Groundwater modeling and flow simulations

Groundwater modeling:
flow simulations
A. Dassargues
References: Dassargues A., 2018. Hydrogeology: groundwater science and
engineering, 472p. Taylor & Francis CRC press (chapters 4, 12 & 13)
Dassargues A. 2020. Hydrogéologie appliquée: science et ingénierie
des eaux souterraines, 512p. Dunod (chapitres 4, 2 et 13)

Groundwater modeling: flow simulation
► Groundwater flow equations
 REV concept
 Potentiometric/piezometric head
 Porosities, hydraulic conductivty, Darcy’s law, transmissivity
 Steady state equations
 Transient equations
 Extending to partially saturated flow
► Flow Boundary Conditions
 Dirichlet BC’s
 Neumann BC’s
 Cauchy BC’s
► Introduction to solving methods
 Finite Difference method
 Time integrations schemas
 Finite Difference method (FD): practical recommendations
 Finite Element method: summary
 Finite Volume method: summary
► References

Assumptions and REV concept
REV is the volume of geological medium considered as
representative for quantification of the value of the different
properties (by averaged, equivalent values)
 the REV depends on the kind of problem being
studied and the study objectives
 the REV is used for groundwater flow and solute
transport … but also in all other fields where a
quantification is needed for properties of the
geological medium
 large enough to be relevant with regard to the studied problem
(avoid microscopic scale for a real case study)
 small enough for avoiding too smoothed values that will corrupt
our description of the processes
… this REV concept implicitly implies that the
medium is considered as continuous (and porous)
3
(de Marsily 1986, Dagan 1989, Bear & Verruijt 1987)

Balance models, black-box models, transfer
functions, etc.
Assumptions, REV concept and scale
Fluid dynamic and pore studies
Lab tests (from dm to a few m)
Detailed numerical models – Physically
consistent simulations of the reservoir
Homogenisation
Homogenisation
Homogenisation
Micro
Macro
Mega
Giga
Scale
4
(Dassargues & Monjoie 1993, Dassargues 2018 and 2020, Hoffmann et al. 2021)
parameter values can be quite different in function of the considered scale

Hydrostatic
… the potential energy is expressed usually in ‘water head’
or hydraulic head or piezometric head:
g
p
z
g
h
.





5

Water pressure vs piezometric head
… a direct link between hydraulic/piezometric
head h and water pressure p:
g
z
h
p .
)
( 


t
h
g
t
p





.

for groundwater flow problems, the main variable is the
piezometric head or the water pressure
piezometric heads can be compared only if groundwater has
everywhere the same temperature and the same salt content
if it is not the case, … density will vary … and to a same water
pressure correspond different piezometric heads (of groundwater with
different salt content)
corrections in function of the salinity must be done 6
Rmq: if density effect (salwater) is considered
work with pressure or with ‘equivalent
freshwater piezometric heads’
(Carabin and Dassargues 1999)

Porosity
… two components in the total porosity:
cinematic porosity
retention capacity’ or ‘specific retention’)
c
n
r
S
n
S
n r
c 

t
m
c
V
V
n 
t
im
r
V
V
S 
with
the cinematic porosity is difficult to be measured in practice
(… after which duration ? … at which pressure ? … )
drainage porosity
= effective porosity
corresponding to the drainable water by gravity
(also called mobile water or free water)
…. also called = “specific yield”
c
e n
n 
y
S
7

8
Porosity and granular analysis
Dassargues, 2018
(Eckis 1934, Castany 1963)

… experimental law
quantity of water per time unit through a porous medium:
L
h
A
K
Q

 .
.
K permeability coefficient, hydraulic conductivity,
water permeability (by abuse of language:
permeability) of the porous medium (m/s)
… specific flux or flow rate
(specific discharge):
A
Q
q 
in m3/(m2.s) … in m/s 9
Hydraulic conductivity and Darcy’s law

This specific discharge is often called ‘Darcy’s velocity’ … it is only a
flow rate Q divided by a surface A
… to obtain a mean (averaged/equivalent on the REV)
groundwater velocity :
the actual groundwater flow section is : e
n
A.
L
h
n
K
n
q
v
e
e
e


 . m/s ‘ advection velocity’
(effective velocity)
this surface is not the groundwater flow
section
10
Hydraulic conductivity and Darcy’s law

the ‘mobile water porosity’ to be considered for groundwater flow is
typically higher than the ‘mobile water porosity’ acting in solute
transport processes (Payne et al. 2008, Hadley and Newell, 2014)
useful
porosity for
solute
transport is
‘effective
transport
porosity’
< drainage
porosity
11
…about mobile water porosities
(Dassargues 2018 and 2020)

 fluid properties:
viscosity
density
 porous medium properties:
granular proportions,
grains shapes,
pore distribution and
shapes
intergranular porosity
depends on:
intrinsic permeability or permeability (m2)
K

 g
k
K
.
.

volume mass of the fluid (kg/m3)
gravity acceleration (m/s2)
dynamic viscosity (kg/(m.s), N.s/m2 or Pa.s
12
Hydraulic conductivity and intrinsic permeability

hydraulic conductivity and intrinsic permeability described by
tensors:
and
elevation of the considered point
(with regards to the reference datum)
13
Generalisation of the Darcy’s law in 3D
𝑲 =
𝐾𝑥𝑥 𝐾𝑥𝑦 𝐾𝑥𝑧
𝐾𝑦𝑥 𝐾𝑦𝑦 𝐾𝑦𝑧
𝐾𝑧𝑥 𝐾𝑧𝑦 𝐾𝑧𝑧
𝑲 = 𝒌𝜌𝑔 𝜇
𝒒 = −𝑲 ∙ 𝛻ℎ = −
𝒌𝜌𝑔
𝜇
∙ 𝛻ℎ = −
𝒌
𝜇
∙ 𝛻𝑝 + 𝜌𝑔𝛻𝑧
𝒈𝒓𝒂𝒅 ℎ = 𝛻ℎ =
𝜕ℎ
𝜕𝑥
,
𝜕ℎ
𝜕𝑦
,
𝜕ℎ
𝜕𝑧

Transmissivity
mean value of the hydraulic
conductivity on the vertical of the
concerned point
transmissivity
(m2/s) in a point
thickness of the confined
aquifer at the concerned
point
… for a confined aquifer
14
𝑇 𝑥, 𝑦 =
0
𝑏 𝑥,𝑦
𝐾 𝑥, 𝑦, 𝑧 𝑑𝑧 = 𝐾𝑎𝑣𝑔 𝑥, 𝑦 𝑏 𝑥, 𝑦
(Dassargues, 2018 & 2020)

… for an unconfined aquifer


)
,
(
0
).
,
(
)
,
( dz
y
x
K
y
x
T
the saturated thickness of the unconfined aquifer
at the point (of horizontal coordinates x and y )
depends on the
piezometric head!
15
Transmissivity
(Dupuit 1863, Delleur 1999)

16
Groundwater flow equation in steady state
terms are kg/(m3s )
in indicial notation
if density is assumed constant and the principal anisotropy
directions of the K tensor are known and aligned with the
selected coordinate system – terms are in s-1
𝛻 ∙ 𝜌 𝑲 ∙ 𝛻ℎ + 𝜌𝑞′
= 0
𝜕
𝜕𝑥𝑖
𝜌𝐾𝑖𝑗
𝜕ℎ
𝜕𝑥𝑗
+ 𝜌𝑞′𝑖 = 0
𝜕
𝜕𝑥
𝐾𝑥𝑥
𝜕ℎ
𝜕𝑥
+
𝜕
𝜕𝑦
𝐾𝑦𝑦
𝜕ℎ
𝜕𝑦
+
𝜕
𝜕𝑧
𝐾𝑧𝑧
𝜕ℎ
𝜕𝑧
+ 𝑞′ = 0
𝜕
𝜕𝑥
𝐾𝑥𝑥
𝜕ℎ
𝜕𝑥
+
𝜕
𝜕𝑧
𝐾𝑧𝑧
𝜕ℎ
𝜕𝑧
+ 𝑞′
= 0
if 2D vertical flow, terms are in s-1
𝜕
𝜕𝑥
𝑇𝑥𝑥
𝜕ℎ
𝜕𝑥
+
𝜕
𝜕𝑦
𝑇𝑦𝑦
𝜕ℎ
𝜕𝑦
+ 𝑞′′ = 0
if 2D horizontal flow, terms are in m/s

17
Equation in transient conditions
… in transient flow,
storage variation in function of the time:
t
n


 )
(
 
t
h
S
t
h
n
n
g
t
n
s
w
s















 2
)
(
specific storage coefficient (m-1)
Volume (REV)
compressibility
of the porous
medium
(Pa-1)
Solid grain
compressibility (Pa-1)
Water
compressibility
(Pa-1)

 g
Ss 

… often, the influence of the water compressibility and the solid grain
compressibility can be neglected with regards to the volume
compressibility of the porous medium (as a whole)

 g
Ss 
… this link between the volume compressibility and the
specific storage coefficient is showing clearly the direct
coupling between saturated transient groundwater flow
and geomechanical behaviour in compressible porous
media
the volume compressibility is dependent on
effective stress variation
the effective preconsolidation stress of the porous
medium
18
Specific storage coefficient
Rmq: Ss is actually variable as it depends on compressibility that is
dependent on the effective stress state that is dependent on the water
pressure,… (Dassargues et al. 1991 and 1993, Dassargues 1995, 1997, 1998)

19
Storage coefficient = water volume (m3) stored or drained
per aquifer surface unit (m2) for a unit variation of
piezometric head (m)
… vertical integration


)
,
(
0
).
,
(
)
,
(
y
x
e
s dz
y
x
S
y
x
S confined aquifer
e
S
S s.

dz
S
n
S
h
z
s
e .
1


 unconfined aquifer
the most important part of the storage is due to
saturation/drainage of the porous medium
Equation in transient conditions
(Dassargues 1995b, 2018, 2020)

reference datum
= bottom of the
aquifer layer
h
S
n
S s
e .


e
n
S 
20
Storage coefficient

21
confined aquifer
(horizontal flow)
2D groundwater flow equations in
transient conditions
𝛻 ∙ 𝑻 ∙ 𝛻ℎ + 𝑞′′ = 𝑆
𝜕ℎ
𝜕𝑡
𝜕
𝜕𝑥𝑖
𝑇𝑖𝑗
𝜕ℎ
𝜕𝑥𝑗
+ 𝑞′′𝑖 = 𝑆
𝜕ℎ
𝜕𝑡
𝜕
𝜕𝑥
𝑇𝑥𝑥
𝜕ℎ
𝜕𝑥
+
𝜕
𝜕𝑦
𝑇𝑦𝑦
𝜕ℎ
𝜕𝑦
+ 𝑞′′ = 𝑆
𝜕ℎ
𝜕𝑡
unconfined aquifer
principal anisotropy directions aligned
with the selected coordinate system
terms are in m/s
𝛻 ∙ 𝑻(ℎ ∙ 𝛻ℎ + 𝑞′′ = 𝑛𝑒
𝜕ℎ
𝜕𝑡
= 𝑆𝑦
𝜕ℎ
𝜕𝑡
𝜕
𝜕𝑥𝑖
𝑇𝑖𝑗
𝜕ℎ
𝜕𝑥𝑗
+ 𝑞′′𝑖 = 𝑛𝑒
𝜕ℎ
𝜕𝑡
= 𝑆𝑦
𝜕ℎ
𝜕𝑡
𝜕
𝜕𝑥
𝑇𝑥𝑥
𝜕ℎ
𝜕𝑥
+
𝜕
𝜕𝑦
𝑇𝑦𝑦
𝜕ℎ
𝜕𝑦
+ 𝑞′′ = 𝑆
𝜕ℎ
𝜕𝑡
principal anisotropy directions
aligned with the selected coordinate
system
terms are in m/s

22
vertical integration of the 3D equation on the thickness of
the confined aquifer
L
R
t
h
S
y
h
T
y
x
h
T
x
yy
xx 



























where T : transmissivity [L2 T-1], K  thickness
S : storage coefficient [-], Ss  thickness
R : recharge rate [L3 L-2 T-1] = [L T-1]
L : leakance [L T-1]
Thickness (b) Groundwater flow
Recharge (R)
Particular case: 2D horizontal confined flow

23
Multi-aquifer system
Leakance (vertical flow) through the aquitard
Darcy’s law
'
'
)
(
b
h
h
K
L s
z 


où K'z : vertical hydraulic conductivity of the aquitard
b' : aquitard thickness
hs : surperfical aquifer piezometric head
h : main aquifer piezometric head (main variable)
Aquifer
Aquitard
Aquifer
b', K'z
hs
h
Groundwater flow
Particular case: 2D horizontal confined flow
… leakance calculation for
a quasi 3D approach

24
with water pressure head as main variable
Groundwater flow equations including the
partially saturated zone
𝛻 ∙ 𝜌 𝑲 ℎ𝑝 ∙ 𝛻ℎ𝑝 + 𝑲 ℎ𝑝 ∙ 𝛻𝑧 + 𝜌𝑞′ = 𝜌𝐶 ℎ𝑝
𝜕ℎ𝑝
𝜕𝑡
terms are kg/(m3s )
(Celia et al. 1990,
Dassargues 1997)
𝛻 ∙ 𝜌 𝑲 𝜃 ∙ 𝛻ℎ𝑝 + 𝑲 𝜃 ∙ 𝛻𝑧 + 𝜌𝑞′ = 𝜌
𝜕𝜃
𝜕𝑡
in a mixed way as a function of the water content and the
pressure head
(Richards 1931)
needs relations between
- 𝜃 and hp
- 𝜃 and K
…
… van Genuchten relations and others
𝑝𝑐 𝜌𝑔 = −ℎ𝑝 = −𝑝 𝜌𝑔 = 𝑧 − ℎ

25
Flow Boundary Conditions
 Dirichlet conditions: prescribed piezometric head
 Neumann conditions: prescribed flux
 Cauchy or mixed conditions: flux depending on
piezometric head
Practical examples where groundwater flow BCs are discussed on various
practical cases are available from the experience of the author and researchers
from his team in the following references:
Dassargues et al. 1988, Dassargues 1991, Stefanescu & Dassargues 1996, Dassargues 1997, Carabin & Dassargues 2000, Brouyère et al. 2004,
Peeters et al. 2004, Rojas & Dassargues 2007, Rocha et al. 2007, Rojas et al. 2008, Jusseret et al. 2009, Goderniaux et al. 2009, Brouyère et al.
2009, Rojas et al. 2010, Wildemeersch et al. 2010, Orban et al. 2010, Goderniaux et al. 2011, Wildemeersch et al. 2014, César et al. 2014, Pujades et
al. 2016)

26
Prescribed piezometric head
(Dirichlet condition)
Prescribed piezometric head on the concerned boundary:
'
f can vary in space and time
(one value per node and per time step)
a flux will be computed per concerned node
Flow BC’s
ℎ 𝑥, 𝑦, 𝑧, 𝑡 = 𝑓′(𝑥, 𝑦, 𝑧, 𝑡

27
Prescribed flux (Neumann condition)
The first derivative of the piezometric head is
prescribed on the concerned boundary:
piezometric gradient normal to the concerned boundary,
its value can vary in space and time
(one value per concerned node and per time step)
Applying the Darcy’s law, it is a way of prescribing the water
flux through the boundary:
'
'
f
: precribed flux through the boundary (m/s)
'
'
q
0
'
' 
f
particular case:
Flow BC’s
𝛻ℎ ∙ 𝒏 =
𝜕ℎ
𝜕𝑛
𝑥, 𝑦, 𝑧, 𝑡 = 𝑓′′ 𝑥, 𝑦, 𝑧, 𝑡
𝐾
𝜕ℎ
𝜕𝑛
𝑥, 𝑦, 𝑧, 𝑡 = 𝑞′′(𝑥, 𝑦, 𝑧, 𝑡

28
Flow BC’s
(Dassargues, 2018)

29
Flow BC’s
(Dassargues, 2018)

30
A combination (linear relation) of the piezometric head
and its first derivative is prescribed on the boundary:
Flux depending on the piezometric head
(mixed condition or Cauchy condition)
)
,
,
,
(
'
'
'
)
,
,
,
(
.
)
,
,
,
(
. t
z
y
x
f
t
z
y
x
h
b
t
z
y
x
n
h
a 



'
'
'
f can vary in space and in time
(one value per concerned node and per time step)
 interactions between surface water bodies and
groundwater
 interactions between different aquifers
Flow BC’s

31
Flow BC’s
𝑞′′
= −𝐾
𝜕ℎ
𝜕𝑛
=
𝐾′
𝑏′
(ℎ𝑟 − ℎ
𝑞′′
= −𝐾
𝜕ℎ
𝜕𝑛
=
𝐾′
𝑏′ (ℎ𝑠 − ℎ
conductance
conductance

32
Flow BC’s
𝑞′′ = −𝐾
𝜕ℎ
𝜕𝑛
=
𝐾′
𝑏′
(ℎ𝑏 − ℎ
𝑞′′
= −𝐾
𝜕ℎ
𝜕𝑛
=
𝐾′
𝑑′
(ℎ𝑟 − ℎ
conductance
conductance
prescribing an ‘external head’ (i.e. not on the true boundary but outside the
modelled zone) so that a groundwater flux across the boundary is computed
from the difference between this ‘external head’ and the piezometric head on
the model boundary using a given conductance

33
Flow BC’s
conductance
𝑞𝐸𝑣𝑇 =
𝑅𝐸𝑣𝑇
𝑑𝑒𝑥𝑡
(ℎ 𝑥, 𝑦, 𝑧, 𝑡 − ℎ𝑐𝑟𝑖𝑡 𝑥, 𝑦, 𝑧, 𝑡
represent an evapotranspiration flux leaving the model but dependent on the ‘depth
to water’ (i.e. the land surface elevation minus piezometric head). An extinction
depth 𝑑𝑒𝑥𝑡 corresponding to a critical head ℎ𝑐𝑟𝑖𝑡 can be defined so that EvT occurs
only if the water table is higher
(Anderson et al. 2015)
in arid and semi-arid zones

34
Simple case…
 steady state
 no infiltration, no sink/source term
 homogeneity of the medium
 2D problem
0



t
h
0

I 0

q
Cst
T 
0
2
2
2
2






y
h
x
h
GW flow equation
in steady state
Assumptions:
Solving a gw flow problem with a Finite
Difference schema

35
Definition of a partial derivative of a function
of the variable :
)
(x
h
x














 x
x
h
x
x
h
x
h
x
)
(
)
(
lim
0
spatial discretisation with a grid
the nodes are the central points of rectangular cells
(‘Block Centered Finite Difference method’ = BCFD)
the cells are homogeneous ... the continuous variation
of the variable is replaced by a discrete variable defined
at the central points of the cells
the approximation of the differential equation is better as
the cells are small
Simple case…
Solving a gw flow problem with a FD schema

36
the nodes are numbered sequentially,
index and piezometric head values
are attributed
j columns
i lines
j
i, ij
h
Simple case…

37
Taylor series for a continuous function )
(x
h
n
n
n
x
x
h
n
x
x
x
h
x
x
x
h
x
x
x
h
x
x
h
x
x
h




















)
(
.
!
)
(
...
)
(
.
!
3
)
(
)
(
.
!
2
)
(
)
(
.
)
(
)
( 3
3
3
2
2
2
2
2
2
1
1
1 .
2
)
(
).
(
x
h
x
x
x
h
x
x
h
h i
i
i
i
ij
j
i








 


2
2
2
1
1
,
1 .
2
)
(
).
(
x
h
x
x
x
h
x
x
h
h i
i
i
i
ij
j
i








 


2
2
1
1
1
2
)
(
)
( x
h
x
x
x
h
x
x
h
h i
i
i
i
ij
j
i








 


2
2
1
1
1
1
1
1
).
(
2
1
)
(
)
(
)
(
)
(
x
h
x
x
x
x
h
h
x
x
h
h
i
i
i
i
ij
j
i
i
i
ij
j
i















… terms of the 3rd
order and more
are neglected
… for hij and the x direction:
2
2
1
1
1
2
)
(
)
( x
h
x
x
x
h
x
x
h
h i
i
i
i
ij
j
i








 


and
… for x > 0
… for x < 0
Simple case…

38
if
































 )
(
.
)
(
1
)
(
1
)
(
.
)
(
2
1
1
1
1
1
1
1
1
2
2
i
i
j
i
ij
i
i
i
i
i
i
j
i
i
i x
x
h
h
x
x
x
x
x
x
h
x
x
x
h
)
(
)
( 1
1 i
i
i
i x
x
x
x
x 



 

2
1
1
2
2
)
(
.
2
x
h
h
h
x
h j
i
ij
j
i





 

2
1
1
2
2
)
(
.
2
y
h
h
h
y
h j
i
ij
j
i





 

Cst
m
y
x 





if
  0
4
.
)
(
. 1
1
1
1
2
2
2
2
2























 ij
j
i
j
i
j
i
j
i h
h
h
h
h
m
T
y
h
x
h
T
)
.(
4
1
1
1
1
1 


 


 j
i
j
i
j
i
j
i
ij h
h
h
h
h
Simple case…

39
Introduction to solving methods: FD
𝜕ℎ
𝜕𝑥
≈
ℎ 𝑥 + ∆𝑥 − ℎ(𝑥
∆𝑥
𝜕ℎ
𝜕𝑥
≈
ℎ 𝑥 + ∆𝑥 − ℎ(𝑥 − ∆𝑥
2∆𝑥
𝜕2
ℎ
𝜕𝑥2
≈
ℎ𝑖+1𝑗 − 2ℎ𝑖𝑗 + ℎ𝑖−1𝑗
∆𝑥 2
1D spatial approximation of the gradient
by a finite difference:
Forward FD
Central FD
In 2D, with a 2nd order accurate FD:
𝜕2
ℎ
𝜕𝑥2
+
𝜕2
ℎ
𝜕𝑦2
≈
ℎ𝑖+1𝑗 + ℎ𝑖−1𝑗 + ℎ𝑖𝑗+1 + ℎ𝑖𝑗−1 − 4ℎ𝑖𝑗
∆𝑚 2
= 0

40
C
A
A
y
A
B
x
BB
y
AB
x
Q
dx
y
h
T
dy
x
h
T
dx
y
h
T
dy
x
h
T
















'
'
'
'
C
A
A
C
S
C
S
ySC
A
B
C
W
C
W
xWC
B
B
C
N
C
N
yNC
A
B
C
E
C
E
xEC
Q
x
x
y
y
h
h
T
y
y
x
x
h
h
T
x
x
y
y
h
h
T
y
y
x
x
h
h
T
















































)
(
)
(
)
(
)
(
'
'
'
'
b
y
a
x 


 et
A
C
A’
B’ B
E
N
S
W
… if rectangular cells :
C
C
S
ySC
C
W
xWC
C
N
yNC
C
E
xEC Q
h
h
b
a
T
h
h
a
b
T
h
h
b
a
T
h
h
a
b
T 






 )
(
)
(
)
(
)
(
Generalisation
and

41
A
C
A’
B’ B
E
N
S
W
C
C
S
ySC
C
W
xWC
C
N
yNC
C
E
xEC Q
h
h
b
a
T
h
h
a
b
T
h
h
b
a
T
h
h
a
b
T 






 )
(
)
(
)
(
)
(
b
a 
… if
C
C
S
ySC
C
W
xWC
C
N
yNC
C
E
xEC Q
h
h
T
h
h
T
h
h
T
h
h
T 






 )
(
)
(
)
(
)
(
a ratio of maximum 1/10 for dimensions of the
rectangular cells (for good computation
conditions)
Generalisation

42
‘averaged’ or ‘equivalent’ values between cells
… on the basis of the continuity principle
xC
xE
xC
xE
xEC
T
T
T
T
T


.
.
2
N
W
C E
S
TEC
C
E
C
E
EC
T
T
T
T
T


.
.
2
Equivalent values for parameters

43
… for more complex mesh (here nested mesh), the water flux
on the boundary :
I
A
B
J
K
)
.(
)
.
.
.
(
.
.
.
4
).
.
( I
K
I
J
I
K
K
J
J
I
K
J
I
AB
x h
h
h
h
T
T
T
T
T
T
T
T
T
dy
x
h
T 









44
… for Control Volume
Finite Elements:
same principle
I
A
B
J
M
)
.(
)
.
.
(
.
.
.
.
. I
J
I
J
J
I
AB
y
x h
h
T
MJ
T
IM
T
T
AB
dS
n
y
h
n
x
h
T 

















45
Introduction to solving methods: BCFD
𝑇𝑒𝑞𝑖+
=
2𝑇𝑖+1𝑗𝑇𝑖𝑗
𝑇𝑖𝑗 + 𝑇𝑖+1𝑗

46
Introduction to solving methods: time
integration scheme
𝜕ℎ
𝜕𝑡
=
ℎ 𝑡 + ∆𝑡 − ℎ(𝑡
∆𝑡
ℎ𝑖𝑗 𝑡 + ∆𝑡 = ℎ𝑖𝑗 𝑡 +
𝑄𝑖𝑗∆𝑡
𝑆
+
𝑇∆𝑡
∆𝑚 2𝑆
ℎ𝑖+1𝑗(𝑡 + ℎ𝑖−1𝑗(𝑡 + ℎ𝑖𝑗+1(𝑡 + ℎ𝑖𝑗−1(𝑡 − 4ℎ𝑖𝑗(𝑡
𝑇
∆𝑚 2
ℎ𝑖+1𝑗 + ℎ𝑖−1𝑗 + ℎ𝑖𝑗+1 + ℎ𝑖𝑗−1 − 4ℎ𝑖𝑗 + 𝑄𝑖𝑗 = 𝑆
ℎ𝑖𝑗 𝑡 + ∆𝑡 − ℎ𝑖𝑗(𝑡
∆𝑡
𝑇
𝜕2
ℎ
𝜕𝑥2
+
𝜕2
ℎ
𝜕𝑦2
≈ 𝑇
ℎ𝑖+1𝑗 + ℎ𝑖−1𝑗 + ℎ𝑖𝑗+1 + ℎ𝑖𝑗−1 − 4ℎ𝑖𝑗
∆𝑚 2
= S
𝜕ℎ
𝜕𝑡
Explicit
at what time do we consider the piezometric head values?
Cst
T 
Cst
m
y
x 





 physically: not so accurate
 numerically: stability problem when the time step becomes larger
 respect a stability criterion

Explicit method
Example:
- squared island
- initial value h = 10 m
- BC’s : h = 10 m
- infiltration: 0.002 m/day
- S = 0.4 ; T = 100 m2/day
- t =10 days m =50m
 
 
)
(
4
)
(
)
(
)
(
)
(
.
.
.
.
)
(
)
( 1
1
1
1
2
t
h
t
h
t
h
t
h
t
h
S
m
t
T
S
t
I
t
h
t
t
h ij
j
i
j
i
j
i
j
i
ij
ij 










 



10 10 10 10 10
10 10 10 10 10
10 10 10 10 10
10 10 10 10 10
10 10 10 10 10
05
.
0
.


S
t
I
 
25
.
0
.
.
2



S
m
t
T
… computation:
- 1st time step;
- 2nd time step;
- 3rd time step;
- …
10.05 10.05 10.05
10.05 10.05 10.05
10.05 10.05 10.05
10.075 10.09 10.075
10.075 10.09 10.075
10.09 10.10 10.09
10.10 10.12 10.10
10.10 10.12 10.10
10.12 10.14 10.12
… no problem

Explicit method
… now with a t = 40 days
 
 
)
(
4
)
(
)
(
)
(
)
(
.
.
.
.
)
(
)
( 1
1
1
1
2
t
h
t
h
t
h
t
h
t
h
S
m
t
T
S
t
I
t
h
t
t
h ij
j
i
j
i
j
i
j
i
ij
ij 










 



10 10 10 10 10
10 10 10 10 10
10 10 10 10 10
10 10 10 10 10
10 10 10 10 10
2
.
0
.


S
t
I
 
1
.
.
2



S
m
t
T
… computation:
- 1st time step;
- 2nd time step;
- 3rd time step;
- …
10.2 10.2 10.2
10.2 10.2 10.2
10.2 10.2 10.2
10.0 10.2 10.0
10.0 10.2 10.0
10.2 10.4 10.2
10.6 10.0 10.6
10.6 10.0 10.6
10.0 9.8 10.0
… numerically not stable
time

Explicit method: stability criterion (example)
… worst case
 
 
)
(
4
)
(
)
(
)
(
)
(
.
.
.
.
)
(
)
( 1
1
1
1
2
t
h
t
h
t
h
t
h
t
h
S
m
t
T
S
t
I
t
h
t
t
h ij
j
i
j
i
j
i
j
i
ij
ij 










 



… for obtaining the stability :
10 - 
10 + 
10 + 
10 + 
10 +  )
8
(
0
)
10
(
)
( 

 




 t
t
hij
0
.


S
t
I
 




S
m
t
T
.
.
2
)
10
(
)
( 


t
hij

 )
1
8
(
10
)
( 



 t
t
hij


 
 )
1
8
(
4
/
1


 
4
/
1
.
.
2





S
m
t
T

the stability of the computations depends on the size
of the time step with regards to the size of the grid cells
and of the parameters values
stability criterion :
 physical propagation of rounding errors
 numerical errors
 long CPU time
  4
1
.
.
2



m
S
t
T
2
1
)
(
)
(
. 2
2














y
t
x
t
S
T
y
x 


Additional drawbacks:
Explicit method: stability criterion

51
… at the time t
t 

implicit equation
Time integration scheme
Implicit
ℎ𝑖𝑗 𝑡 + ∆𝑡 1 + 4𝛼 = ℎ𝑖𝑗 𝑡 +
𝑄𝑖𝑗∆𝑡
𝑆
+
𝑇∆𝑡
∆𝑚 2𝑆
ℎ𝑖+1𝑗(𝑡 + ∆𝑡 + ℎ𝑖−1𝑗(𝑡 + ∆𝑡 + ℎ𝑖𝑗+1(𝑡 + ∆𝑡 + ℎ𝑖𝑗−1(𝑡 + ∆𝑡
 physically: not so accurate (error increases with time step)
 numerically: unconditional stability
 mathematically: more complex/heavy
the unknown cannot be deduced from one equation
you need the whole system to be solved
(Bear and Cheng 2010)

Implicit method
 
S
t
I
t
h
t
t
h ij
ij






.
)
(
.
4
1
).
( 
 
)
(
)
(
)
(
)
(
. 1
1
1
1 t
t
h
t
t
h
t
t
h
t
t
h j
i
j
i
j
i
j
i 










 




10 10 10 10 10
10 10 10 10 10
10 10 10 10 10
10 10 10 10 10
10 10 10 10 10
… even with a t = 40 days
2
.
0
.


S
t
I
 
1
.
.
2



S
m
t
T
… computation:
- 1st time step;
- 2nd time step;
- …
10.125 10.135 10.125
10.125 10.135 10.125
10.135 10.158 10.135
10.2 10.2 10.2
10.2 10.2 10.2
10.2 10.2 10.2
… numerical stability

Implicit method: stability can be proven
 
S
t
I
t
h
t
t
h ij
ij






.
)
(
.
4
1
).
( 
 
)
(
)
(
)
(
)
(
. 1
1
1
1 t
t
h
t
t
h
t
t
h
t
t
h j
i
j
i
j
i
j
i 










 




for obtaining stability :
… the worst case
10 - 
10 + 
10 + 
10 + 
10 + 
)
10
(
4
0
)
10
(
)
4
1
)(
( 


 






 t
t
hij
0
.


S
t
I
 




S
m
t
T
.
.
2
)
10
(
)
( 


t
hij
)
4
1
(
)
10
(
4
)
10
(
)
(










 t
t
hij




 10
)
( t
t
hij











10
)
4
1
(
)
10
(
4
)
10
(





 4
40
10
4
40
10 






…always the case

54
Crank-Nicholson method
 physically: more accurate
 numerically: implicit procedure, unconditional stability
… at the time 2
t
t 

Galerkin method
… at the time 3
2 t
t 

 physically: most accurate
 numerically: implicit procedure, unconditional stability

55
𝑇
∆𝑚 2 ℎ𝑖+1𝑗 + ℎ𝑖−1𝑗 + ℎ𝑖𝑗+1 + ℎ𝑖𝑗−1 − 4ℎ𝑖𝑗 + 𝑄𝑖𝑗 = 𝑆
ℎ𝑖𝑗 𝑡 + ∆𝑡 − ℎ𝑖𝑗(𝑡
∆𝑡
𝑇
∆𝑚 2
(1 − 𝜃 ℎ𝑖+1𝑗(𝑡 + ℎ𝑖−1𝑗(𝑡 + ℎ𝑖𝑗+1(𝑡 + ℎ𝑖𝑗−1(𝑡 − 4ℎ𝑖𝑗(𝑡
+
𝑇
∆𝑚 2
𝜃 ℎ𝑖+1𝑗(𝑡 + ∆𝑡 + ℎ𝑖−1𝑗(𝑡 + ∆𝑡 + ℎ𝑖𝑗+1(𝑡 + ∆𝑡 + ℎ𝑖𝑗−1(𝑡 + ∆𝑡 − 4ℎ𝑖𝑗(𝑡 + ∆𝑡
0


1


2
/
1


3
/
2


Full explicit time integration
Full implicit time integration
Crank-Nicholson implicit
Galerkin implicit
stability criterion only for explicit schemes 2
/
1


time integration schemas used in all numerical techniques

56
Introduction to solving methods: FD practical
recommendations
 an initial field of values for the main unknown variable (piezometric head)
needed for initiating the iterative solving
 accuracy increases with the number of cells but portability (i.e. computing
efficiency) decreases
 use smaller cells where a steep gradient of the main variable is expected
 spatial discretization: nodes located at pumping wells and observation
piezometers
 avoid distances between nodes greater than 1.5 the previous one
 avoid ratios greater than 1/10 for the cell dimensions (bad numerical
conditions for solving the system of equations)
 boundaries with a prescribed head should correspond to nodes (central
points of the cells, if BCFD)
 boundaries with a prescribed flux should correspond to sides of the cells
(where the flux condition is calculated) if BCFD
 …

57
Introduction to solving methods: Finite Elements
- discrete elements, unstructured
FE mesh
- better for irregular boundaries,
spatial variations, and exact
locations for stress-factors and
observation measurements
- optimized mesh generation to
reduce the needed memory
space
(refs among others: Narasimhan et al. 1978, Huyakorn and Pinder 1983, Bear and
Verruijt 1987, Wang and Anderson 1982, Fitts 2002, Rausch et al. 2005, Bear and
Cheng 2010, Anderson et al. 2015, Diersch 2014, Pinder and Celia 2006, Dassargues
2018 and 2020)

58
Introduction to solving methods: FE
- the continuous field of the variable (i.e.
piezometric head) approximated typically
by interpolation functions (here also
referred to as basis functions)
- piezometric field described in each finite
element by a plane
- the discrete unknowns are the nodal
values
- an integral approach expressing the weak
formulation (i.e. a variational form
integrating the governing partial differential
equation of the process with its BCs and
initial conditions) for obtaining a global
continuum balance statement
- two ways:
(1) minimum of a natural variational
functional (when it exists)
(2) method of weighted residuals
(applicable to all types of partial
differential equations)

59
Introduction to solving methods: FV
- common features with FD and FE
- FD for unstructured grids
- if triangles: similarities with triangle FE
- as for FE, FV approximates the main
variable using basis functions in the
triangular element
- Finite Volume refers to the volume
surrounding each node point in a mesh
with nodal basis function = 1 only at the
considered node and 0 at all others
- conservation law is satisfied locally for a
given control volume with respect to its
neighboring volumes (similar to FD not to
FE)
- balance relies on evaluation of surface
integrals on the boundaries (i.e. the
conservation must be satisfied across the
boundaries of the adjoining control
volumes)
(refs among others: Patankar 1980, Baliga and Patankar 1983, Chung 2002, Diersch
2014, Narasimhan and Witherspoon 1976, Rausch et al. 2005, Fletcher 1988,
Idelsohn and Onate 1994, Forsyth et al. 1995, Therrien and Sudicky 1996, Pinder
and Celia 2006, Therrien et al. 2010)

60
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