1 NN31545.1077 X «OTA 1077 augustus 1978 Instituut voorCultuurtechniek en Waterhuishouding Wageningen [ A NUMERICAL MODEL FOR NON-STATIONARY SATURATED GROUNDWATER FLOW IN A MULTI-LAYERED SYSTEM ir. P.J.T. van Bakel S£ÎÎ r*"" &*: I Nota's van het Instituut zijn in principe interne communicatiemidde- len, dus geenofficiële publikaties. Hun inhoud varieertsterk en kan zowel betrekking hebben op een een- voudige weergave van cijferreeksen, als op een concluderende discussie van onderzoeksresultaten. In de meeste gevallen zullen de conclusies van voorlopigeaard zijn omdat het onderzoek nog niet is afgesloten. Bepaalde nota's komen niet voor verspreiding buiten het Instituut in aanmerking E LANDBOUWCATALOGUS 0000 0044 7090
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1 NN315451077
X laquoOTA 1077 augustus 1978
Instituut voor Cultuurtechniek en Waterhuishouding Wageningen
[
A NUMERICAL MODEL FOR NON-STATIONARY SATURATED
GROUNDWATER FLOW IN A MULTI-LAYERED SYSTEM
i r P J T van Bakel
SpoundIcircIcirc r
amp
I Notas van het Instituut zijn in principe interne communicatiemiddeshylen dus geen officieumlle publikaties Hun inhoud varieert sterk en kan zowel betrekking hebben op een eenshyvoudige weergave van cijferreeksen als op een concluderende discussie van onderzoeksresultaten In de meeste gevallen zullen de conclusies van voorlopige aard zijn omdat het onderzoek nog niet is afgesloten Bepaalde notas komen niet voor verspreiding buiten het Instituut in aanmerking
E LANDBOUWCATALOGUS
0000 0044 7090
C O N T E N T S
b i z
I INTRODUCTION 1
II GENERAL DESCRIPTION OF THE PROBLEM 2
21 Physical background 2
211 The law of linear resistance (Darcys law) 2
212 The law of continuity 3
213 Combination of Darcys law and the law of
continuity 3
22 Hydraulic properties and types of aquifers 3
23 Schematization of flow 5
24 Water balance terms 6
25 Solution of groundwater flow problems 9
III NUMERICAL SOLUTIONS 10
31 Finite element method 10
32 Application of the finite element method on the
flow in groundwater basins 13
33 Computer program 17
IV INPUT EXECUTION AND OUTPUT 19
41 Input 19
411 Endogenous variables 19
412 Exogenous variables 21
413 Data input to Program FEMSAT 21
42 Running the program on the Cyber 7200 25
43 Output 26
44 Computing time 27
mdash 1
biz
V NUMERICAL EXPERIMENTS 27
51 Flow to a well in an unconfined circular-
shaped aquifer 27
511 Steady state flow in an unconfined aquifer 28
512 Unsteady-state flow in an unconfined aquifer 30
52 Flow to a well in a 3-layered system 33
VI SUMMARY AND CONCLUSIONS 37
VII REFERENCES 39
i
I INTRODUCTION
The groundwater system is part of the hydrological cycle and
changes in this system affects phenomena such as plant production
river flow etc Generally one tries to use groundwater resources
in such a way that the utility for the community as a whole is
maximal
Numerical flow models can be valuable tools in the management
of water resources In many situations these models are the only
possible way to indicate effects of certain interferences in the
groundwater system
In this paper a numerical model is presented which simulates
non-stationary saturated groundwater flow in a certain region
Purpose of the model is to predict effects of certain operations
upon the various terms of the water balance of the groundwater basin
and on the hydraulic head
The basic idea of the model is that groundwater is flowing
horizontally in waterbearing layers and vertically in less-permeable
layers Each layer is discretized into a number of elements To
each element both Darcys law and the law of continuity are applied
(chapter II)
In this way a set of equations is obtained which are solved
using a finite element method and the Gauss-Seidel iteration method
(chapter III)
In chapter IV the input and output of the model are treated
while in chapter V some numerical experiments are presented
Finally in chapter VI an evaluation of the model is given
II GENERAL DESCRIPTION OF THE PROBLEM
21 P h y s i c a l b a c k g r o u n d
Throughout this report water potential is expressed on a unit
weight basis and is being called hydraulic head with symbol h
h = 2- + z [L] Pg
where
mdash = piezometer head |L | Pg u -1
z = gravitational head |_Lj
The hydraulic head is an expression for the capacity of a unit
weight of water to do work as compared to the work capacity of the
same weight of water with the same chemical composition at a certain
reference level and under atmosferic pressure
The problem is restricted to saturated flow of groundwater with
one constant density
211 The law of linear resistance (Darcys law)
According to Darcys law the rate of flow through a porous medium
is proportional to the gradient in hydraulic head It may be written
as
ocirc h ltSh ocirch N q = _k Tmdash q = -k vmdash q = -k -rmdash ( 1 ) x y ocircx y y lt5y nz z ocircz
where
q = volume flux of water passing through a unit area per x y z
per unit time perpendicular to x y z direction
respectively |_LT J
k = hydraulic conductivity in x y and z direction x gt y gt z _ _ i _
respectively | LT J
The coefficient k in the Darcy flow equation is taken to be
a constant depending on both the properties of the porous medium
and the fluid
212 The law of continuity
The other important physical law states that no matter can be
lost or created
The conservation equation can be found by applying the principle
of continuity to an infinite small volume
6v 6v lt5v 60 x y z TT = T + x + ~T7 + 1 (2)
Ot OX OcircX oz
where
0 = volume of water per unit bulk volume of soil |_-J
q = sink or source term T J
213 Combination of Darcys law and the law of continuity
Darcys law combined with the law of continuity gives
60 lt5 oh lt5 ocirch 6 oh _ ocirct ocircx x ocircx ocircy y oy ocircz z ocircz
or
c 5h 6 ocirch ocirc Ocirch 6 ocirch c ot ox x ocircx ocircy y oy laquoz z 6z
where
S = volume of water released or stored per unit bulk c I
volume of soil per unit change in hydraulic head |L I
22 H y d r a u l i c p r o p e r t i e s a n d t y p e s o f
a q u i f e r s
The flow in a natural groundwater basin takes place in layers
with different hydraulic properties Usually the layers are divided
into good permeable layers or waterbearing layers (aquifers) and
less permeable layers (aquitards)
The most important hydraulic properties are
a Transmissiviy - the product of the average hydraulic conductivity
k and the saturated vertical thickness d of the waterbearing
layer
It is a measure for the transport capacity of aquifers
Notation T = kd |_ L T ~
b Hydraulic resistance - the ratio of vertical thickness of
aquitards d and hydraulic conductivity k It is a measure for
the resistance capacity of aquitards denoted as c = d k _ T J
c Specific yield - the volume of water released or stored in the
phreatic zone per unit surface area of an unconfined aquifer per
unit change in hydraulic head
It is also called effective porosity
The symbol used for specific storage will be S j_-J
d Specific storage - the volume of water released or stored per
unit volume of the saturated parts of an aquifer or aquitard per
unit change in hydraulic head It is a measure for the elasticity
of the soil material and the fluid Excluded are irreversible
processes Notation S _ L J
e Storage coefficient - the volume of water released or stored
pe7 nit surface of a layer per unit change in hydraulic head
It is the sum of specific yield and the product of specific
storage and thickness of the aquifer or aquitard
S = S + S d r - 1 c y s - -1
Note in confined layers S = 0 y
In fig 1 a 3-layered configuration is given of two aquifers
separated by an aquitard Aquifer I has a free water surface and is
an unconfined aquifer Aquifer II is a completely saturated aquifer
Dependent on the hydraulic resistance of the aquitard KRUSEMAN and
DE RIDDER (1976) distinguish between confined semi-confined and
semi-unconfined aquifers For our purpose it is not useful to make
this distinction We restrict the aquifer types to unconfined and
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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mdash1 0 0 t-t w 0 lt- - n
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
z ltc laquo LU U EC LU ltC
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LUX
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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en - n
es
u bull-bull
bull-1 m c -z bull ^ ^ - z r w --lt - 0 - raquo w yO 1mdash1
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J
0 11
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CS
r
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h ) ^ f m n f i V n i gt t n
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3 0 0 m
t o H
^ 00 X
iuml gt - 4
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3 m
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t n
t o
bullbull0 x
z
3
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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mdash1 0 0 t-t w 0 lt- - n
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
z ltc laquo LU U EC LU ltC
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LUX
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
where h and h are values of h at time t and t 4n 4n-1 4 n n-1
respectively
14
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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en - n
es
u bull-bull
bull-1 m c -z bull ^ ^ - z r w --lt - 0 - raquo w yO 1mdash1
3gt Z Xgt n - f n 0 n 1-1 [Tl r-i Z 03 Z
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- z o 0 - f-1 bull-- z z
J
0 11
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CS
r
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3 0 0 m
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iuml gt - 4
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t o
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3
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
TERMS OF WATERBALANCE PER LAYER DURING LAST TIMESTEP AND
AND SINCE BEGINNING IN M3DAY M3
FLOU TO UNSATURATED ZONE
FLOU TO TERTIARY SYSTEM
0000
0000
0000
0000
LAYER NO 1
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS)
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-000
0000
000
000
0000
-000
0000
000
000
LAYER NO 2
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS1
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
-378
0000
381
0000
0000
-055
0000
056
0000
LAYER NO 3
FLOU TO SECONDARY SYSTEM
FLOU TO AD3ACENT LAYERCS5
ARTIFICIAL FLOU AND PRESCRIBED BOUNDARY FLOU
ACCUMULATION TERMS
LATERAL FLOU THROUGH PRESCR HEAD BOUNDARY
0000
370
50000
49376
0000
0000
055
12979
12909
0000
If the h-values belonging to the A-values are taken at time
t = t we get an equation with one unknown h for time t = t n-1 4n n
The equation can then be solved directly
However this is an explicit formulation which puts strong limitations
on the magnitude of the time step t - t (REMSON etal 1971) n n-1
Values larger than a certain value cause instability With an implicit method the values of h are taken at time
t = t This method is unconditionally stable Now one gets an equation n
with 7 unknowns h through h If one has 7 equations for every 1 n n
point 1 through 7 we can solve the values h i = 17
in
Of all the methods possible the iterative method of Gauss-Seidel
has been chosen The Gauss-Seidel method starts by choosing a first estimate of h at time t = t (usually equal to h ) At each in n j -i in-l iteration step a new value of h is calculated using the last
in calculated values of h If one passes the nodal points from 1 to
i n 7 eq (14) becomes
h-j + 1 = A h^ + 1 + A bdquo h | + 1 + A _ h | + 1 + A h j + A r h j + 4 n 4 1 1 n 4 2 2 n 43 3 n 45 5 n 4 6 6 n
B h B + A h J + - 4 4 4 n ~ + 0 J (A + mdash) A 4 7 h 7 n At n U 4 4 At
(15)
where
h-i = in
j = iteration step or number of iteration
i = number of nodal point n = time = t
n
The iteration process converges if the successive differences k+1 k
between h and h become continuously smaller in in
The iteration process is stopped if at each nodal point the k+1 k
difference between h and h is smaller than a certain prescribed in in
value
15
To accelerate the convergence an overrelaxation factor W is used
(for more information about W see FORSYTHE and WASOW 1960)
k+1 k T7 k+1 k h = h + W (h - h )
in in in in (16)
If W = 1 there is no overrelaxation Usually W varies between 1 and
2 and the optimum value very often is found by trial and error
Eq (15) is valid for points in aquifers The connection with
adjacent layers is made by the term Q because in this term the
flow from adjacent aquitards q is included (two aquifers are always
separated by an aquitard)
Aquitards
We dont apply the finite element method on points in aquitards
As stated before there is no horizontal flow in aquitards So it has
no sense to replace the continuous variation of h in the aquitard
by equation (11)
The nodal points in aquitards represent a certain volume of an
aquitard hich transmit water from one aquifer to another and which
has storage capacity The nodal points are included in the Gauss-
Seidel scheme -s follows (see fig 6 )
1 aauifer
a -_vS laquo 2 [~^-t^yM^^^-^r~ aqutarci l-Z
aquifer
Fig 6 Schematic situation for points in aquitards
Point 2 is situated in the middle of the aquitard with thickness
d storage coefficient S and vertical hvdraulic conductivity k c J v
Suppose the Gauss-Seidel iteration starts in the upper layer and
ends in the bottom-layer Then the flux from point 1 in aquifer 1 to
16
point 2 in the aquitard at time t = t and in iteration step j + 1
is
k q = (hf -hf )-iumlmdash f (17)
1 n 2 n OSxd1
and the flux from point 3 in aquifer 2 to point 2 in the aquitard
is
1 c 1
qbdquo = (h - hf) mdashHmdashr- (18) 2 3 n 2 n OSxd
The net flux ltl-q9 must be equal to the change in storage per unit
area per unit time This change in storage can numerically be
approximated by
te - h-
n n-1
Combina t ion of ( 1 7 ) (18) ana (19) g i v e s
l tr1 - h (hJ + 1 + hj - 2 hj+1) - - - - - - S1 - ilaquoS L u l l = o (20)
] i 3 n 2 n 1 c t - t j j x d n n -1
T + 1
From t h i s e q u a t i o n h^ kan be s o l v e d
3 3 C o m p u t e r o g r a m
The problem outl^ugrave so far has been translated into a computer
program written in FORTRAN IV
The flow chart ot Uis program is given in fig 7 whereas in
appendix A the complete listing of the program can be found
In the flow chart the 5 main parts of the program are
indicated
A Reading and printing of input data
B Determination of the matrices A and B and the vector Q per layer
17
M
1 A
FLOH to unsaturated zono
FLOW to tertiary syster
FLOW to secondnrv system
Leakage to adjacent aquitards
SOLVE equation (14)
yes
CALCULATE flux through constant head boundnr
CALCULATE waterbalance n p r 1 OAro r r J j - ~
PRINT of output
Ier node per layer
k = k+1
t = t + At
7 DUMP variables on tape
-gt-
( STOP )
Fig 7 Flow chart of the program FEMSAT with diviation in 5 main
parts A B C D and E
C START PROGRAM FEMSAT
f READ general information
PRINT general information
f READ element information
PRINT element information 7 READ stationary nodal point information
PRINT stationary nodal point information 7
READ non-stationary nodal point information for t = t
FRINT non-stationary nodal point information for t = t
L 2 (
^ yes R E A D non-stat input for t gt t
J PRINT non-stat input for t gt t (- 1 -- M degY
GENERATE A B and AREA-matrix for confined aquifers
_L
GENERATE A B and AREA-matrix for unconfined aouifer
bull
C Solving h with Gauss-Seidel
D Determination of the water balance terms per layer
E Printing of results
The program exists of a main program with several subroutines
The main program calls the subroutines In the subroutines most
calculations are performed
In this way an easy adaptation to different problems is
obtained (see also section 44) Comments in the program will
help the reader to understand the program
In the next chapter the preparation of the input data executing
of the program and the output will be treated
Acknowledgement
Some parts of the computer program were taken from the program
UNSAT2 of NEUMAN etal (1974)
IV INPUT EXECUTION AND OUTPUT
41 I n p u t
The input is divided into endogenous variables (chosen by the
user or calculated by the program during execution) and exogenous
variables (based on physical data)
411 Endogenous variables
Translation of groundwater basin into a nodal point network
Probably the most important but also the most difficult stage
in the process of simulating groundwater flow is the translation
of the natural groundwater basin into a nodal point network Questions
which arise are
a In how many layer the geo-hydrological system mustcan be
schematized
b The locations of the boundaries and the nature of these boundaries
(impervious prescribed head prescribed flux)
c The configuration of the nodal points
19
The answers of these questions depend on the problem under
consideration
- Schematization into layers
Usually our knowledge about the geo-hydrological situation is
such that no more than 5 different layers can be distinguished
(remember contrast rule)
- Boundary conditions
The top boundary condition of a groundwater basin usually is
a prescribed flux (effective precipitation)
The bottom boundary condition usually is a prescribed flux
boundary with flux zero (impervious hydrological base)
For the boundaries at the other sides of the system one sees
either prescribed head boundaries (eg a fully penetrating
river) or prescribed flux boundaries (including zero flux)
- Nodal point configuration
Nodal points are situated
on the boundaries
along internal boundaries which indicate changes in
hydraulic properties within the layer
along the tertiary surface water system
on places where artificial recharge or discharge wiii tgtke
place
elsewhere
Construction of the finite element mesh
Assume we have schemetized the groundwater basin iatu a finite
number of layers with different direction of flow (horizontal and
vertical) Within one layer the hydrological properties may vary
from place to place
Now a nodal grid is superimposed on each layer in a way already
described in section 32 Thus each layer has the same number of
nodal points situated on the same places in the horizontal plane
but with different z-coordinates
We are free to construct quadrilaterals or triangles because
the program automatically divides a quadrilateral into two triangles
having identical material properties The dimensions of the elements
20
should be small at places where large changes in hydrological
properties occur or where large gradients in hydraulic head are
expected to occur
Coding of nodal points
Each node in the finite element network is assigned an integer
code which indicates the type of calculation for this node
The coding is as follows
- Code = 1 nodal points with described head as function of time
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
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HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
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HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
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McGraw-Hill London pp 521
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APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU
ZIENKIEWICZ OC 1971 Finite Element Method in Engineering Science
McGraw-Hill London pp 521
40
z ltc laquo LU U EC LU ltC
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HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
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HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
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I 03 f - NI r -fc copy bull 3D -lt t n t n -il X ~i X ~i -raquo - n - n - T U mdash n r z r - - t r c c 0 cn a z 3 c - e - - n a i gt -s - n - - n -c 0 r 0 r
mdashf c - i cn c z m 2 33 en H xgt i - H 3gt C 33 33 -lt X
_ 3
-k
33 s c e s t n
j t i T T i - i i n n A Z O S O C O O X d 33
U) 3gt V Z N O bull
t o
Z n m
00 m 0
z z
z C l
z
raquo J J
- -n bullbull-gt r w 33 J Z 2 1gt n bullbull r n
3 Z 3 3 - 1 -H 1 gt - H 1 gt B H H H
V t n Ni
mdash1 m 33 3 t n
- n
c xgt
m x 55
f j
M zgt -n
O)
_ D t 3 - T) C bulln c 0 3= 0 t o 0 x
3= w gt0 3 raquo-i 3 3 r x raquo i gt r - r 0 3 Z u 3 X 3 Z 33 -lt 33
- 1 - i Z Z m bull 33 lt-raquo -lt t n iuml gt
o ^ r - r - c c o z m z m z CD t o t o F I H
t n t n
- n as - T] S) -raquobull bull œ w raquo - t o ro bull-gt CD X
en - n
es
u bull-bull
bull-1 m c -z bull ^ ^ - z r w --lt - 0 - raquo w yO 1mdash1
3gt Z Xgt n - f n 0 n 1-1 [Tl r-i Z 03 Z
-lt -lt icirc-f
- z o 0 - f-1 bull-- z z
J
0 11
w z -0 c 33 3 w Z Z O H -
t gt bull t o -
CS
r
-lt z
7Z O O
H - t S) 3gt - 1
- 1 r o r - r - lt-raquo r o
h ) ^ f m n f i V n i gt t n
J X I - 3gt z c i r c m i gt 3 ~lt r m bulljgt raquo
-lt rgt z raquo 0 3 3 0 m m 0
3 0 0 m
t o H
^ 00 X
iuml gt - 4
-H
3 m
- n
t n
t o
bullbull0 x
z
3
3gt
C -lt Igt - t Ci
APPENDIX 8 EXAMPLE OF OUTPUT
HEAD AND TERMS OF UATERBALANCE PER NODE PER LAYER
AT TIME 260 IN II AND I13DAYS
T
LAYER NODE KODE HEAD FLOUUN FLOUTS FLOUSS LEAKAGE STORAGE LAT FLOU