Post on 09-Jan-2023
Wolfgang KinzelbachInstitute of Environmental Engineering
ETH Zürich
Groundwater modeling:Basics and equations
Groundwater II
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
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)
Why models?• Relative inaccessibility of object of interest• Scarce data in geosciences required conceptual
model 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
Types of groundwater models• Flow models (saturated flow, homogeneous fluid)
– Modeled quantity: Piezometric head h– Derived quantity: Darcy-velocity v
• Solute transport models (hydrodynamicallyinactive, dissolved substances)– Modeled quantity: concentration c– Derived quantity: mass flux j
• Unsaturated flow and multiphase flow• Coupled models
– density, soil mechanics, temperature– complex chemistry
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 Φ (volume, dissolved mass) – Intensive quantity ξ = Extensive quantity per unit volume
(porosity, concentration)– Flux j (of volume, diss. mass) (Quantity per unit area and time)– Source-sink distribution σ (volume per geometric volume and
time, diss. mass per water volume and time)
General principle: in 1D (for simplicity)
x
Δx
Fluxin Fluxout
Storage is change in extensive quantity
Gain/loss from volumesources/sinks
Conservation law in words:
Cross-sectional area AVolume V = AΔx
x+Δx
Time interval [t, t+Δt]
( ) _in outFlux Flux A t source density V t Storage− ⋅ ⋅Δ + ⋅ ⋅Δ =
General principle in 1D
)()())()(( ttttVtAxxjxj Φ−Δ+Φ=Δ⋅⋅+Δ⋅⋅Δ+− σ
tttt
xxxjxj
Δ−Δ+
=+Δ
Δ+− )()()()( ξξσ
Division by ΔtΔxA yields:
In the limit Δt, Δx to 0:
txj
∂∂
=+∂∂
−ξσ
Water balance: in 1D
x
Δx
Storage can be seen as changein intensive quantity
Gain/loss by volumesources/sinks
Conservation equation for water volume
x+Δx
Time interval [t, t+Δt]
)(vx xjin = )(vx xxjout Δ+=
)()())(v)(v( xx tVttVtVwtAxxx waterwater −Δ+=Δ⋅⋅+Δ⋅⋅Δ+−
Density assumed constant!
V=AΔx
Water balance: in 1D continued
th
dhdnw
x ∂∂
=+∂
∂− xv
xhKandS
dhdn
x ∂∂
−== v0
tn
tVVw
xxxx waterx
ΔΔ
=Δ
Δ=+
Δ−Δ+
−)/()(v)(vx
Using the definition of the storage coefficient and Darcy‘s law:
In the limit
thSw
xhK
x ∂∂
=+∂∂
∂∂
0)(
(Vwater/V=n)
Generalization to 3D
thSw
zyx ∂∂
=+∂∂
−∂
∂−
∂∂
− 0zyx vvv
vv vv vyx zwith - h andx y z
∂∂ ∂= ∇ + + = ∇ ⋅
∂ ∂ ∂K
thSwh)(
∂∂
=+∇⋅∇ 0K
Multitude of aquifers through distributions of K, S0 and wand boundary and initial conditions
+ Initial conditions+ Boundary conditions
In practical applications: often 2D and 2.5 D models
thSqh)(T
∂∂
=+∇⋅∇
2D confined Aquifer: by integration over z-coordinate
2D unconfined Aquifer: by integration over z-coordinate
thnqh)Bottomh(K
∂∂
=+∇−⋅∇ )(
with T=K(Top-Bottom) andS= S0(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)λ λ− −= − + − +(λ=Kvert/Δz)
• 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 ∂∂⋅+⋅ /βα
• 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. At downstream 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 flux between the two streamlines stay small against total flow through the stream tube.)
Practical rules for setting boundaryconditions in flow models
Prescribed flux
Prescrib
ed flux
Impermeable boundary
Imperm
eable boundary
Witi
Constant head (Aare)
Grenchen
Witibach
Staadkanal
Aarmattenkanal
Example
And now the same for transport of a solute...
• The flux is more complicated:• It is composed of
– Advective Flux – Diffusive Flux– Dispersive Flux
• Total flux:
cjconv v=
cnDj mdiff ∇−=
cnjdisp ∇−= D
dispdiffadvtot jjjj ++=
(v ist Darcy-velocity)
Mass balance: in 1D
x
Δx
Storage of dissolved mass
Losses from degradationand reaction according to 1st order reaction.Gains by injection
Conservation equation for dissolved mass
x+Δx
)(, xj xtotal )(, xxj xtotal
time interval [t, t+Δt]
Δ+
))()(())()((
tmttmtVcwtVnctAxxjxj intotal,xtotal,x
−Δ+
=Δ⋅⋅⋅+Δ⋅⋅⋅⋅−Δ⋅⋅Δ+− λ
V=AΔx
Mass balance: in 1D continued
tncwcnc
xj
intotal,x
∂∂
=+−∂
∂−
)(λ
tVtcVttcVwcnc
xxjxxj waterwater
intotal,xtotal,x
Δ⋅−Δ+⋅
=+−Δ
−Δ+−
/))()(()()(λ
In the limit:
Substitution of expression into fluxes:
tncwcnc
xcDDn
xxc in
Lmx
∂∂
=+−∂∂
+⋅∂∂
+∂
∂−
)(1
))(()v( λ
With n = constant and pore velocity u = v/n:
tc
nwcc
xcDD
xxc in
Lmx
∂∂
=+−∂∂
+∂∂
+∂
∂− λ))(()u(
Mass balance: generalization to 3D
Again: + boundary conditions + initial conditions
tc
nwcccDc in
m ∂∂
=+−∇+⋅∇+⋅∇− λ))(()u( D
With )()(u)(u)(u zyx cuzc
yc
xc
⋅∇=∂
∂+
∂
∂+
∂∂
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=∇+=+
T
T
L
mdispdiff
DD
DundcDjj
000000
)( DD
In a system, in which u isparallel to x-axis!
und
• Still more complete version
• required: Flow field u and• parameter: n, D (d.h. αL, αT, αTv,Dm), R, λ
)()(1 ccnRwccD
Rc
Ru
tc
in −+−∇∇+∇−=∂∂ λ
Storage
Advection
Dispersion
Degradation
External sources/sinks
Retardation
Transport equation
Illustration of transportFlow direction
t=0
t=ΔtAdvection
Advection and dispersion
Advection, dispersion and adsorption
Advection, dispersion, adsorption and degradation
x
x
x
x
x
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 ∂∂⋅+⋅ /βα
0/ 22 =∂∂ nc
c=0 impervious0/ =∂∂ nc
transmission
impervious
first or third type
Analysis of type of equations
• Flow equation (K and S0 constant)
th
xh
SK
∂∂
=∂∂
2
2
0
Type: Parabolic equation, diffusion equation withdiffusion coefficient K/S0
• Transport equation (n and u constant)
0+∂∂
tc
2
2
=∂∂xcD−
∂∂xcu
Type: mixed, two types of terms
Has consequences for solution method!
Special properties of transportequation
Tdisp = L2/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 to boundaries. Slowing down due to decreasinggradients, typical time scale
• Invariant: Peclet-number = Tdisp/Tadv = uL/D
L correspondsto Δx in numericalmodel!