t

V V w

x

x x x water x

=

=+

+ ) / ()(v)(v x

Using the definition of the storage coefficient and Darcys law:

In the limit

t h

S w xh

K x

=+

0)(

(V water /V=n)


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Generalization to 3D

t h

S w z y x

=+

0zyx vvv

vv vv v y x zwith - h and

x y z

= + + =

KG G

t

hS wh)(

=+0

K

Multitude of aquifers through distributions of K, S 0 and w

and boundary and initial conditions

+ Initial conditions+ Boundary conditions


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In practical applications: often 2Dand 2.5 D models

t hS qh)(T

=+

2D confined Aquifer: by integration over z-coordinate

2D unconfined Aquifer: by integration over z-coordinate

t

hnqh) Bottomh(K

=+ )(

with T=K(Top-Bottom) andS= S 0(Top-Bottom)

with n= porosity

2.5 D by vertical coupling of 2D layers throughleakage-term

1 1( ) (i layer i layer i i layer i layer iq h h h h 1 ) = + + ( =K vert / z)


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First type: h on boundaryspecified

Second type:

given on boundary Third type:given on boundary

Further: Free surface p=0 ,evaporation boundary,moving boundary

nh

Boundary conditions for flow

/

nhh + /


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Use natural hydrogeological boundaries (e.g. rivers, waterdivides, end of aquifers)

Use as few fixed head boundaries as possible (at least one isnecessary in a steady state model)

At upstream boundaries use fixed flux boundary. Atdownstream boundaries use fixed head boundary.

Use third type boundary conditions for distant fixed heads.

Streamlines are boundaries of the second type. Careful: If used together with pumping, check whether changes in fluxbetween the two streamlines stay small against total flowthrough the stream tube.)

Practical rules for setting boundaryconditions in flow models


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Prescribed flux

P r e s c r

i b e d f

l u x

Impermeable boundary

I m p e r m

e a b l e b o u n d a r y

Witi

Constant head (Aare)

Grenchen

Witibach

Staadkanal

Aarmattenkanal

Example


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And now the same for transport of asolute...

The flux is more complicated: It is composed of

Advective Flux Diffusive Flux Dispersive Flux

Total flux:

c jconv vG

= cnD j mdiff =cn j

disp= D

dispdiff advtot j j j jGGGG

++=

(v ist Darcy-velocity)


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Mass balance: in 1D

x

x

Storage of dissolved mass

Losses from degradationand reaction according to1st order reaction.Gains by injection

Conservation equation for dissolved mass

x+ x

)(, x j xtotal )(, x x j xtotal

time interval [t, t+ t]

+

))()((

))()((

t mt t m

t V cwt V nct A x x j x j intotal,xtotal,x

+=++

V=A x


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Mass balance: in 1D continued

t nc

wcnc x

jin

total,x

=+

)(

t V t cV t t cV

wcnc x

x j x x j water water in

total,xtotal,x

+=+

+ / ))()(()()(

In the limit:

Substitution of expression into fluxes:

t

ncwcnc

x

c D Dn

x x

c in Lm

x

=+

+

+

)(

1))((

)v(

With n = constant and pore velocity u = v/n:

t c

nwc

c xc

D D x xc in

Lm x

=+

+

+

))(()u(


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Mass balance: generalization to 3D

Again: + boundary conditions + initial conditions

t

c

n

wccc Dc inm

=+++ ))(()u( DK

With )()(u)(u)(u zyx cu

zc

y

c

xc G

=

+

+

=+=+T

T

L

mdispdiff

D

D

D

und c D j j

00

00

00

)( DDGG

In a system,in which u isparallel to x-axis!

und


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Still more complete version

required: Flow field u and parameter: n, D (d.h.

L,

T,

Tv,D

m), R,

)()(1

ccnR

wcc D

Rc

R

u

t

cin

++=

G

Storage

Advection

Dispersion

Degradation

External sources/sinks

Retardation

Transport equation


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Illustration of transportFlow direction

t=0

t=tAdvection

Advection and dispersion

Advection, dispersion and adsorption

Advection, dispersion, adsorption and degradation

x

x

x

x

x


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Boundary conditions for thetransport equation

First type: c prescribed on boundary,

determines advective flux Second type:prescribed on boundary, determinesdiffusive-dispersive flux (usually usedto make a boundary impervious fordiffusion/dispersion, e.g. on streamlinealong impervious boundary

Third type:prescribed on boundary, determines

total flux Transmission boundary

on boundary (or D=0 on boundary)

nc /

ncc + /

0 / 22 = nc

c=0 impervious0 / = nc

transmission

impervious

first or third type


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Analysis of type of equations Flow equation (K and S 0 constant )

t h

xh

S K

=

2

2

0

Type: Parabolic equation,diffusion equation withdiffusion coefficient K/S 0

Transport equation (n and u constant)

0+

t c

2

2

= x

c D

xc

uType: mixed, two types of terms

Has consequences for solution method!


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Special properties of transport

equation

Tdisp = L 2 /D

If only advection: Hyperbolic equation, describingthe propagation of a shock front along characteristiclines (trajectories), typical time scale

Tadv

= L/u If only diffusion: Parabolic equation, describing

diffusive spreading in all directions of space up toboundaries. Slowing down due to decreasinggradients, typical time scale

Invariant: Peclet-number = T disp /T adv = uL/D

L correspondsto x innumericalmodel!

gwii 1 basics

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