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!
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