1. No category

Journal of Science and Technology
3 (2), 2009, 161 - 164
©BEYKENT UNIVERSITY
A Numerical Solution of the One Dimensional
Groundwater Flow by Transfer Matrix Method
Rasoul DANESFARAZ1 and Birol KAYA 2
2
department of Civil Eng., University of Maragheh, Maragheh, Iran.
Department of Civil Eng., University of Dokuz Eylül, Buca, Izmir, Turkey.
daneshfaraz@yahoo.com
birol.kaya@deu.edu.tr
Received: 01.04.2009, Revised: 07.04.2009, Accepted: 13.07.2009
Abstract
Transfer matrix method is applied to one dimensional initial value problems.
The use of this method in hydraulic engineering is not widespread and only
limited studies are available. The method is applied to one dimensional
groundwater flow problems for the first time. In this study, a new approach via
transfer matrix method has been presented for the solution of one dimensional
groundwater flow representing parabolic type partial differential equation. A
confined aquifer of constant or varying boundary conditions is selected as on
illustrative example. The problem was solved for heterogeneous soil. Finite
element method, finite difference method and transfer matrix methods
proposed in this study were utilized for numerical solution. The outcomes of
the three methods are compared. Regarding to the outcomes of the transfer
matrix method, it can be revealed that the model is quite easily programmable
and the solution obtained is seldom divergent.
Keywords: Groundwater hydraulics, Transfer matrix, Confined
Unsteady flow, Parabolic differential equation.
aquifers,
Özet
Transfer
matrisi
yöntemi
tek
boyutlu
akım
problemlerine
uygulanmıştır.Yöntemin hidrolik problemlerine uygulanması yaygın olmayıp
bir kaç çalışmayla sınırlıdır. Yöntem tek boyutlu yeraltı suyu problemlerine bu
çalışmayla ilk defa uygulanmıştır. Yöntem, tek boyutlu parabolik kısmi
diferansiyel denklem içeren yeraltı suyu problemlerine uygulanmıştır
Örneklerde Akifer için sabit veya değişken sınır koşulları dikkate alınmıştır..
Problem heterojen zeminler için uygulanmıtır. Çalışmanın sonunda yöntemin
uygunluğunu göstermek üzere çözülen örneklerde Sonlu farklar ve sonlu
elemanlar yöntemleri de karşılaştırma amacıyla kullanılmıştır.. Örneklerden
yöntemin yeter uygunlukta sonuç verdiği gözlenmiştir.
229
A Numerical Solution of The One Dimensional Groundwater Flow by
Transfer Matrix Method
Keywords: Yeraltı suyu hidroliği, Taşıma matrisi, Basınçlı aquifer, Kararsız
akım, Parabolic denklemler.
1. Introduction
Investigating the studies related to the solution of groundwater flow in the
literature, underlined that the transfer matrix methodology is applied to 1D
groundwater flow problems for the first time. Some other related studies are as
follows.
Developed analytical solutions based on the Laplace transform method in
order to describe the rise and fall of the water table and the groundwater
velocities in finite and semi-infinite stream-aquifer systems. Sinusoidal and
asymmetric flood wave in the stream were considered to determine transient
solutions[5]. Employed a similar approach, with a discrete number of time
steps and a finite difference scheme of numerical integration, in order to solve
several transient unconfined groundwater problems. The axisymmetric flow to
a pumping well and the free surface behavior within a dam when the reservoir
water level is suddenly lowered, were determined with isoparametric elements
in the prior case and linear approximated elements in the latter case, for
different times[16]. Investigated a one-dimensional groundwater transport
equation with two uncertain parameters, groundwater velocity and longitudinal
dispersivity. They used Monte Carlo simulation to investigate the uncertainty
of one-dimensional groundwater flow field in non-uniform homogeneous
media[12]. Used the perturbation approach to analyze the groundwater flow
field by taking the spatial variability of the hydraulic conductivity into
consideration. Method of Taylor series expansion is more straightforward than
the perturbation method[2]. Fatullayev and Can used, an inverse problem is
formulated to determine water capacity of porous media. They used Finite
Difference Method for solution of parabolic equation[8]. Koutitas (1982)
solved nonlinear groundwater flows with the use of Finite Element and Finite
difference methods.
Allen and Murphy presented a finite element collocation method for variably
saturated flows in porous media[1].
Saenton et al. and Saenton developed an explicittly coupled finite difference
mass transfer model based on existing groundwater flow program,
MODFLOW[17,18].
Khebchareon and Saenton used Crank-Nicolson and Galerkin finite element
methods for 1D groundwater[10]. Srivastava, Serrano and Workman used
stochastic modeling of transient stream-aquifer interaction with the nonlinear
Boussinesq equation. They offered analytical solution for nonlinear
Boussinesq equation[20].
230
Rasoul DANESFARAZ and Birol KAYA
Time dependent parabolic equations are representative models of unsteady
flows in nature. These equations have main importance in engineering. The
equations explaining the 1D groundwater flow are parabolic partial
differential. The transfer matrix method utilized in this study is not widely
known in hydraulic engineering. In transfer matrix method, which is also
known as the method of initial values where, the aim is to convert problem of
boundary values into problem of initial value prevent new constant values and
to express equations of the problem by initial constant values. With the help of
this method, many problems in hydraulics can be solved. Limited applications
in the literature are listed below.
Baume, Sau and Malaterre expressed a need for using linear control theory
while pointing out the difficulty of complex hydraulic systems to control. They
obtained a reach transfer matrix by liberation of Saint Venant equations near a
steady flow regime. In this study Saint Venant equations were used in an
hydraulic application for the first time. However, the equations were used
between two given points only[3].
Litrico and Flomionused Transfer matrix method to obtain a frequency domain
model of Saint-Venant equations, linearized around any stationary regime,
including backwater curves[13,14].
Litrico and Flomioninvestigated the control of oscillating modes occurring in
open channels, due to the reflection of propagating waves on the boundaries.
They characterized the effect of a proportional boundary control on the poles
of the transfer matrix by a root locus which derived to an asymptotic result for
high frequencies closed loop poles[15].
Chaudry used Transfer Matrix Method for waterhammer calculation in
neglecting v.dv/dx in dynamic equation[4].
Shimada et al. introduced an exact method of assessing numerical errors in
analyses of unsteady flows in pipe networks. A pipe polynomial transfer
matrix had been developed and was analogous to transfer function matrices
used in free oscillation theory[19].
Daneshfaraz and Kaya have applied the transfer matrix method to 1D pipe
hydraulic problems. The authors have applied the method on two problems and
compared the solutions with FE method. They have also emphasized that the
method is easily programmable and stable[6].
Daneshfaraz and Kaya have used Transfer Matrix method for waves
propagation in open channel flow[7].
231
A Numerical Solution of The One Dimensional Groundwater Flow by
Transfer Matrix Method
2. Groundwater Flow Equations
The basic equations of groundwater flow are obtained by means of mass and
energy conservation principles combined with Darcy law [9].
Altering flow and soil properties in three directions in space, namely x, y, z;
and in the case of unsteady flow, the change in mass by time is:
àqx + dqy
dx
dy
S dh
+ dqz
dz
b
(1)
dt
In this equation, x, y and z are coordinates, t is time, S is the storage
coefficient, b is the aquifer thickness, h is the piesometric height, qx, qy and q z
are the unit discharge values in x, y and z directions, respectively. kx, ky and k z
are permeability coefficients in three directions. Using Darcy's law the
following equations can be written.
,
qx = kx
dh
(2)
y
dx
, dh
q
y = ky dy
dh
q
z
dz
(3)
(
=
4
)
Assuming that the soil is homogeneous and isotropic (Kx=Ky= K z =K); the flow
in a confined aquifer, the combination of Eqs 1, 2, 3 and 4 gives:
d2h d2h d2h
ô—+ ÎT + ÎT
dx 2 dy 2 dz 2
S dh
- • —
T
dt
(5)
In this equation T is the transmissivity capacity which is expressed as follows:
T=b.k
(6a)
In the heterogenous soil, the transmissivity capacity can be calculate as:
T=b1k1+b2k2+b3k3+
(6b)
Equation (5); written for flow in 3D homogeneous and isotropic pore media is
transformed into equation (7), in the case of 1D flow;
232
Rasoul DANESFARAZ and Birol KAYA
2
d h
dx
2
S dh
T ' dt
(7)
3. USE OF TRANSFER MATRIX METHODS
Different numerical methods can be used to solve parabolic equations. In this
study, finite difference, finite element and transfer matrix methods were used.
In finite difference solution, explicit approach is preferred. In finite element
method, Galerkin method is applied with quadratic elements[21].
In engineering problems, the calculation procedure is laborious and prone to
errors if the number of constants to be determined by boundary conditions is
very high. Therefore, it is of main importance to keep the constants of
integration at their minimum numbers. The transfer matrix method is very
efficient in this procedure. In this method, which is applicable to 1D problems,
it is aimed to transform the boundary value problems into initial value
problems.
The use of transfer matrix method enables the user to find out the solution of
many problems in a shorter time and an easier way, in comparison with other
methods.
For a while, by neglecting the time variable term:
a 2h
dx
2
0
(8)
The solution depending on x is obtained as follows;
dh
— =C '
dx
h = Q x + C2.
(9)
(10)
From boundary conditions, we have C i = (
dh
dx
)Q and C 2 = h g .
Substituting the constants and considering time derivative, the following
equations can be written for node(i+1) at time step n:
un
h
i+1
A
V
dh r x + h. n
un i+1
' dh
V dx
(11)
dx Ji
i+1
V dx
h
i
h
un-1
i+1 S A
— •Ax
T
Dt
233
(12)
A Numerical Solution of The One Dimensional Groundwater Flow by
Transfer Matrix Method
The unknown values at point (i+1) may be found from
,
n
Dx
( dh \
— v j nn
h
i+1 +
T Dt
v dx J i+1
^S
( dh
V dx Ji
h + 1 1 . £ .Dx
Dt T
(13)
Equations (11) and (13) can be written by using matrix form;
1
0
0
hin
hn
i +1
1 x
0 1
dh
+
(14)
_ £ Dx
1
( dd xh )ni
(
-£-A x
_ T Dt
Dt
T
dx
and finally, making the necessary arrangements Equation 15.1 can be obtained:
(15.1)
"
0
1
x
h
i+1
" hn '
n
—)i+1
+
=
£ Dx £ Dx
(dh
n
hT+l1 S D x
x + 1 (d h ) n
(
^~)i+1
'
Â
7
T
Dt
_
_ _T
_ dx ' _
_ Dt
T
dx
_
The transfer matrix of the system is
1
x
(15.2)
t
i
S Dx
S Dx
T
T Dt
Dt
x +1
and F presents the time effect on node point the time-dependent values in
addition node points
0
(15.3)
F+1 =
hn+11. S . Dx
Dt T
The equations (15.1), (15.2) and (15.3) can be rewritten as
(
+1
_ dx
^n
+1
_
II
" f
df
it
f
(
df
_ dx
'
)) n
i
+ Fi+1
(16)
_
The indeterminates at initial and final points of the system equation are
determined by boundary conditions. Afterwards, the indeterminates at other
points are determined by using these values.
234
Rasoul DANESFARAZ and Birol KAYA
4. Numerical Examples
The interval between the nodes (Ax) is taken as 10m. In order to achieve a
convergent solution,
TAt
S (Ax)2
value is ensured to be under 0.5. In this case,
At should be selected below 5 minutes[21].
The pressure aquifer; which is given in Figure 1. has a storage coefficient(S) of
0.002 and length (L) of 250m. The aquifer has different k values that taken as
it is given below:
0 <h<4m
^ k1=0.001 m/sec
4 < h < 10 m
^ k2=0.0004 m/sec
10< h < 15 m
^ k3=0.002 m/sec
The boundary conditions for the solution of Equation 7 which is written for 1D
flow are as follows.
*) at t=0 s, all points have a constant piesometer level (h(x, 0)=20m).
*) h(0,t)=-1.0135t+21 m if t<120 sec;
*) h(0,t)=16
if t>= 120 sec;
*) h(L, t)=16 m
x=0
x=25Iirri
Figure 1. Confined aquifer
The water level changes are calculated using transfer matrix method (TMM);
at t=60, t=120, t=200 and t=500 sec; which are compared with the finite
element method (FEM) and finite difference method (FDM) solutions of
Wang, Anderson(1995) for the same problem. The results are given in Table 1
and Figures 2, 3, 4 and 5. Differences between TMM and other methods are
given Figures 6 and 7.
235
A Numerical Solution of The One Dimensional Groundwater Flow by
Transfer Matrix Method
1=60 sec
FDM
18,764
19,645
19,938
19,992
19,999
19,999
19,989
19,903
19,482
18,226
16
FEM
18,764
19,644
19,947
19,995
20
19,999
19,990
19,908
19,494
18,243
16
TMM
18,764
19,690
19,931
19,986
19,996
19,994
2
19,874
19,502
18,373
16
1=120 sec
FDM
16
18,558
19,606
19,906
19,977
19,97!
19,884
19,602
18,920
17,679
16
FEM
16
18,521
19,612
19,915
19,982
19,976
19,891
19,608
18,919
17,675
16
TMM
16
18,694
19,602
19,883
19,959
19,952
19,865
19,604
18,977
17,764
16
1=200 sec
FDM
16
17,651
18,870
19,544
19,812
19,830
19,647
19,212
18,446
17,336
16
FEM
16
17,633
18,858
19,547
19,823
19,840
19,654
19,213
18,442
17,332
16
TMM
16
17,732
18,922
19,542
19,786
19,805
19,639
19,237
18,496
17,381
16
1=500 sec
FDM
16
16,889
17,679
18,287
18,658
18,765
18,608
18,206
17,598
16,839
16
FEM
16
16,887
17,676
18,285
18,657
18,765
18,608
18,206
17,597
16,838
16
TMM
16
16,903
17,700
18,309
18,676
18,780
18,623
18,223
17,613
16,848
16
Table 1. Results obtained in the case of constant boundary conditions
• • TMM
-FDM
-FEM
Figure 2. Results obtained from TMM,
FDM and FEM methods at t=60 sec
Figure 3. Results obtained from TMM,
FDM and FEM methods at t=120 sec
Figure 4. Results obtained from TMM,
FDM and FEM methods at t=200 sec
Figure 5. Results obtained from TMM,
FDM and FEM methods at t=500sec
236
Rasoul DANESFARAZ and Birol KAYA
Figure 6. The differences between TMM
and other methods at t=60 sec
Figure 7. The differences between TMM
and other methods at t=200 sec
At x=75 m., the variation of h values and the differences between TMM and
other methods are shown Figures 8 and 9.
When the results that taken by TMM are compared with the results that taken
with FEM and FDM, it will seen that the differences are too small. Therefore,
it will be possible to say that the results that have taken with the TMM are so
satisfactory. While flow gets closer to steady position, naturally it seems that
the differences between them are less and less obvious.
5. Conclusion
In this study, a new approach, the transfer matrix method is used for the
solution of 1D groundwater flow problems, which is described by parabolic
partial differential equations.
In the solution of Eq.7, the equation is integrated for the steady flow case, and
the time dependency is introduced at the nodes.
The illustrative examples given at the end of the study show that the method is
quite stable and convergent. An efficient and easily computerized method also
provides a fast and practical solution, because the dimension of the matrix
237
A Numerical Solution of The One Dimensional Groundwater Flow by
Transfer Matrix Method
explaining the 1D flow never changes. The transfer matrix method solution
can be applied to solve various 1D flow systems possessing different boundary
conditions.
In the case of the transfer matrix method; being independent of the number of
nodes and the computation time, the problem can be solved by using a matrix
of 2 X 2 which is a noticeable advantage.
As can be seen in the example, the results are quite similar with the outcomes
of the other methods. In this scope, the ease of application stimulates the use of
TMM.
As in other numerical solution methods, dx and dt calculation ranges are
essential for a steady solution. Selection of low dx values can increase the total
error because of consequent matrix multiplications.
238
Rasoul DANESFARAZ and Birol KAYA
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Finite