NN31545.1077 X - WUR E-depot home

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

confined aquifers

WIHtUtttWttllttlt(ltmtttltHlttltlil(lltlittlfllnl ) bull (

d V U i l i l i ii i Hi il i M M bull f i m

i tMi| i i | J i j j i _ I I bull

aq ui tard

aquifer II

Fig 1 Schematic representatipn of flow direction and variation

of hydraulic head h in a 3-rlayered system

In an unconfined aquifers ( I ) poundhe vertical thickness pf the

waterbearing layer d is a variable while in a confined aquifer II

this is a constant (denoted as D ) Other important properties of

unconfined aquifers are the hydraulic head being the same as the

phreatic surface and the specific yield being different from zero

23 S c h eacute m a t i s a t i o n o f f l o w

The groundwater flow in a groundwater basin is schematized

into a horizontal flow in aquifers and into a vertical flow in

aquitards Consequently the vertical variation of the hydraulic

head is as drawn in figl

The validity of this assumption has )een investigated by

NEUMAN and WITHERSPOON (1969) The found that if the contrast

in hydraulic conductivity between two adjacent layers is bigger

than a factor two the relative error caused by this assumption

is usually smaller than 5

This schematization gives a simplification of the general flow

equation (eq 4 ) In an aquifer the vertical is zero and eq (4)

then reduces to

-Sbdquo -r- 7 - (k d -r-) + 7 mdash (k d 7-) + qd c lt5t ox x ox oy y oy

or

bull(S + S d) g y s ot

-r- (k d -=-) + -T- (k d -=-) + Q ocircx x ox oy y ocircy

(5)

where

Q = volume of water substracted from or added to unit surface

of aquifer per unit time |_ L T J

In an aquitard eq (4) reduces to

-sQ |pound = (k Q) s 6t oz Z OcircZ

(6)

As will be discussed in the next paragraph the term q in an

aquitard is set equal to zero

24 W a t e r b a l a n c e t e r m s

In eq (4) the term q is mentioned With this sink or source

term different terms of the water balance are taken into account

Fig 2 depicts schematically the situation for an unconfined aquifer

(for other layers some terms dont exist)

recharge- - discharge

laquo-b

Fig 2 Schematic representation of the various terms involved in the

water balance

Description of the different terms

a The flow through the phreatic surface q f

If this term is negative (water goes out of the saturated system)

it is called capillary rise If positive it is called percolation

(or drainage)

This flow can be calculated as a rest term of the water balance

of the unsaturated zone Very often this term is expressed as a

flow per unit surface (flux) and is called effective precipitation

For the time being this flux is a given input

b Flow to adjacent aquitards q

The flux to adjacent aquitards can be calculated by applying eq

(6) to the aquitard

c Flow to the primary surface water system q

The primary surface water system penetrates fully the groundwater

basin So it acts as a boundary with given values for the

hydraulic head (is equal to the phreatic level of the primary

system)

d Flow to the secondary surface water system q

The secondary surface water system does not penetrate fully the

groundwater basin which causes a resistance against flow between

groundwater and surface water The flow q can be calculated as

follows

h - h

- mdash f i i gt

where

h = hydraulic head of the groundwater at the place of

the secondary system |_ L |

h = phreatic level of secondary system |_ L J i s -

R = radial and entrance resistance of the

secondary system per unit surface |_ T J

According to ERNST (1962) the radial resistance H per unit

surface can be caKulated with

Uuml = 4 i n 4 e8) TTk B

where

1 = area drained by 1 m conduit L ^ J

d = thickness of the layer near the conduit _ L J

B = wet perimeter of the conduit |_ L J

The entrance resistance is caused by a layer which shows a

lower conductivity around the conduit than its surroundings

In practise radial and entrance resistancies are difficult to

deduct from hydraulic properties They can be found indirectly

by measuring simultaneously q h and h

t s

e Flow to the tertiary surface water system q

In smaller conduits usually the phreatic level is not known

So eq (7) cant be applied The flow to these conduits however

is sometimes very important (eg in areas with shallow groundwater

tables) One may a-sume (ERNST 1978) that a relationship exists

between the hydraulic head of an aquifor and the flow q

According to Ernst the tertiary system can be divided into a

number of categories with respect to their mutual distance and

bottom height For each category the flow is h - h (b)

q ( h b ) - mdash (9)

where

h = average hydraulic head | L J

h (b) = phreatic level in the category under

consideration a function of bottom height |_L J

a = geometry factor to convert h into h _ - J

Y = drainage resistance _ T J

hm = hydraulic head midway between two

conduits |_ L J m

Summation of the q(hb)-relations of the different categories

yields the overall relation between hydraulic head and flow to

the tertiary system we are looking for

f Artificial recharge or discharge q and prescribed boundary

flux qfo

The magnitudes of these flows are not influenced by the conditions

inside the groundwater basin but remain on a prescribed level

Thus q is an internal boundary condition q an external boundary cl D

condition

For confined aquifers some of the above mentioned terms dont

exist These terms are the flow through the phreatic surface qf

and the flow to the tertiary surface water system q

Aquitards are considered only as transmitters of water from one

aquifer to another with possibilities of storage (eq 6 ) This means

that for aquitards the water balance terms ar through f are set

equal to zero

25 S o l u t i o n o f g r o u n d w a t e r f l o w p r o b l e m s

The basic equations for saturated groundwater flow are eq (5)

for aquifers and eq (6) for aquitards The solution of these

equations require knowledge of the physical system and the auxiliary

conditions describing the system constraints

These auxiliary conditions are (REMSON etal 1971)

a Geometry of the system

Eg the diviation of the system into a restricted number of

layers and the horizontal extension of the system under

consideration

b The matrix of hydraulic properties including in-homogeneity

and an-isotropy

c Initial conditions Because we are dealing with non-stationary

flow the values of pertinent system variables (such as hydraulic

head at the initial time) must be known

d Boundary conditions They describe the conditions at the boundaries

pf the system as a function of time

One can distinguish between

- hydraulic head boundary

- flow (including zero flow) boundary

- both head and flow boundary

After specifying the auxiliary conditions an unique solution of

the variation of the hydraulic head in space and time can be

obtained

For special shapes of flow and assumptions exact analytical

solutions exist In nature however one is usually confronted with

problems like

- spacial variation in the hydraulic properties of the soil

- irregular shape of the groundwater basin

- empirical relations between q and h

- d not being a constant

For such situations by numerical and analogy models only an

approximation of the actual situation can be obtained In the next

chapter a numerical solution called the finite element method is

treated

III NUMERICAL SOLUTIONS

In this chapter the numerical solution of the aquifer equation

(eq 5) and the aquitard equation (eq 6) will be treated The

method of solving is the finite element method In section 31 only

a short mathematical description of this method will be given For

more mathematical background the reader is referred to publications

on this subject (eg ZIENKIEWICZ 1971 NEUMAN etal 1974 and

WESSELING 1976)

In section 32 the application of the finite element method on

the flow in groundwater basins is treated while in section 33 the

translation into a computer program is given

3 1 F i n i t e e l e m e n t m e t h o d

For reasons of simplicity the finite element method will be

explained for the stationary version of eq 5 ie

10

(k d^) + (k d ^ ) - Q = O (10) ocircx x 6x 6y y Sy

The basic idea of each numerical solution of eq (10) is to

apply this equation on a small but finite volume The flow region

of interest R is divided into a finite number of volumes called

elements The elements can have different shapes but for reasons

of conveniency we only use triangles- Quadrilaterals can also be

used because they can be divided into two triangles The sizes of

the triangles may vary They can be adapted according to the

geometry of the region and the accuracy desired (fig 11 gives an

example of the diviation of a region into triangles and quadrilaterals)

The corner points of the elements represent the nodal points

If there are N nodal points then the continuous variation of h

in region R is approximated by the expression

h (x_v) ~ h f (xy) (11) a n n

where

h (AV) = approximation of u (xy) a

h = value of h in nodal point n n

f ltxgty) = global coordinate function

In nodal point n a must be equal to h so f (x y ) = 1 and a n n n n

f (xy) = 0 i n Between f (x v ) and f (xy) of the nodal n j ] n n n n j j

points surrounding nodal point n f (xy) varies linearly from

1 to 0

In fig 3 a picture of f (xy) is given n

If one knows the values h in the nodal points n we know the approximation h of h We can find these values of b bv using for

a n

example the Galerkin method

Galerkins principle states that an approximate solution of h in

eq (10) can be obtained from the following system of equations

N ~~ (k d -A + -^- (k d T-y E h f (xy) f(xy)dxdy +

-ox x ox ocircy y oy - _ n n i R n~

J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1

Fig 3 The global coordinate function f (xy)

The integration can be performed element by element with the

aid of the coordinate functions f (NEUMAN etal 1974)

For non-stationary problems it is not possible to apply the

Galerkin principle on the time derivative But it is possible

to discretize this derivative and to set up the system of equation

(eq 12) for each finite time step

One obtains in this way a series of sets of laquocriations tha s

an approximation of the non-stationary flow problem

Treatment of boundary conditions

a Initial conditions

In non-stationary flow problern gtne must start with given

values of a at time t = t ri c

b Boundary conditions

- Prescribed head boundary

Nodal points which lie on a prescribed head boundary have a

prescribed value for h bull r Hgte r umber of unknown h s is n n

reduced and accordingly no the number of equations

- Prescribed flux through z-- boundary

If one knows the flux (iuumlr jw or outflow per unit area) and

the area of the boundary oi-r element the inflow or outflow

through the boundary per flemlt=nt can be calculated

One may assume this flow to J bull part of the sink term Q in

eq (12)

12

32 A p p l i c a t i o n o f t h e f i n i t e e l e m e n t

m e t h o d o n t h e f l o w i n g r o u n d w a t e r

b a s i n s

From hydrogeological investigations the geometry and the matrix

of the hydraulic properties are known Knowing the geometry the

flow can be split up into a finite number of layers This holds

also for the nature of the boundaries

Now a nodal grid is superimposed on each layer in such a way

that the nodal points of the different layers are situated right

above each other (see fig 4) and in each layer the nodal point

lies in the middle of this layer

-nodal point

Fig 4 Schematic representation of the nodal points in the

xyz-space

Each layer has N nodal points of which P nodal points have a

prescribed head In each layer remain N-P = N nodal points with

unknown head Eq (12) can be applied to each point with unknown

head in each separate layer

As was stated in the last paragraph for each time step one

can write

13

Ah bdquo ~ -r1 + Q = O in = 12N in n m At yi A h + B (13)

where

A = in

ocircf ocircf ocircf of (k d f mdashri- ~ + k df - ~ -jS) dxdy

J x n ox ox y n oy oy

in (S d + S ) ff dx dy O if i icirc n

s y i n

Q f dx dy

The matrices A B and the vector Q have been defined in in l

elsewhere (NEUMAN etal 1974) A is the conductivity matrix in

B the storage matrix and Q the source vector in l

In order to clarify these rather complicate expressions apply

equation (13) poundo point 4 of fig 5

6 7

Fig 5 Top view of part of the nodal grid

lhl + A42h2 + A43h3 + A44h4 + A45h5 + A46h6 + A47h7 +

h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)

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 outlûgrave 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

- Code 2 internal nodal points

- Code = 3 internal nodal points which represent secondary surface

water system

- Code = 5 nodal points with prescribed artificial recharge or

discharge and points with prescribed boundary flow

(both as functions of time)

Other important endogenous variables are the magnitude of the

time step the value of the variable which stops the iteration

process and the maximum number of iterations allowed per time step

(see also group A and B of section 413)

412 Exogenous variables

After constructing the nodal grid each node represents part of

the groundwater basin Geo-hydrological properties can be assigned

to each mode based on physical data (group C D and E of section

413) In group F of section 413 time-dependent variables can

be assigned to each node

413 Data input to Program FEMSAT

The data input must be punched on cards according to the

following instructions

GROUP COLUMNS FORMAT SYMBOL DESCRIPTION

A 1-80 20A4 HEAD Desired heading to be printed at

the beginning of the execution

1-5 15 NUMNP number of nodal points

21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers

maximum difference in node number

between two adjacent nodes

number of time steps between a

change in non-stationary input data

number of time steps between two

successive outputs

maximum number of iterations per

time step

time to start with the execution

initial time interval

maximum time to be reached before

execution is stopped

maximum allowed absolute change in

the values of h between any two

successive iterations in a given

time step

Iterations continue until all changes

are less than or equal to TOL or

until MAXIT is exceeded

1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)

sequential number of i-th corner of

element N

sequential number of j-th corner

sequential number of k-th corner

sequential number of 1-th corner

(see fig 8)

Fig 8 Numbering of element corner nodes

22

GROUP COLUMNS FORMAcircT SYMBOL DESCRIPTION

One card must be provided for each element in sequential order

starting with N = 1 and ending with N = NVJMEL

I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)

angle in degrees between CI and

the laquo-coordinate to he used in

nodal print N in layer LY (see

poundig 9)

First principal conductivity of

material in node N and layer LY

idem second principal conductivity

thickness of layer LY at nodal

point N

ig V Definition of SANGCI and C2

One card icircuiPt be provided for each model point ctarting wih M = 1

and ending with N = NUMNP If there is norlt3 than 1 layer for each

layer a set must be provided starting with the upper layer LY - 1

and ending with the bottom layer LY = NIJMLAY The variables SANG

M and 2 are not relevant tor aquitards but for reasons of

conveniency this procedure is maintained (making SANG CI and C2

dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)

coding of nodal points (see

paragraph 42)

number of tertiary relation in node N

The q-h relationship of the flow to

23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)

the tertiary system as discussed

in par 24 are schematized in a small

number of different relations

x-coardinate of node N

y-coordinate of node N

height ground level in node N

height bottom secondary conduit in

node N If node N is not a secondary

conduit this variable is a dummy

variable

length of secondary conduit in area

represented by node N

One card must be provided for each node starting with N = 1 and

ending with N = NUMNP

E 1-10 FlOl P(NLY1) initial value of h at node N in

layer LY

11-20 FlOl VERRES(NLY) vertical resistance at node N and

layer LY If the layer is an aquifer

this value is a dummy variable

21-30 F101 RADRES(NLY) radial resistance for flow to

secondary system at node N in layer

LY

31-40 FlOl ENRES(NLY) entrance resistance for flow to


LY

41-50 FlOl WP(NLY) wet perimeter of secondary system at

node N in layer LY

51-60 FlOl P0R(NLY) specific yield of the material at

node N in layer LY

61-70 FlOl SS(NLY) specific storage of the material

at node N in layer LY

24



ending with N = NUMNP If there is more than 1 layer for each layer

a set of cards must be provided starting with LY = 1 and ending

with LY = NUMLAY

F 1-10 F101 P(NLY1) value of h at different times at

node N in layer LY Node N lies

on a prescribed head boundary

11-20 F101 WTSC(N) water table in secondary system at

node N at different times

21-30 F 101 ARRED(NLY) artificial recharge to or discharge

from node N in layer LY or

prescribed boundary flow at

different times

31-40 F 101 QPHR(N) flux through phreatic surface at


The data in group F are non-stationary input data One card must be

provided for each node starting with N = 1 and ending with N = NUMNP

In case of more than 1 layer for each layer a set must be provided

Not relevant variables are taken as dummy variables

After a certain number of time steps (NTSPI) the data of group E

are read So the set cards E must contain NxLYxC cards where C =

(TMAXNTSPIXDELT) + 1 if DELT is a constant

42 R u n n i n g t h e p r o g r a m o n t h e C y b e r 7200

The program has been executed on a Cyber 7200 computer of

IWIS-TNO in the Hague

- The demanded input is punched on cards according to the instructions

of the last paragraph and is put on a permanent file

- For each specific problem inthe program only the dimension statements

in the main program have to be adjusted according to the instructions

in the lines of comment

- After compilation the program together with the input file is

executed

25

- Intermediate data necessary for continuation of the calculations

are saved on another permanent file This gives the opportunity

to restart the program at a time the calculation was stopped

The only change in the input file is to adjust the starting time

Remark

With the aid of a terminal small changes in the program and in the

input file can be made easily and the output immediately appears

on the terminal In this way it is possible to stop erroneous

calculations This is especially useful when being in the testing

phase of the program

43 O u t p u t

Immediately after reading the input data they are printed so

that possible errors can be detected During execution after each

certain number of time steps (NTSPO) calculated values of h and

water balance terms are printed In appendix B an example is given

The output data can be devided into two groups

A Data per node per layer

- Coding of the nodal points KODE

- Hydraulic head HEAD

- Flow through phreatic surface at the time indicated in head

of table FLOWUN

- Flow to tertiary surface water system FLOWTS

- Flow to secondary surface water system FLOWSS

- Flow to adjacent layers LEAKAGE

- Artificial recharge or discharge and prescribed boundary

flow ARRED

- Change in storage during last time step STORAGE

- Flow through prescribed head boundary LATFLOW

B Data per layer

Summation of the terms already mentioned under A gives the

magnitude of the water balance terms per layer at the time

indicated (instanteneous water balance terms) Summation of

these instanteneous terms multiplied by their respective time

intervals since time is zero gives the cumulative water balance

terms per layer

26

After each print out we are able to check the calculations

because the sum of the water balance terms per layer must be

approximately zero

44 C o m p u t i n g t i m e

The computing time depends on the number of nodal points and

the number of layers But it depends also on the magnitude of the

changes in hydraulic head and on the value which terminates the

Gauss-Seidel iteration (TOL) To give an impression TOL = 001 m

Costs per day simulation (in hlfl) on the Cyber 7200

number of nodes number of layer bull bdquo bull bull _ 1mdash x a x b (21)

where

a = costs in Dutch guilders per system second (ƒ 027)

b = empirical factor with magnitude about 001

V NUMERICAL EXPERIMENTS

In general there are two possibilities to test the validity of

numerical models The first and most commonly used method is to

compare numerical results with analytical solutions The second

possibility is to compare numerical results with measured field

data Since the last alternative requires many data that are

difficult to obtain we restrict ourselves for the time being to

comparison with the analytical solutions

5 1 F l o w t o a w e l l i n a n u n c o n f i n e d

c i r c u l a r - s h a p e d a q u i f e r

Flow to a well can be divided into steady-state flow and

unsteady-state flow (KRUSEMAN and DE RIDDER 1976)

27

511 Steady state flow in an unconfined aquifer

For this case the formula of Thiem-Dupuit is valid (see fig 10)

Q bull k lth2-hicircgt ln(r2r])

(22)

where

hhbdquo = hydraulic head in 1 and 2 respectively

Q = discharge from well

rbdquo = distance from well

CL31

hmdash^ri --r2

Fig 10 Schematic cross-section of a pumped unconfined aquifer

Fig 11 gives the nodal grid superimposed on a well in a

circular shaped aquifer Near the well the nodal point distance

is made smaller Quadrilaterals as well as triangles are used

Numerical calculations with this finite element network and 3 -1

an extraction rate of 200 m day were performed for 10 days

At that moment a steady-state situation was almost achieved

28

2tmdashnodal point number

160 180 X-axesim)

Fig 11 Nodal point configuration for flow to a well in a circular-

shaped aquifer The outer boundary is taken to be constant

The well is situated in nodal point 42

29

In fig 12 the results of the simulation are compared with the

analytical solution The agreement between numerical results and

the analytical solution is excellent

draw- 15irS (m) down

analytical solution imdashx calculated values

distance from well r (m)

Fig 12 Comparison between analyt ical and calculated s teady-state

drawdown for the problem of f ig 11 with 1

-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s

512 Unsteady-state flow in an unconfined aquifer

Analytical solutions for unsteady-state flow in an unconfined

aquifer only exist if the aquifer extends infinitely For the

aquifer given in fig 11 this condition is not fulfilled However

during the first one or two days after starting the extraction the

influence of a constant hydraulic head at the outer boundary on the

30

drawdown near the well can be neglected (see fig 13) After one

day the lateral inflow is only 25 of the extraction the remainder

being the change in storage due to drawdown

2 0 0

m 3 day 1

Q=200m 3 day

100-

vraquomdash Lateral inflow

N amp storage

I I ^ V - W 10

t (days)

Fig 13 Values of Q lateral inflow and change in storage as

functions of time

The nonsteady-state or Theis equation can be written as

i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt

r = distance from well

S = storage coefficient

Q = constant discharge from well

D = thickness of the waterbearing layer

at the beginning of pumping

t = time after starting the extraction

s = corrected drawdown

= s - 322D

s = calculated drawdown

W(u) = Theiss well function

[J

CIAT1

[T]

]

31

In fig 14 the analytical unsteady-state solution for the case

described in section 511 is given together with the numerical

results at times 05 and 10 days The agreement between numerical

results and the analytical solution is rather good

Drawdown s1(m)

10

10

10

Theiss solution

x caculated values t=Q5

bdquo t= lO

10 10- 10 10 t r2(daym2)

Fig 14 Comparison between analytical and calculated steady-state

drawdown for the problem of fig 11 with

Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m

These results indicate that the assumption of a linear variation

of h between two nodal points (see fig 3 ) even with the rather

rough element network of fig 11 is acceptable So there is no

need to improve for example the finite element network by introducing

higher order coordinate functions

32

52 F l o w t o w e l l i n 3-1 a y e r e d s y s t e m

In fig 15 a schematic cross-section of a 3-layered system of

two aquifers separated by an aquitard is given A well of

infinitisemal radius is completed in the lower aquifer and discharges

at a constant rate QTT Each layer is homogeneous isotropic and

extending radially infinite

bdquo bull - bull bull ~ i

mdash Q II

aquifer I

lugraveî~IcircIuml7i-i~~ aquitard --r-rl~-

-II--I i - i

aquifer

Fig 15 Schematic cross-section of a 3-layered system

For this situation NEUMAN and WITHERSPOON (1969a) give rather

complicated analytical solutions In the special case where the

hydraulic properties of the two aquifers are identical they give

graphical illustrations of their solutions for three different

cases

These graphical solutions give the opportunity to verify

roughly numerical results of a 3-layered system Therefore a nodal

grid is superimposed on a circular shaped groundwater flow system

with the extraction in the centre of the lower aquifer and with

a constant head on the outer boundary Fig 16 gives the nodal

point situation in the x - y plane of 1 layer The nodes in the

other two layers are situated in the middle of the layers at the

same places in the x - y plane only their z-coordinate is different

33

A

-Jy

bull33

V 5

iumlff

Fig 16 Nodal point configuration for the 3-layered problem (top view)

34

In fig 17 numerical results are compared with the analytical

solution The agreement is rather bad especially for the two

aquifers But in terms of the water balance the difference between

aquifers and aquitard is not so big because the aquitard has

about 16 times bigger storage capacity than the aquifers The

small value of the specific storage of the aquifers is the reason

that small errors in the components of the water balance have a

big influence on the calculated values of the hydraulic head

Eig 17 Dimensionless drawdown vs dimensionless time in a

3-layered system with

II = 50 m day

3 m (distance from well)

aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1

The discrepancies are partly caused by errors in discretization

In the z-direction these errors can be eliminated by taking the

35

vertical resistance of the aquitard to be infinite Then the

calculated drawdown in the pumped aquifer can be compared with

the analytical Theiss solution

This is done in fig 18 for different distances from the well

One sees that the agreement is better when the distance is bigger

Apparently the linearization of the logarithmic drawdown curve

near the well is responsible for the differences The differences

for r = 3 m are of the same magnitude as for the pumped aquifer in

the 3-layered case

joo gt A 50

20

Theiss solution

bull calculated values r=3m (distance f rom well) x bdquo bdquo r=7m

1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t

Fig 18 Dimensionless drawdown vs dimensionless time in a

confined aquifer with

Q =500 m day

k = 10 mday

D = 100 m

S = 001 s

The differences between numerical results and analytical solutions

are not caused by taking a finite area because changing the outer

boundary into an impermeable boundary has only minor effects

36

VI SUMMARY AND CONCLUSIONS

In this report a numerical model for non-stationary saturated

groundwater flow in a multi-layered system has been treated

Groundwater flow can be described in different ways

a Physical-mathematical by combining Darcys law and the law of

continuity (section 21)

b Geo-hydrological the flow in a natural groundwater basin takes

place in layers with different hydraulic properties Definitions

of the most important hydraulic properties are given in section

22

c Schematical groundwater flows horizontally in aquifers and

vertically in aquitards (section 23)

d Approximating by dividing the region of flow into a finite

number of elements and applying Darcys law and the law of

continuity on each of these elements In these ways one gets

a set of equations which can be solved given (initial and)

boundary conditions (section 25)

e Numerical the set of equations has been solved using a Galerkin

type finite element method and the implicit Gauss-Seidel method

with overrelaxation factor (section 31 and 32)

f Computer-solvable the numerical solution has been translated

in a FORTRAN IV program called FEMSAT (section 33)

In chapter IV the input execution and output of the program

FEMSAT are treated

Finally in chapter V some numerical applications are given

simulation of flow to a well in unconfined aquifer and simulation

of flow to a well in a 3-layered configuration

Comparison of hydraulic heads obtained by numerical calculations

and analytical true solutions showed an excellent agreement in case

of stationary flow and a good to moderate agreement in case of non-

stationary flow However the diviations caused by the discretization

process must be seen in relation to the errors made in the

establishment of the hydraulic properties

37

As was stated in the introduction numerical models can be

valuable tools in the management of water resources Therefore

numerical models must be suitable to simulate effects of certain

operations upon the spreading of water over saturated groundwater

unsaturated groundwater and surface water The model described in

this report is a model for saturated groundwater flow The relation

with the surface water can be taken into account via radial and

entrance resistances or via a relation between flow to the surface

water system and the hydraulic head The relation with the unsaturated

zone has been restricted to the input factor effective precipitation

In practise however the unsaturated and saturated system are

interconnected Therefore the model should be improved by taking

into account the unsaturated zone This can be done in at least

two ways

a Replacing the unsaturated zone by a parametrical model

b Coupling the unsaturated zone in an iterative way with the

saturated system

In order to find out the usefulness of the numerical model for

water resources management it will be applied to actual regional

groundwater systemslaquoAlso sensitivity analyses will be carried out

The computer model itself can be improved by applying standard

programs eg automatical generation of a nodal grid and graphical

plotting of the results

38

VII REFERENCES

ERNST LF 1962 Grondwaterstromingen in de verzadigde zone en hun

berekening bij aanwezigheid van horizontale evenwijdige open

leidingen Thesis University Utrecht pp 189

1978 Drainage of undulating soils with high groundwater

tables (in press)

FORSYTHE GE and WR WASON 1960 Finite Difference Methods for

Partial Differential Equations John Wiley and Sons New York

pp 444

KRUSEMAN GP and NA DE RIDDER 1976 Analysis and evaluation of

pumping test data ILRI-bulletin No 11 pp 200

LAAT PJM DE and C VAN DEN AKKER 1976 Simulatiemodel voor grond-

waterstroming en verdamping in Modelonderzoek 1971-1974 ten

behoeve van de waterhuishouding in Gelderland deel II

Grondslagen pp 145-197

NEUMAN SP RA FEDDES and E BRESLER 1974 Finite element

simulation of flow in saturated -unsaturated soils considering

water uptake by plants Hydraulic Engineering Laboratory

Technion Israel pp 104

and PA WITHERSPOON 1969a Theory of Flow in a confined

two Aquifer System Water Resources Research Vol 5 No 4

pp 803-816

^_ and PA WITHERSPOON 1969b Applicability of Current Theories

of Flow in leaky aquifers Water Resources Research Vol 5

No 4 pp 817-829

PINDER GF and EO FRIND 1972 Application of Galerkins Procedure

to Aquifer Analysis Water Resources Research Vol 8 no 1

pp 108-120

REMSON I GE HORNBERGER and FJ MOLZ 1971 Numerical Methods

in Subsurface Hydrology John Wiley and Sons New York

pp 389

WESSELING JG 1977 Toepassing van Finite Difference- en Finite

Element Methoden voor het oplossen van een aantal grondwatershy

stromingsproblemen (met computerprogrammas) ICW-nota 944

pp 63

39

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

13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

009

003

003

0000 0000 0000 0000 0000 0000 0000 0000

1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000

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 AD3ACENT LAYERCS1


ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000

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



shaped aquifer 27





VII REFERENCES 39

i

I INTRODUCTION





maximal




groundwater system










(chapter II)



(chapter III)








h = 2- + z [L] Pg

where












as


where



respectively |_LT J


respectively | LT J



and the fluid



lost or created



6v 6v lt5v 60 x y z TT = T + x + ~T7 + 1 (2)

Ot OX OcircX oz

where






or


where




a q u i f e r s








layer




















S = S + S d r - 1 c y s - -1









confined aquifers




aq ui tard

aquifer II




















then reduces to


or



(5)

where





(6)









laquo-b


water balance





(or drainage)







(6) to the aquitard





system)





follows

h - h

- mdash f i i gt

where









where









t s










q ( h b ) - mdash (9)

where







conduits |_ L J m





flux qfo




condition







equal to zero










consideration


and an-isotropy












obtained



problems like








treated








WESSELING 1976)







10














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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T


13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

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1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000

mdash 1

biz



shaped aquifer 27





VII REFERENCES 39

i

I INTRODUCTION





maximal




groundwater system










(chapter II)



(chapter III)








h = 2- + z [L] Pg

where












as


where



respectively |_LT J


respectively | LT J



and the fluid



lost or created



6v 6v lt5v 60 x y z TT = T + x + ~T7 + 1 (2)

Ot OX OcircX oz

where






or


where




a q u i f e r s








layer




















S = S + S d r - 1 c y s - -1









confined aquifers




aq ui tard

aquifer II




















then reduces to


or



(5)

where





(6)









laquo-b


water balance





(or drainage)







(6) to the aquitard





system)





follows

h - h

- mdash f i i gt

where









where









t s










q ( h b ) - mdash (9)

where







conduits |_ L J m





flux qfo




condition







equal to zero










consideration


and an-isotropy












obtained



problems like








treated








WESSELING 1976)







10














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


c z O E 0C 3 GLZ

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en


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o LU M

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n n n


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ce Lii V

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bull E K J I _J Ul


s s Ul

gt-laquo O

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en CD

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CC Ul a

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bullc en CD

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cy Il H H H

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II

Ul

sect r-

z o ltr t -ltt

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laquo _J 3 U

i u u a u z raquo H

1 -

a 3

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s r -z

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u

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u u r t


bull U 11

gt- laquo w

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1 -- 1 Ul

a r-

laquo Z 3 - 1 U U

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il K

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laquo E r-

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laquo _ l E 3 Z

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ce a

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m agt 39 -C

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raquo Hlaquo

m ti -t ro c 33 2

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- bull H -X bull



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x X

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bull 2 3 j j - o r


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OD CD CD

N OJ N LOW UI

CD CD 00 0

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CD ^

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xraquo H z - r -H

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3

3gt

C -lt Igt - t Ci




T


13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

009

003

003

0000 0000 0000 0000 0000 0000 0000 0000

1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000

I INTRODUCTION





maximal




groundwater system










(chapter II)



(chapter III)








h = 2- + z [L] Pg

where












as


where



respectively |_LT J


respectively | LT J



and the fluid



lost or created



6v 6v lt5v 60 x y z TT = T + x + ~T7 + 1 (2)

Ot OX OcircX oz

where






or


where




a q u i f e r s








layer




















S = S + S d r - 1 c y s - -1









confined aquifers




aq ui tard

aquifer II




















then reduces to


or



(5)

where





(6)









laquo-b


water balance





(or drainage)







(6) to the aquitard





system)





follows

h - h

- mdash f i i gt

where









where









t s










q ( h b ) - mdash (9)

where







conduits |_ L J m





flux qfo




condition







equal to zero










consideration


and an-isotropy












obtained



problems like








treated








WESSELING 1976)







10














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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T


13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

009

003

003

0000 0000 0000 0000 0000 0000 0000 0000

1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000





h = 2- + z [L] Pg

where












as


where



respectively |_LT J


respectively | LT J



and the fluid



lost or created



6v 6v lt5v 60 x y z TT = T + x + ~T7 + 1 (2)

Ot OX OcircX oz

where






or


where




a q u i f e r s








layer




















S = S + S d r - 1 c y s - -1









confined aquifers




aq ui tard

aquifer II




















then reduces to


or



(5)

where





(6)









laquo-b


water balance





(or drainage)







(6) to the aquitard





system)





follows

h - h

- mdash f i i gt

where









where









t s










q ( h b ) - mdash (9)

where







conduits |_ L J m





flux qfo




condition







equal to zero










consideration


and an-isotropy












obtained



problems like








treated








WESSELING 1976)







10














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

009

003

003

0000 0000 0000 0000 0000 0000 0000 0000

1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000



and the fluid



lost or created



6v 6v lt5v 60 x y z TT = T + x + ~T7 + 1 (2)

Ot OX OcircX oz

where






or


where




a q u i f e r s








layer




















S = S + S d r - 1 c y s - -1









confined aquifers




aq ui tard

aquifer II




















then reduces to


or



(5)

where





(6)









laquo-b


water balance





(or drainage)







(6) to the aquitard





system)





follows

h - h

- mdash f i i gt

where









where









t s










q ( h b ) - mdash (9)

where







conduits |_ L J m





flux qfo




condition







equal to zero










consideration


and an-isotropy












obtained



problems like








treated








WESSELING 1976)







10














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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X - e - o

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LUX

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u

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bullc u

v v PJ LTIcirc



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LU 03

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13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

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- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

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1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

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000

001

007

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020

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1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

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0000

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50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

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0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000




layer




















S = S + S d r - 1 c y s - -1









confined aquifers




aq ui tard

aquifer II




















then reduces to


or



(5)

where





(6)









laquo-b


water balance





(or drainage)







(6) to the aquitard





system)





follows

h - h

- mdash f i i gt

where









where









t s










q ( h b ) - mdash (9)

where







conduits |_ L J m





flux qfo




condition







equal to zero










consideration


and an-isotropy












obtained



problems like








treated








WESSELING 1976)







10














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

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125000 125000 124998 124972 124895 124787 124647 124358

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50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


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000

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0000

-000

0000

000

000

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ACCUMULATION TERMS


0000

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0000

381

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0000

370

50000

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aq ui tard

aquifer II




















then reduces to


or



(5)

where





(6)









laquo-b


water balance





(or drainage)







(6) to the aquitard





system)





follows

h - h

- mdash f i i gt

where









where









t s










q ( h b ) - mdash (9)

where







conduits |_ L J m





flux qfo




condition







equal to zero










consideration


and an-isotropy












obtained



problems like








treated








WESSELING 1976)







10














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

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6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

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- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

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1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

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1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

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50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

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0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

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12979

12909

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(5)

where





(6)









laquo-b


water balance





(or drainage)







(6) to the aquitard





system)





follows

h - h

- mdash f i i gt

where









where









t s










q ( h b ) - mdash (9)

where







conduits |_ L J m





flux qfo




condition







equal to zero










consideration


and an-isotropy












obtained



problems like








treated








WESSELING 1976)







10














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

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6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

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cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

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000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

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0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

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001

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1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

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0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

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0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000





(or drainage)







(6) to the aquitard





system)





follows

h - h

- mdash f i i gt

where









where









t s










q ( h b ) - mdash (9)

where







conduits |_ L J m





flux qfo




condition







equal to zero










consideration


and an-isotropy












obtained



problems like








treated








WESSELING 1976)







10














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

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0000 0000 0000 0000 0000 0000 0000 0000

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- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

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003

003

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1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000


where









t s










q ( h b ) - mdash (9)

where







conduits |_ L J m





flux qfo




condition







equal to zero










consideration


and an-isotropy












obtained



problems like








treated








WESSELING 1976)







10














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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f j

M zgt -n

O)

_ D t 3 - T) C bulln c 0 3= 0 t o 0 x




t n t n


en - n

es

u bull-bull



-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

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t o H

^ 00 X

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3 m

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t n

t o

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z

3

3gt

C -lt Igt - t Ci




T


13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

009

003

003

0000 0000 0000 0000 0000 0000 0000 0000

1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000


flux qfo




condition







equal to zero










consideration


and an-isotropy












obtained



problems like








treated








WESSELING 1976)







10














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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T


13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

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- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

009

003

003

0000 0000 0000 0000 0000 0000 0000 0000

1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000





obtained



problems like








treated








WESSELING 1976)







10














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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T


13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

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- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

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1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

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000

001

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020

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1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

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50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

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0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

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0000

381

0000

0000

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0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000














h (x_v) ~ h f (xy) (11) a n n

where







1 to 0



a n






J J Q f(xy) dx dy = 0 i = 1 N (12)

1 1
























eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

009

003

003

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1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

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0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

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12979

12909

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eq (12)

12



b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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X - e - o

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LUX

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13 19 25 31 37

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13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

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50000

0000 182

1982 3862 1674 419 078 041

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0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


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-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

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0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

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12979

12909

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b a s i n s









-nodal point


xyz-space






can write

13


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

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- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

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1 7

13 19 25 31 37

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13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

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50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000


where

A = in




s y i n

Q f dx dy






6 7



h - h

+ B _iuumli i ^ + Q = o 44 t - t x4

n n-1

(14)


respectively

14










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

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= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

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m day

m






32








mdash Q II

aquifer I


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cases









33

A

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bull33

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iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

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1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

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0000 0000 0000 0000 0000 0000 0000 0000

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- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

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000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

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000

001

007

023

020

009

003

003

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1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000










in







(15)

where

h-i = in



n





value

15



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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z c 3



z n C v 3 Z

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z z + -u r

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s

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ro



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T


13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

009

003

003

0000 0000 0000 0000 0000 0000 0000 0000

1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000



k+1 k T7 k+1 k h = h + W (h - h )

in in in in (16)







Aquitards




by equation (11)





1 aauifer


aquifer






16


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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T


13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

009

003

003

0000 0000 0000 0000 0000 0000 0000 0000

1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000


is


1 n 2 n OSxd1


is

1 c 1




approximated by

te - h-

n n-1



] i 3 n 2 n 1 c t - t j j x d n n -1

T + 1








indicated



17

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


c z O E 0C 3 GLZ

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z

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3gt

C -lt Igt - t Ci




T


13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

009

003

003

0000 0000 0000 0000 0000 0000 0000 0000

1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000

M

1 A





SOLVE equation (14)

yes



PRINT of output

Ier node per layer

k = k+1

t = t + At


-gt-

( STOP )


parts A B C D and E









L 2 (




_L


bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

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- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

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0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

009

003

003

0000 0000 0000 0000 0000 0000 0000 0000

1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000









L 2 (




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bull












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

-1 day

m day

m






32








mdash Q II

aquifer I


-II--I i - i

aquifer






cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

Thi (m)

100

25

100

S s

001

0642

001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

Theiss solution


1 L

I l_ 001 003 0 0 5 Ol Q2

- I I 5 10 t - K t



Q =500 m day

k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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13 19 25 31 37 43

125008 125000 12S000 12S000 12S000 12S000 12S000 12S000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

0000 0000 0000 0000 0000 0000 0000 0000

- 000 - 000 -000 - 000 - 000 - 000 - 000 0000

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0000 000 000 000 000 000 000 000

000 0000 0000 0P100 0000 0000 0000 0000

1 7

13 19 25 31 37

12S000 125000 125000 125000 125000 12S000 125000 125000

0000 0000 0000 0000 0000 0000 0000 0000

- 001 -007 - 023 - 020 - 009 - 003 0000

0000 0000 0000 0000 0000 0000 0000 0000

000

001

007

023

020

009

003

003

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1 7

13 19 25 31 37 43

125000 125000 124998 124972 124895 124787 124647 124358

0000 0000 0000 0000 0000 0000 0000 0000

- 000 001 007 023 020 009 003

0000

0000 0000 0000 0000 0000 0000 0000

50000

0000 182

1982 3862 1674 419 078 041

0000 0000 0000 0000 0000 0000 0000 0000





0000

0000

0000

0000

LAYER NO 1




ACCUMULATION TERMS


0000

-000

0000

000

000

0000

-000

0000

000

000

LAYER NO 2




ACCUMULATION TERMS


0000

-378

0000

381

0000

0000

-055

0000

056

0000

LAYER NO 3




ACCUMULATION TERMS


0000

370

50000

49376

0000

0000

055

12979

12909

0000












Acknowledgement




41 I n p u t









which arise are


schematized




19


consideration















on the boundaries





place

elsewhere





from place to place








20



expected to occur








water system





















21


6-10 15

11-15 15

16-20 15

21-25 15

26-30 15

31-35 15

NUMEL

NUMLAY

MBAND

NTSPI

NTSPO

MAXIT

1-10 F103 ST

11-20 F103 DELT

21-30 F103 TMAX

31-40 F105 TOL

number of elements

number of layers






successive outputs


time step








time step




1- 5 15

6-10 15

11-15 15

16-20 15

KX(N1)

KX(N2)

KX(N3)

KX(N4)


element N




(see fig 8)


22




I-K 1103 SANG(NLY)

1i-iO MO3 C1(NLY)

21-30 F 103 C2(NLY)

21-4- 1-104 THI(NLY)




poundig 9)





point N








dummy variables)

1-10 110 KODE(N)

11-20 II0 NTERR(N)


paragraph 42)



23


21-30 FlOl X(N)

31-40 FlOl Y(N)

41-50 FlOl HGL(N)

51-60 FlOl HBSC(N)

61-70 FlOl LESC(N)










variable






layer LY






LY



LY


node N in layer LY


node N in layer LY



24





with LY = NUMLAY









different times



















executed

25





Remark






43 O u t p u t










of table FLOWUN





flow ARRED



B Data per layer






terms per layer

26



approximately zero








where















27




(22)

where




CL31

hmdash^ri --r2








28


160 180 X-axesim)




29









-1 Q bull - 200 m 3 d a y _ 1

k = 10 mday

d = 100 m

S = 2 y

S = 0 s







30




2 0 0

m 3 day 1

Q=200m 3 day

100-


N amp storage

I I ^ V - W 10

t (days)


functions of time


i _ Q s 4iTkD

dy = 4irkD

W(u) (23)

where

u = r2S

4irkDt








= s - 322D



[J

CIAT1

[T]

]

31





Drawdown s1(m)

10

10

10

Theiss solution


bdquo t= lO

10 10- 10 10 t r2(daym2)



Q

k

d

S y

s s

=

=

=

=

=

200

1

10

3 m

0

0

2

0

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m day

m






32








mdash Q II

aquifer I


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cases









33

A

-Jy

bull33

V 5

iumlff


34











II = 50 m day


aquifer I

aquitard

aquifer II

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100

25

100

S s

001

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001

k(mday )

1

0277

1



35









the 3-layered case

joo gt A 50

20

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1 L

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k = 10 mday

D = 100 m

S = 001 s




36










22














FEMSAT are treated










37














two ways



saturated system







38

VII REFERENCES





tables (in press)



pp 444













pp 803-816



No 4 pp 817-829



pp 108-120



pp 389




pp 63

39



40


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1 7

13 19 25 31 37

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LAYER NO 1




ACCUMULATION TERMS


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LAYER NO 2




ACCUMULATION TERMS


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LAYER NO 3




ACCUMULATION TERMS


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370

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49376

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12979

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NN31545.1077 X - WUR E-depot home

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Transcript of NN31545.1077 X - WUR E-depot home