Methodologies for the Simulation of Groundwater Age and ... - FEFlow

Methodologies for the Simulation
of Groundwater Age and
Travel Time Distributions in Aquifers
~ Practical Applications ~
fabien.cornaton
fabien cornaton@unine.ch
Centre for Hydrogeology, University of Neuchâtel, Switzerland

OUTLINE OF THE PRESENTATION
(1) Groundwater Age (GWA) Concepts, Variables and Models
(2) Applying the Reservoir Theory to GW Resource Protection
(3) Forward and Backward Travel Times as Vulnerability and
Risk Assessment Criteria
(4) Application Example

(1) Groundwater Age
~ Concepts, Variables and Models ~

Frequency
Age

− Q
(i) Groundwater Groundwater
Groundwater
Groundwater « Ages Ages
Ages
Ages » - Definitions
Definitions
Definitions
Definitions
• Age Age Age
Age
Age [gA (x,τ), GA (x,τ)]: :
:
:
Time elapsed since water particles entered the aquifer
• Lifetime Lifetime Lifetime Lifetime Lifetime expectancy expectancy
expectancy
expectancy [gE (x,τ), GE (x,τ)]: :
:
:
Time remaining prior to leaving the aquifer
• Total Total Total Total transit transit
transit
transit (residence) (residence)
(residence)
(residence) time time
time
time [gT(x,τ), GT(x,τ)]: :
:
:
Total time required to travel from recharge to discharge
E
T = A + +
+
+ E
A
+Q

(ii ii) A A
A
A general general general
general
general theory theory
theory
theory for for for for for the the
the
the property property property
property
property “age age age” age
age
A theory for evaluating age in numerical (analytical analytical) ) models such that: that
● Advection – mixing (dispersion/diffusion) – production/destruction
processes are accounted for;
● Age is a time– time and position–dependent position
random variable;
● Age transport ⇔ transport of the bulk-volumetric
bulk volumetric water mass density: density
ρg(x, t, τ)dVdτ : mass of the « age species » in dV, lying in the interval
[τ−dτ/2, τ+dτ/2] (dτ→0)
g : « age concentration » distribution function [T −1 ]
● Equations governing the complete distribution of GWA (space and time);
● …where age mass is the conserved property.

Groundwater Groundwater Groundwater Groundwater flow
flow
flow
flow
Advection Advection-Dispersion Advection
Advection
Advection Dispersion Dispersion Dispersion Equation Equation Equation Equation (ADE)
(ADE)
(ADE)
(ADE)
(iii iii) ) Mathematical Mathematical
Mathematical
Mathematical models
models
models
models
∂φ
+ ∇ ⋅ q = qI − qO , q = −krK∇H ∂t
∂φ C
+ ∇ ⋅ J = 0 , J = qC − D∇C
∂t
Governing Governing Governing Governing equation equation equation equation for for for for the the the the distribution distribution distribution distribution of of of of GWA
GWA
GWA
GWA
Governing equation for the distribution of GWA gA(x, t, τ) (5
∂φg ∂φg
= −∇ ⋅ g + ∇ ⋅ ∇ g + r −
∂t ∂τ
A A
A A A
(5-D) (5 D)
D)
q D , τ = age dimension
Rates of production/destruction Rate of ageing =
Rate of ageing: ageing Process by which the age of every fluid parcel tends to increase by a
certain amount of time as time progresses by the same amount of time ≈ advection
process with a unit velocity in the age direction (rate of increase of a particle’s age
per unit time in GW is 1) ⇒ jA = = φ φ v gA = = φ φ gA ∂jA
∂τ
Age pdf at the
points (x, t)

● Fickian Fickian
Fickian
Fickian dispersive dispersive flux flux, flux , , , Macro Macro-dispersion
Macro
Macro
Macro dispersion dispersion: dispersion
dispersion
d
q ⊗ q
J = −D∇ C , D = ( α L − α T ) + α T q I + φDmI
q
● Boussinesq Boussinesq
Boussinesq
Boussinesq approximation
approximation
approximation
approximation
∂φg
∂t
A
(iii iii) Mathematical Mathematical
Mathematical
Mathematical models
models
models
models
MAIN ASSUMPTIONS (Present work)
● Steady Steady-state
Steady
Steady
Steady state state
state
velocity fields: q = q(x)
● Stationary Stationary
Stationary
Stationary reservoirs, reservoirs, ADE ADE
ADE
ADE with time time-independent
time
time
time independent independent
independent
parameters
parameters
“Steady Steady Steady-state
Steady
Steady
state state” state
state
equation equation equation equation for for for for the the the the distribution distribution distribution distribution of of of of GWA GWA, GWA
GWA
GWA,
,
,
= −∇ ⋅ qg + ∇ ⋅ D∇
g + r , r = q δ( t) − q g
● Age Age
Age
Age pdf pdf
pdf
pdf = =
=
= response response to:
to:
A A A A I O A
Unit Unit Unit Unit instantaneous instantaneous instantaneous instantaneous pulse pulse
pulse
pulse input input - δ(t)
, τ = t, g A = g A (x,t):

J = qg − D∇g
A A A
Groundwater age pdf : The forward ADE
∂φg
A = −∇ ⋅ qg ∂t
+ ∇ ⋅ D∇
g + q δ( t) − q g
g ( x,0) = g ( x,
∞ ) = 0 in Ω
A A
Age flux:
Zero-age BC: 3 rd kind (Cauchy)
J ⋅ n = ( q ⋅n) δ(
t)
A
No flow: 3 rd kind (Cauchy)
J ⋅ n =
A
0
on Γ 0
A A I O A
on Γ −

Groundwater lifetime expectancy pdf : The backward ADE
∂φg
∂t
E
= ∇ ⋅ qg + ∇ ⋅ D∇g
− q g
g ( x,0) = g ( x,
∞ ) = 0
E E
Lifetime expectancy flux:
J = −qg − D∇g
E E E
J ⋅ n = −( q ⋅n) δ(
t)
E
D∇g ⋅ n =
E
0
E E I E
Zero LE BC: 3 rd kind (Cauchy)
on Γ +
No flow: 2 nd kind (Neumann)
Sink of lifetime expectancy
on Γ 0
in Ω
● Adjoint Adjoint equation
equation
● “Backward Backward Backward-in
Backward in in-time in time time” time
● Reversed Reversed flow flow-field flow field
● Derived Derived from from standard standard diffusion diffusion theory
theory
(FPE/ (FPE/Kolmogorov
(FPE/ (FPE/ Kolmogorov
Kolmogorov)
Kolmogorov
q I

Total residence time pdf
g T (x,t) = pdf of the sum of A and E :: Convolution Product:
g ( x, t) = g ( x, t) = g ( x,
u, t − u) du
T A+ E
−∞
A, E
t
= g ( x, u) g ( x,
t − u) du
∫
0
A E
∫
+∞

:: Convenient Numerical Implementation ::
Laplace aplace Transform ransform Galerkin alerkin:
(see Sudicky E. A., 1989 – WRR)
Laplace transformed ADE:
Convolution product:
Implicit mplicit Neumann eumann Condition: ondition:
(GW program – Cornaton, 2003-2006)
st
gˆ( s) = L g( t), t, s = ∫ e g( t) dt
sφ gˆ ( x, s) + ∇ ⋅ Jˆ
( x, s) = rˆ ( x,
s)
ˆ ( , ) ⋅ = ± ⋅ , L δ ( ), , = 1
Zero-age boundary condition: J x s n q n { t t s}
gˆ ( x, s) = gˆ ( x, s) gˆ ( x,
s)
T A E
(see Cornaton F., Perrochet P., Diersch H.-J., 2004 – IJNME)
{ } 0
−D∇gˆ( x, s)
⋅n ∈ ℤ
Γ
+ / −
∞ −

∂φg
∂t
:: Backward solutions using FEFLOW ::
Solve for the cdf (using G = = 1 instead of g = = δδ(t)
δ as BC) and differentiate the
obtained breakthrough curves: g(t) = ∂G(t)/∂t.
E
= ∇ ⋅ qg + ∇ ⋅ D∇g
− q g
E E I E
Sink
⎧∇ ⎪ ⋅ qgE = q ⋅∇ gE + gE∇ ⋅q ∂ gE
⎨ ⇒ φ = q ⋅∇ g + ∇ ⋅ D∇g
⎪⎩ ∇ ⋅ q = qI ∂t
C-form
D-form
C-form
!
E E

Forward equation
Backward equation
TEMPORAL MOMENT EQUATIONS
+∞
U n n ∂ gˆ U ( x,
t, s = 0)
µ n ( x, t) = ∫ τ gU ( x,
t, τ) dτ = ( − 1) , U = A or E
n
−∞
∂s
Mean age equation (n = = = 1)
Mean lifetime expectancy
equation (n = = 1)
A
1
〈 A〉 = µ ( x,
t)
(iii iii) Mathematical Mathematical
Mathematical
Mathematical models
models
models
models
A
n
n
∂φµ
∂t
= −∇ ⋅ qµ + ∇ ⋅ D∇µ
+ φ µ − µ
E
∂φµ n
E E E E
= ∇ ⋅ qµ n + ∇ ⋅ D∇µ
n + φ nµ n−1 − qIµ
n
∂t
A A A A
n n n n−1 qO
n
∂φ〈 A〉
= −∇ ⋅ q〈 A〉 + ∇ ⋅ D∇〈
A〉 + φ − qO 〈 A〉
∂t
∂φ〈 E〉
= ∇ ⋅ q〈 E〉 + ∇ ⋅ D∇〈
E〉 + φ − qI 〈 E〉
∂t
Mean age: Mean lifetime expectancy: Mean total residence time:
E
1
〈 E〉 = µ ( x, t)
〈 T〉 = 〈 A〉 + 〈 E〉

:: Solving mean age/lifetime expectancy with FEFLOW ::
(1) Set your transport problem (parameters)
(2) Copy porosity to mass sources
(3) Prescribe C = = 0 at inlet (age) or outlet (lifetime expectancy)

● Mixing at outlet: 〈A〉 is not maximum
● Mixing at inlet: 〈E〉 is not maximum
〈E〉 = = 〈T〉 on the recharge area:
● Mapping of mean transit times to outlets
● 〈T〉 ∼ streamline model
● 〈T〉 : Direct visualization:
Rapid flow zones
Low flow zones
Zones of mixing

(2) Applying the Reservoir Theory to
Groundwater Resource Protection

Residence Residence Residence Residence Time Time Time Time Theory
Theory
Theory
Theory
Chemical engineering
[Danckwerts, 1953, 1958;
Zwietering, 1959]
Fundaments:
Fundaments
- Characterizes the dynamic
process of fluid particles
entering a reservoir,
flowing within it, and
leaving it;
- Based on simple mass
balance formulations;
- Can be seen as a result of
the Gauss divergence
theorem. theorem
Q 0
RESERVOIR RESERVOIR
RESERVOIR
RESERVOIR V0 Flow processes
Internal distribution
of ages
Q 0
Distribution of ages
Reservoir Reservoir Reservoir Reservoir Theory
Theory
Theory
Theory
Environmental sciences
[Eriksson, 1961, 1971]
Hypotheses: Hypotheses
- Open reservoirs with
well-defined
well defined geometry;
- The reservoir limits must
enclose a group of objects
with similar property; property
- Each particle must leave
the reservoir sooner or
later : no nil velocities; velocities
- Steady-flow
Steady flow conditions;
- Stationary state; state
- No mixing?

(i) Reservoir Reservoir
Reservoir
Reservoir Theory Theory
Theory
Theory : : : : : Mass Mass Mass Mass Balance Balance Balance Balance formulations
formulations
formulations
formulations
V(τ) = volumes of water in Ω with an age τ τ or less [L3 Q0 = system steady state discharge rate [L
]
3 V0 = system total pore volume [L
/T]
3 ]
Q(τ) = flow of water leaving Ω with an age τ τ or less [L3 /T]
ψ(τ) = internal frequency distribution of ages [T −1 ]
ϕ(τ) = frequency distribution of ages at outlet [T −1 ]
● Recharge volume entering the
reservoir during the time interval τ:
Q0τ = V ( τ ) + ∫ Q( t) dt
● Difference between total discharge Q 0
and partial discharge Q(τ) = discharge
occuring after a residence time greater
than τ, i.e:
∂V ( τ)
Q0 − Q( τ ) = = V0ψ(
τ)
∂τ
0
τ
or
V
ϕ( τ ) = −
Q
0
0
∂ψ( τ)
∂τ

(i) Reservoir Reservoir Reservoir
Reservoir
Reservoir Theory Theory
Theory
Theory : :
:
: Analogy Analogy Analogy
Analogy
Analogy with with
with
with Human Human
Human
Human Population
Population
Population
Population
∂V ( τ)
Q0 − Q( τ ) = = V0ψ(
τ)
∂τ
V0 = total nb. of people
V(τ) = nb. of individuals with age < τ
Q0 = birth rate
Q(τ) = yearly nb. of deaths among the part
of the population younger than τ
V0ψ(τ) = population age distribution
measured as the nb. of people in
each class of age
Nb. of deaths among people older than τ
= = Nb. of people reaching age τ each year
or
V
ϕ( τ ) = −
Q
0
0
∂ψ( τ)
∂τ

Distribution Distribution of of ages ages (transit transit times times) times ) at at outlet outlet
outlet
pdf: pdf:
pdf:
1
A( t) A d
Q 0 + Γ
ϕ = ∫ J ⋅n Γ
1
E ( t) E d
Q0 − Γ
ϕ = ∫ J ⋅n Γ
cdf:
cdf:
QA( t)
t
A
Q0
0
A
f ( t) = = ∫ ϕ ( τ) dτ
Distribution Distribution of of of lifetime lifetime expectancies
expectancies expectancies (transit transit times times) times ) at at inlet
inlet
pdf: pdf:
pdf:
(ii ii) Reservoir Reservoir
Reservoir
Reservoir Theory Theory
Theory
Theory : :
:
: Characteristic Characteristic
Characteristic
Characteristic Distributions
Distributions
Distributions
Distributions
v 1
v 2
v 3
Γ− Γ+
Γ+
cdf:
cdf:
QE ( t)
t
E
Q0
0
E
f ( t) = = ∫ ϕ ( τ) dτ

cdf:
cdf:
pdf:
pdf:
Internal Internal age age & & Internal Internal lifetime lifetime expectancy
expectancy
expectancy
t
VU ( t)
1 ⎛ ⎞
vU ( t) = = ∫ φ ( x) ⎜∫ gU ( x, u) du ⎟dx
, U = A or E
V0 V0 Ω ⎝ 0 ⎠
∂v
( t)
1
ψ ( t) = = ∫ φ ( x) g ( x, t) dx , U = A or E
∂
U
U
t V0 Ω
U
v 1
v 2
v 3
- Monotonically Monotonically decreasing
decreasing
Γ− Γ+
Γ+
Monotonically
Monotonically
- increasing
increasing
- decreasing decreasing increments
increments

cdf: cdf:
cdf:
pdf:
pdf:
Internal Internal total total transit transit time time
time
t
VT ( t)
1 ⎛ ⎞
vT ( t) = = ∫ φ( x) ⎜∫ gT ( x, u) du ⎟dx
V0 V0 Ω ⎝ 0 ⎠
∂v
( t)
1
ψ ( t) = = ∫ φ(
x) g ( x, t) dx
∂
T
T
t V0 Ω
T
Transit Transit times times modes
modes
(ii ii) Additional Additional Additional Additional Distributions Distributions
Distributions
Distributions :
v 1
v 2
v 3
Γ− Γ+
Γ+

F:
B:
(iii iii) Reservoir Reservoir Reservoir Reservoir Theory Theory Theory Theory and and
and
and ADEs
ADEs
ADEs
ADEs
∂
[ qg A − D∇g A] ⋅n dΓ + φg A dΩ = − [ qg A − D∇g A]
⋅n dΓ
∂t
∫ ∫ ∫
Γ Ω Γ
+ −
= Q 0 ϕ A(t) = V 0 ψ A(t)
∂
− ∫ [ qgE + D∇gE ] ⋅n dΓ + ∫ φg E dΩ = ∫ [ qgE + D∇gE ] ⋅n dΓ
Γ ∂t
− Ω Γ+
= Q0ϕE(t) = V0ψE(t) = Q0 δ(t)
V
with the turnover time
τ 0 =
∂ψ(
t)
ϕ ( t) + τ 0 = δ(
t)
Q
∂ t
⎧ϕ
( t) = ϕ A( t) = ϕE
( t)
with
⎨
⎩ψ
( t) = ψ A( t) = ψE
( t)
MIXING MIXING MIXING MIXING MIXING MIXING MIXING MIXING IS IS IS IS IS IS IS IS ACCOUNTED ACCOUNTED ACCOUNTED ACCOUNTED ACCOUNTED ACCOUNTED ACCOUNTED FOR!!! FOR!!! FOR!!! FOR!!! FOR!!! FOR!!! FOR!!! FOR!!!
= Q 0 δ(t)
0
0

+∞
2
2[ ] ∫ t ( t) dt 2 0 a 0 r
0
µ ϕ = ϕ = τ τ = τ τ
2
(iv iv) Reservoir Reservoir Reservoir Reservoir Theory Theory
Theory
Theory : : : : Characteristic Characteristic Characteristic Characteristic Times
Times
Times
Times
σ [ ϕ ] = τ (2 τ − τ ) = τ ( τ − τ )
Mean Mean Mean Mean transit transit transit transit time time
time
time at at at at outlet outlet outlet
outlet
outlet = turnover turnover turnover turnover turnover time
time
time
time
V
τ = ϕ = − = = τ
+∞ +∞
∫
0
t t ( t) dt ∫ [1 f ( t)] dt 0
0 0 Q0
Mean Mean Mean Mean Mean residence residence residence residence time time
time
time (average (average (average (average internal internal internal internal internal age/lifetime age/lifetime age/lifetime age/lifetime age/lifetime expectancy)
expectancy)
expectancy)
expectancy)
+∞
0 a 0 0 r 0
+∞ 2
τ ⎛
0 σ [ ϕ]
⎞
∫ a t ( t) dt ⎜1 2 ⎟
0 2 ⎜ τ ⎟
0
τ = ψ = +
⎝ ⎠
Mean Mean Mean Mean internal internal internal internal total total total total transit transit transit transit transit time
time
time
time
1
τ = t ψ ( t) dt = φ〈 T〉 dΩ
= 2τ
∫ ∫
r T
a
0 V0
Ω
since
〈 T 〉 =〈 A〉 + 〈 E〉
Transit Transit Transit Transit time time time time pdf pdf pdf pdf variance variance variance
variance
variance
Transit Transit Transit Transit Transit time time time time pdf pdf pdf pdf 3 3
3
3 rd moment moment
moment
moment
moment
3
+∞
3
t
0
t dt 0 2
µ [ ϕ ] = ∫ ϕ ( ) = 3 τ µ [ ψ]

Accuracy Accuracy Accuracy Accuracy of of of of the the the the Reservoir Reservoir Reservoir Reservoir Theory Theory Theory Theory Approach
Approach
Approach
Approach
Mean age representativity ???

(v) Quantification Quantification Quantification Quantification of of of of Internal Internal Internal Internal Groundwater Groundwater Groundwater Groundwater Volumes
Volumes
Volumes
Volumes
V A,T = V 3 + V 4 : GW volume of age ≤ t, but transit time > t
V T = V 1 + V 2 : GW volume of transit time ≤ t
V A,T
V 5 : GW volume of age superior to t V A (t) + V 5 (t) = V 0
E 1 : Vol. of exfiltrated water with total transit time t or less
V A,T (t) + V T (t) = V A (t)
E 2 : Vol. of exfiltrated water having traveled in a time ranging between t and t max
V T

Outlet: Outlet
ANALYSIS OF SYSTEM WATER AGE QUANTITIES
τ
1
2
PΓ ( τ < t < τ ) = ∫ ϕ ( t) dt =− τ ( ψ( τ ) − ψ( τ ))
1 2 0 2 1
τ
E ∆t = volume of exfiltrated water with a residence time ranging between τ 1 and τ 2
during an observation period ∆t of any length = (Q(τ 2 ) − Q(τ 1 )) ∆t
Reservoir: Reservoir
1 2 1 2
α 1+ε1 α 1+ε2 α 2 +ε1 α 2 +ε2
P ( α < α < α and ε < ε < ε ) = ψ( t) dt − ψ(
t) dt
Ω
1 2 1 2
α +ε α +ε
2 1 2 2
V ( α < α < α and ε < ε < ε ) = Q( t) dt − Q( t) dt
Ω
∫ ∫ Useful to address
∫ ∫
α +ε α +ε
1 1 1 2
decontamination
problems

∂φg
∂t
E
= ∇ ⋅ qg + ∇ ⋅ D∇g
− q g
1
ψ ( t) = ∫ φ( x) g ( x,
t) dΩ
i E
V0, i Ω
∂ψi(
t)
ϕ i( t) + τ 0, i = δ(
t)
∂t
E E I E
JE ⋅ n = −( q ⋅n) δ(
t)
on Γi ⋅ = 0 J n on Γ +
E
D∇g ⋅ n = 0
on Γ0 E
(vi vi) Reservoir Reservoir Reservoir Reservoir Theory Theory Theory Theory and and and and Outlet Outlet Outlet Outlet Protection Protection Protection Protection Problems Problems
Problems
Problems
Problems
~ ~ Flow Flow systems systems with with multiple multiple multiple outlets outlets outlets and and sub sub-systems sub systems ~
~
Lifetime Lifetime Lifetime expectancy
expectancy-to
expectancy to to-outlet to outlet Γi pdf pdf: pdf
pdf
pdf :
Travel Travel times times to to to outlet outlet quantification
quantification
Boundary value problem:
RT for drainage basin of outlet Γ i :
t = = tref

Lifetime Lifetime expectancy
expectancy-to
expectancy to to-outlet to outlet Γi cdf cdf: cdf
cdf
cdf :
Outlet Outlet drainage drainage basin basin basin (capture (capture zone) zone) quantification
quantification
p ( x, t) = ∫ g ( x,
u) du ≤1
i E
0
Absolute lifetime expectancy-to
expectancy to-outlet outlet cdf: cdf
+∞
∞
i E
0
t
● Time-dependent probability of exit at Γ i
⇒ Evaluate time-dependent
time dependent capture zones
∂φ pi
= ∇ ⋅ q p + ∇ ⋅ D∇p
− q p
∂t
[ −q p − D∇p ] ⋅ n = −q ⋅n
i i
i i I i
[ −q pi − D∇pi ] ⋅ n = 0 on Γ +
D∇p ⋅ n = 0
on Γ0 i
Boundary value problem:
p ( x) = ∫ g ( x,
u) du ≤1
● Absolute probability of exit at Γ i
on Γ i
⇒ Evaluate steady-state
steady state capture zones
t = = tref Drainage basin pore volume:
∞
V0, i = ∫ φ( x) pi ( x)
dΩ
Ω

Designing Designing Protection Protection Zones
Zones
∫ ∫
Q ( t) = Q −V ψ ( t) = − [ q p ( x, t) + D∇p ( x, t)] ⋅n dΓ + q ( x) p ( x,
t) dΩ
i 0, i 0, i i i i I i
Γ Ω
−
= Q ( t) + Q ( t)
i,limits i,recharge

Low water level
conditions
Seeland porous aquifer (CH)
High water level
conditions
p w (0.9Q 0 ) p w (0.9Q 0 )

(3) Forward and Backward Travel
Times as Vulnerability and Risk
Assessment Criteria

Travel Distance and Travel Time Probabilities
(see Jury & Roth; Neupauer & Wilson)
Initial value
problems
Travel distance:
φ(
x) C ( x,
t)
g x ( x,
t)
=
m
Travel time:
g ( t,
x)
=
t
Q( x) C ( x,
t)
m
Resident Concentration versus Flux Concentration
r
f
3-D D relationship:
relationship
r
f r C
= = −
2 2
C
J ⋅q C
D∇ ⋅q
q q

Forward Forward operator:
operator:
Backward Backward operator:
operator:
Forward Forward Forward equation:
equation:
Backward Backward equation: equation:
equation:
*
FORWARD FORWARD FORWARD FORWARD FORWARD PROBLEM PROBLEM PROBLEM PROBLEM versus versus versus versus BACKWARD BACKWARD BACKWARD BACKWARD PROBLEM PROBLEM
PROBLEM
PROBLEM
PROBLEM
m ( x, t) = δ( x − x ) δ(
t)
T
−3
x ( xT
, ) = [L ]
g t
x
E I −3
E ( xI,
) [L ]
Q0,T
= g t =
Mapping of the Oulet Vulnerability to Source Contamination
I
C( x , t)
M
g ( x , t)
f ∂φR()
L () = + ∇ ⋅q() − ∇ ⋅ D∇
() + φRλ() − qICI + qO
()
∂t
b ∂φR()
L () = − ∇ ⋅q() − ∇ ⋅ D∇
() + φRλ () + qI
()
∂t
f
L ( C( x, t)) = m ( x,
t)
b
L ( g( x,
t )) = 0
(i) Point Point Point Point source source source source problem:
problem:
problem:
problem:
*
: : : : Input Input Input Input mass mass-normalized mass
mass
mass normalized normalized normalized concentration concentration concentration concentration at at at at outlet
outlet
outlet
outlet
: : : : Target Target
Target
Target flowrate flowrate-normalized flowrate
flowrate
flowrate normalized normalized normalized lifetime lifetime lifetime lifetime expectancy expectancy
expectancy
expectancy pdf
pdf
pdf
pdf
Intrinsic vulnerability assessment

(ii ii) Distributed Distributed Distributed Distributed time time-varying time
time
time varying varying varying source source source source problem:
problem:
problem:
problem:
*
−λt
E, λ
E and xI
m ( x, t) = δ( x − x ) m( x,
t)
t ⎛ ⎞
−1
jT ( t) = ∫ δ( x − xI ) ⎜∫ gE , λ ( x, u) m( x, t − u) du ⎟d
x [M ⋅ T ]
∆ ⎝ 0
⎠
g = e g
∈∆
Specific vulnerability assessment
I

- Rainfall intensity on top
- Hydraulic head for a natural outlet
- Pumping-well
3-D D D D heterogeneous heterogeneous heterogeneous heterogeneous example
example
example
example
~1.1 Million nodes
Brick elements
(K-distribution from Renard P.
& de Marsily G., WRR)

Probability of exit at the well and backward particle-tracking
particle tracking solution
Pumping-well

t thresh
t m
C m
µ
Quantifying the Well Vulnerability υ(x)
σ
f δ(
x)
υ ( x) = Cm
( x)
[ −]
t ( x)
m
with C ( x,
t)
=
f m
m
j ( x,
t)
and δ ( x) = [ µ ( x) + nσ( x)] − t ( x)
Q
w
Vulnerability criteria
● Maximum expected concentration C m
● Time to reach C m
● Contamination duration δ
● Time to reach threshold concentration
thresh

Vulnerability Criteria Mapping

Vulnerability Mapping
Multiple realizations – Uncertainty in vulnerability

(4) Application Example
SAFETY SAFETY SAFETY SAFETY (RISK) (RISK) (RISK) (RISK) ANALYSIS ANALYSIS ANALYSIS ANALYSIS OF OF OF OF UNDERGROUND
UNDERGROUND
UNDERGROUND
UNDERGROUND
HIGH HIGH-LEVEL HIGH
HIGH
HIGH LEVEL LEVEL LEVEL RADIOACTIVE RADIOACTIVE RADIOACTIVE RADIOACTIVE WASTE WASTE WASTE WASTE REPOSITORIES
REPOSITORIES
REPOSITORIES
REPOSITORIES

Ontario Ontario Ontario Ontario Power Power Power Power Generation Generation Generation Generation Project
Project
Project
Project
Exploring methods to advance the application of numerical methods for the simulation
and prediction of groundwater flow and mass transport in crystalline Shield settings; settings
Key aspect (1) : developing methods to improve field data traceability within numerical
simulations of heterogeneous and anisotropic flow domains, particularly particularly
those influenced
by fracture zone networks; networks
Key aspect (2) : developing methods to quantify the (human) risk associated to the
elaboration of underground nuclear waste repositories;
repositories
Developing the FRAC3DVS software (Therrien ( Therrien et al., 2001).

● Forward ADE: Evaluation of outlets (biosphere) contamination
● Backward ADE: Optimization of repositories location
t ⎛ ⎞
−1
jT ( t) = ∫ δ( x − xI ) ⎜∫ gE , λ ( x, u) m( x, t − u) du ⎟d
x [M ⋅ T ]
∆ ⎝ 0
⎠
−λt
E, λ
E and xI
g = e g
∈∆
Modelling Procedure
j(t)

Analysis of domain and fracture networks geometry
Regional flow domain
Sub-regional scale Surface lineaments
~84 km 2

Stochastic generation of equally probable discrete fracture networks networks
conditioned by their surface expressions (1)
(Srivastava R. M., 2002)
Generation of fracture zone families Mapping of discrete features in the FE mesh

Stochastic generation of equally probable discrete fracture networks networks
conditioned by their surface expressions (2)
(Srivastava R. M., 2002)
Generation of fracture zone families Mapping of discrete features in the FE mesh

Statistical analysis of fracture zones properties from available
borehole data (Canada - OPG, Finland – Sativa)
- Moving-average type of percentile analysis of K f – z relationship:
because K f decreasing, but still non-zero probability of finding big K f at depth
- Lognormal pdf for fracture zone width.

Flow solution correlated with topographic gradients (1)

Flow solution correlated with topographic gradients (2)

Simulating mean travel times to biosphere:
Selecting potential safe sites (1)

Simulating mean travel times to biosphere:
Selecting potential safe sites (2)

Parameter sensitivity analysis to select relevant parameters
influencing travel times to biosphere
● Fracture zones permeability : major effect on simulated mean travel times
● Change in fracture zones width : significant difference on simulated mean travel times
● Fracture zones porosity has a minor effect

Multiple realizations of contaminant history at the biosphere –
Testing the reliability of selected sites / adding physical (non-linear)
(non linear)
processes to model the fate of contaminants (e.g. brine effects) effects
● Risk index = function of (flux-)concentration at the biosphere
[e.g. see Maxwell et al., 1998; Andricevic et al., 1994]
Global Global responses responses at at the biosphere - 2 2 repository repository locations locations
locations

CONCLUSIONS
√ Variables A, E and T provide complementary information:
E & T pdfs are well-suited for mapping transit times over recharge areas;
T pdf allows direct visualization of the flow dynamics.
√ RT provides:
Refined accuracy to evaluate outlet transit time pdfs;
Fundamental transient information on the system.
√ Backward lifetime expectancy solutions can be post-processed in order to:
Evaluate the origins of the waters flowing out at a discharge zone;
Predict a forward solute transport solution – vulnerability/risk mapping.
√ Presented methodologies can be solved using standard codes like FEFLOW;
√ Open question: question
Extension of the Reservoir Theory to transient velocity fields;

THANK
U
Cornaton, Perrochet 2006. Groundwater Age, life expectancy,
and transit time distributions in advective-dispersive systems:
1. Generalized reservoir theory. Adv. Wat. Res.
Cornaton, Perrochet 2006. Groundwater Age, life expectancy,
and transit time distributions in advective-dispersive systems:
2. Generalized reservoir theory for sub-drainage basins. Adv.
Wat. Res.
Kazemi, Lehr, Perrochet 2006. Groundwater age - John Wiley.