technical report no. 51 september - university of hawaiʻi
TRANSCRIPT
THE RESPONSE TO TIDAL FLUCTUATIONS OF TWO NON-HOMOGENEOUS COASTAL AQUIFER MODELS
by
John A. Williams and
Ta-Chiang Liu
Technical Report No. 51
September 1971
Partial Project Completion Report
of
ANALOG SIMULATION OF TIDAL EFFECTS ON GROUND WATER AQUIFERS
OWRR PROJECT NO. A-020-HI, GRANT AGREEMENT NO. 14-31-0001-3211
PRINCIPAL INVESTIGATORS: JOHN A. WILLIAMS, DOAK C. COX, AND L. STEPHEN LAU
PROJECT PERIOD: JULY 1,1969 to JUNE 30,1972
The programs and activities described herein were supported in part by funds provided by the United States Department of the Interior as authorized under the Water Resources Act of 1964, Public Law 88-379.
-------------------------------------------------------------
ABSTRACT
Mathemati~al and ele~tpie analog (R-C ~ip~uit) models fop two
one-dimensional non-homogeneous eoastal aquifeps have been developed.
The fipst is a semi-infinite aquifep having a dis~ontinuous ~hange in
pe~eability at a distan~e L fpom the ~oastline and the se~ond is an
aquifep of length L whose pepmeability ~hanges lineaply with distan~e
fpom the ~oastline and whose intepiop boundapy peppesents eithep a
~onstant head op a no-flow ~ondition. Both models wepe subje~ted to
a sinusoidal tidal input.
Results, in the fopm of gpaphs of amplitude and phase angle vs
position, show ere~ellent agpeement between the outputs of the math
emati~al and ele~tpi~ analog models. In the ~ase of a dis~ontinuous
pe~eability, these gpaphs indi~ate a positive op negative pefle~tion
fpom the dis~ontinuity fop a de~pease OP in~pease, pespeetively, in
the pe~eability. In the ~ase of the lineaPly vaPying pe~eability
model, the gpaphs of amplitude indi~ate that enepgy is attenuated at
a gpeatep pate neap the eoastline ~hen KL/KO <1 fop both types of
boundapy ~ondition. Gpaphs of the phase angle ape eon~ave downwapds
fop the ~onstant-head boundaPy ~ondition but fop the no-flow eondi
tion they erehibit a point of infle~tion whose position depends on t o and Kr/KO'
Ele~tpie analog model pesults show that foP the semi-infinite
aquifep no signifi~ant eppop will pesult fpom the eipcuit ~onfigupation
if a < A/50 and the ~ip~uit length is equivalent to 2A, fupthe~ope,
"lumped" ~omponents, i.e., a = 1../5, may be used to eretend the ~ip~uit
beyond re = -.41.. if measupements ape pestpi~ted to re > -.2A.
iii
CONTENTS
LIST OF FIGURES ..••••••••. e _ •••• iii It ...................................... v
LIS T 0 F TAB L ES. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • III • • • • • • • • • • • • • • • • •••••• v; i
INTRODUCTION .... •••• " •• '" ••.•••.. 1:1 ••••••••••••••••••••••••••••••••••••• 1
THEORETICAL CONSIDERATIONS ......•..•.............•••....•......•.•.•.• 1
The Mathematical Model •.•••••....•.•..•.••.•.••.•.•.••••.•.•...•.•• l The Electric Analog Model •....•.••.....••.••..•.••••..••••.•••.•.•• 8
EXPERIMENTAL CONSIDERATIONS ............••.•.•.......•.•.....•.••.•.•.• 9
Experimental Set-up ................................................ 9 Experimental Procedure •••.••...•••••.•.•..•.••.•.••••......••••..• l0
PRESENTATION OF THE RESULTS ...•.........•...•..•.....•.•......•....•. 13
DISCUSSION OF THE RESULTS AND THE CONCLUSIONS .......•..•.••.•.•.•.•.. 37
ACKNOWLEDGEMENTS •.••••.•••. ••••••••••.•.••••••••••.••••••••••••••••• • 43
REFERENCES ••..••••••..•••.••.•...••••••••.••••••••.•.••..•••••••....• 44
APPENDICES ••. •••••••..•..•.•..••••..•..•••.••••..••.••....•...•.•.•• • 45
LIST OF FIGURES
1. Photographs of the experimental apparatus •••••••.••.•••••.•...••• l1 2. Circuit diagrams for the discontinuous permeability and the
linearly varying permeability electric analog models .•••••.•.•••• 12 3. Amp 1 i tude and phase ang 1 e VB x/L for a coas ta 1 aqu ifer with a
discontinuous permeability: r 2 = 0.5 ............................ 15 4. Amplitude and phase angle VB x/L for a coastal aquifer with a
discontinuous permeability: r 2 = 0.8 ............................ 16 5. Amplitude and phase angle VB x/L for a coastal aquifer with a
discontinuous permeabi.lity: r 2 = 1.25 ........................... 17 6. Amplitude and phase angle VB x/L for a coastal aquifer with a
discontinuous permeability: r 2 = 1.5 .••..•.........•••......•••• 18 7, Comparison of the math model for an aquifer of infinite extent
(equations 8, with r = 1) with electric analog model results for severa 1 open-end ci rcui. t confi gurati ons .•.....•••••.....••.•....• 19
8. Amplitude and phase angle VB x/L for a coastal aquifer with a no-flow boundary condition at x = 0 and a linearly varying permeability: K = O.l(l+(1/2)x) ft/sec .......................... 21
v
9. Amplitude and phase angle vs x/L for a coastal aquifer with a no-flow boundary condition at x = 0 and a linearly varying permeability: K = 0.3(1-{1/6)x) ft/sec .......................... 22
10. Amplitude and phase angle vs x/L for a coastal aquifer with a no-flow boundary condition at x = 0 and a linearly varying permeabil ity: K = 0.1 (1+(3/4)x) ft/sec ..••..•..•......•...•••... 23
11. Amplitude and phase angle vs x/L for a coastal aquifer with a no-flow boundary condition at x = 0 and a linearly varying permeability: K = 0.4(l-(3/16)x) ft/sec ......................... 24
12. Amplitude and phase angle vs x/L for a coastal aquifer with a no-flow boundary condition at x = 0 and a linearly varying permeability: K = O.l(l+x) ft/sec ............................... 25
13. Amplitude and phase angle vs x/L for a coastal aquifer with a no-flow boundary condition at x = 0 and a linearly varying permeability: K = 0.5(1-(1/5)x) ft/sec .......................... 26
14. Amplitude and phase angle vs x/L for a coastal aquifer with a constant-head boundary condition at x = a and a linearly varying permeability: K = O.1(1+(1/2)x) ft/sec •••••...•.••..•.•• 27
15. Amplitude and phase angle vs x/L for a coastal aquifer with a constant-head boundary condition at x = 0 and a linearly varying permeability: K = 0.3(1-(1/6)x) ft/sec •.•....••••.•••••• 28
16. Amplitude and phase angle vs x/L for a coastal aquifer with a constant-head boundary condition at x = 0 and a linearly varying permeability: K = O.1(1+(3/4)x) ft/sec •...•.......•..••• 29
17. Amplitude and phase angle vs x/L for a coastal aquifer with a constant-head boundary condition at x = 0 and a linearly varying permeability: K = O.4(1-(3/16)x) ft/sec ...••.••.•.....•• 30
18. Amplitude and phase angle vs x/L for a coastal aquifer with a constant-head boundary condition at x = 0 and a linearly varying permeability: K = 0.1 (l+x) ft/sec ....•.••••••••.•••••••. 31
19. Amplitude and phase angle vs x/L for a coastal aquifer with a constant-head boundary condition at x = 0 and a linearly varying permeab-ility: K = 0.5(1-(l/5)x) ft/sec .................. 32
20. Comparison of math model results for aquifers having K = O.l(l+mx) ft/sec: m = 1/2, 0, -1/6, for to = 1.5 sec and a no-flow boundary condi ti on .......... " ................. " ..................... . 33
21. A comparison of the phase angle V8 position relation for several values of r and to for the discontinuous permeability model ..•••• 34
22. Comparison of electric analogs having an image circuit and an open circuit to simulate a no-flow boundary condition with math model DAMP for r = 0 and r = 1 .............................. 35
A-l. Sketch defining the aquifer models analyzed ...................... 49
vi
LIST OF TABLES
1. Summary of conditions for discontinuous permeability mode1 ..•.••.• 14 2. Summary of conditions for linearly changing permeability model •..• 14 3. Penetration lengths for semi-infinite aquifer in region 2 •.••.•.•• 39
vii
I NTRODUCTI ON
Model studies of the response to tidal fluctuations of isotropic
and homogeneous coastal aquifers having a simple boundary geometry have
been made by Williams, et aZ. (1970). These studies indicated that
diffusion theory could be used as the basis for analytical representation
of the physical phenomenon provided that the correct diffusion coeffi
cient was used. This result in turn justified the use of the electric
analog model as a research tool for the confined aquifer and for uncon
fined aquifers where the Dupuit assumptions hold.
The generally good results from the model studies provided the
motivation for this current research which is being conducted in several
phases. This report presents the results of Phase I, which have con
centrated on the development of mathematical and electric analog models
for two non-homogeneous, one-dimensional coastal aquifers, since it is
the non-homogeneous condition which is most likely to be encountered
in practice. The two types considered here are a semi-infinite aquifer
having a discontinuous change in permeability at a distance L from the
coastline and a finite aquifer of length L with a permeability which
varies linearly with distance from the coastline.
THEORETICAL CONSIDERATIONS
The Mathematical Model
Laminar flow through a porous media when streamline curvatures
in any vertical plane are small is represented by the expressions
Il • (Kzllh) :::: zSs
and Il • (Kzllh) :::: £'
ah at
ah 3t
(la)
(lb)
where equations (la) and (lb) apply to a confined and unconfined upper
water surface, respectively. In the above equations z(x, y) is the
thickness of the saturated zone, hex, y) is the piezometric head,
K(x, y) is the Darcy coefficient of permeability, £' is an apparent
porosity, and Ss is the specific storage (i.e., Wo (a + £/S) ).
If the flow does not vary in the y direction, if the thickness
of the saturated zone is a constant z for a confined upper surface or
2
-varies only slightly from an average value z for an unconfined upper
surface, and if the time variation is periodic, i.e.~ hex, t) = R (n (x) e iO't), then equations (la) and (lb) reduce to
~(Z K(x) !.. n(x)) -izO'Dn = ° (2) dx dx
Here, D is either Ss for the confined condition or e'/z for the un
confined condition. Equation (2) provides a basis for the investiga
tion of one-dimensional flows where the medium is non-homogeneous. A
list of definitions of the symbols used together with sketches of the
aquifer models analyzed is included in Appendix A.
First, consider the case where the Darcy permeability changes
suddenly. Thus, in region I, 0 ~ x ~ L, K = Kl and in region 2,
- 00 ::;. x ~ 0, K = K2 and the "coastline" is located at x = L. Equation
(2) must be solved in both regions with the conditions that the
piezometric head must be continuous at x = 0 and that conservation of
mass at x = 0 is satisfied. That is, in both regions 1 and 2
d2 n· iO' D. 0, j I or 2 - J _J = =
dx2 K.
(3a)
J
and at x = 0
nl = n2 (3b)
and
(KZ ~~ ) 1 = (KZ ~~ ) 2 (3c)
The additional boundary condition at x = L is
nl (L) = 1;0 sin O't = R (- 'r;; iO't) ~ oe (3d)
The solution to equation (3a) is known (Williams, et al.~ 1970)
to be
where
n. (x) J
C ~ia.x
= '1 e J + J
C. = a. +ib. J2 J2 J2
- ~ia.x C. e J
J2
D. a. = cr -1. = 21/J . 2
J K. J J
1 This latter definition is introduced for convenience in writing the lengthy equations which follow.
Thus, the expression for the piezometric head becomes
iO't h(x,t) = R (n(x) e )
= 1e~jX (a. cos~.x - b. sin~.x) + e-~jX (a. cos~.x Jl J ]I J J2 J
+ bj2 sin~jx) f cosO't - 1 e~jX (aj1 Sin~jX + bj1 COS~jX)
-~'X t + e J (-aj2sin~jX + bj2COS~jX)! sinO't (4)
3
In region 2 the position variable x may approach negative infinity,
hence, a22 = b22 = O. This leaves six constants of integration, all, a12,
bll, bl2, a21, and b21, to be determined from the three boundary conditions
given by equations (3b), (3c), and (3d). Application of these boundary
conditions generates the following set of equations:
all + al2 + 0 + 0 - ~l + 0 = 0 (Sa)
0 + 0 + bll + bl2 + 0 b21 = 0 (Sb)
all - al2 - bll + bl2 - ra21 + rb21 = 0 (Sc)
all al2 +bll - bl2 - ra2I rb21 = 0 (Sd)
e~IL C(~lL)all + e-~IL C(~lL)aI2 _ e~IL S(~IL)bll+ e-~lL S(~lL)b12 + 0 + 0 = 0 (Se)
e~IL S(~lL)all - e-~IL S(~IL)a12 + e~IL C(~IL)bll + e-~lL C(~lL)b12 + 0 + 0 = -l;o (5£)
where C - cosine, S = sine, and r = (K2z2/K 1z1)1/2 Equations (Sa) and
(Sb) follow from equation (3b), (Sc) and (5d) from (3c), and (Se) and
(Sf) from 3d).
Solving this system for the unknown coefficients yields the rela-
tions: 2
a21 = 1 all r + (6a)
b21 2
bll = r + (6b)
al2 = 1 + r all (6c)
bl2 1 - r
bll = + r
all = -l;oQ p2 + Q2 (6e)
bll = -l;oP p2 + Q2 (6f)
4
where P and Q are given by
p C (l/J 1 L) (el/JI L + 1 - r e-l/JI L) = + r
Q S(l/JIL) (el/JI L _ 1 - r e-l/JIL) = r+r
If equations (6c, d, e, f) are substituted into equation (4) the result
for region 1 is
hI (x, t) = {'(,ol (p2 + Q2)} {el/Jl (x+L) sin(l/Jl (x-L) + at)
+ (1 : ~) el/Jl(X-L) sinCl/Jl(x+L) + at)
_ (11 - r) -l/Jl(x-L) + r e sin(l/JI(x+L) - at)
_ ( 1
1 - r)2 _,I. L (x+L) } + r e 0/1 sin(l/JI(x-L) - at) • (7)
From equation (7) it is apparent that as r approaches 1, only a single
left-traveling disturbance of amplitude t;oel/Jl(x-L) remains, which is
consistent with previous results (Williams, et al.~ 1970) for uniform
aquifers of infinite length.
A second form of equation (7) which is more convenient to use and
to interpret may be formed by substituting equation (6) into equation (4),
and expressing h(x,t) in terms of an amplitude and phase angle. After
considerable algebraic manipulation, the result is
hl(x,t) = '(,op(x,r,L) sin(at - 8p), (8a)
where
p(x,r,L) = Ao + AIr + Azr2 + Aars + A4rlf
[(C2(l/JIL)+Shz(l/JIL))+2rCh(l/JIL)Sh(l/JIL)+r2(SZ(l/JIL)+Sh2(~1L))]~ (Bb)
with
Ao = (C2(l/JIX) + Sh2(l/JIX)) (C 2(l/JIL) + Sh2(l/JIL)) = (Ao,x) (Ao, L)
Al = 2Ch(l/JIX) Sh (l/JIX) (Ao , L) + 2Ch(l/JIL) Sh(l/JIL) (Ao,x)
A2 = 4Ch(l/JIX) Ch(l/J I L) Sh(l/JIX) Sh(l/JIL) + (A4,X) (Ao,L) + (A4, L) (Ao ,x)
As = 2Ch(l/JIX) Sh(l/JIX) (A4,L) + 2Ch(l/JIL) Sh(l/JIL) (A~,x)
Alf = (S2 (l/JIX) + Sh2(l/JIX)) (S2(l/JIL) + Sh2Cl/JIL)) = (AIf,x) CAlf, L)
(Be)
and Ao + AIr + Azr2 tan ep = (8d) So + SIr + Szrz
5
with
Ao = C(1/JIX) Ch(1/JIX) S(1/JIL) Sh(1/JIL) C (1/Jl L) Ch(1/JIL) S(1/JIX) Sh(1/JIX),
Al = C(1/JIX) S(1/JIL) Sh(1/JIX) Sh(1/JIL) + C (1/J lX) Ch(1/JIX) Ch (1/Jl L) S (1/Jl L)
- C(1/JIL) S(1/JIX) Sh(1/JIX) Sh(1/JIL) - C(1/JIL) Ch(1/JIX) Ch(1/JIL) S(1/JIX),
A2 = C (1/JIX) Ch (1/J d,,) S (1/J 1 L) Sh(1/JIX) C(1/JIL) Ch(1/JIX) S(1/JIX) Sh(1/JIL),
80 = S(1/JIX) Sh(1/JIX) S(1/JIL) Sh(1/JIL) + C(1/JIX) Ch(1/JIX) C(1/JIL) Ch (1/JIL),
81 = C(1/JIX) C(1/JIL) Ch(1/JIX) Sh (1/JIL) + C (1/J IX) C(1/JIL) Ch(1/JIL) Sh(1/JIX)
+ Ch(1/JIL) S(1/JIX) S(WIL) Sh(WIX) + Ch(WIX) S(WIX) S(WIL) Sh(WIL),
82 = C(1/JIX) C(1fJIL) Sh(WIX) Sh(WIL) + Ch(1/JIX) Ch(1/JIL) S(1/JIX) S(1/JIL), (Be)
where Sh and Ch represent the hyperbolic sine and hyperbolic cosine,
respectively.
Note that equations (7) and (B) apply not only to aquifers with a
discontinuous change in permeability, but also to unconfined aquifers which
have a discontinuous change in their average depth or a change in both
depth and permeability at the same point, since r = (K2zZ /KIZl)1/2 •
Equations (Sa to Be) may be checked by investigating several
limiting cases:
Case 1. Let r approach O. Then either K2 is very small compared
with KI or Z2 is very small compared with ZI. This corresponds to
an aquifer which is permeable on the range 0 ~ X ~ L and which has
a no-flow condition at X = O. Equations (Sa to Be) reduce to the
appropriate solution for these conditions (see Williams, et al.~
1970).
Ca8e 2. Let r approach infinity. Then either K2 is very large
compared to Kl or Z2 is very large compared to ZI. This corresponds
to an aquifer which is permeable on the range 0 ~ x ~ L and has a
constant-head boundary condition at x = O. Equations (8) reduce
to the appropriate solution for these conditions (see Williams, et
al., 1970).
Case J. Let r approach unity. This corresponds to a single, semi
infinite aquifer of uniform thickness and permeability, as mentioned
above. Equations (8) reduce to the appropriate solution for this
case.
It should be pointed out that equation (7) also reduces to the ap
propriate forms when the above-mentioned limits are taken.
6
If equations (6a and 6b) are substituted into equation (4) with
C22 = a22 + ib22 = 0, the result gives the piezometric head as a function
of (x,t) in region 2. That is,
h2(x,t) = {2to /[(1 + r)(p2 + Q2)]} {e~2(x+rL) sin(~2(x-rL) + crt)
+ 1 - r e~2(x-rL) sin(~2(x+rL) + crt)} . + r
(9)
The alternative form of this expression in terms of amplitude and phase
angles is
h2(x,t) = top(x,r,L) sin(crt - ep), (lOa)
where
p(x,r,L)
and
= [(C2(~lL)+Sh2(~lL))+2rCh(~1L)Sh(~1L)+r2(S2(~lL)+Sh2(~1L))]1/2 (lOb)
C(~2X) S(~lL) Sh(~1L) - C(~lL) Ch(~lL) S(~2X)
tan 8p = + r(C(~2x) Ch(WIL) S(~lL) - C(~1L) S(W2X) Sh(WIL))
C(~2X) C(~lL) Ch(~lL) + S(W2X) S(WIL) Sh(WIL) + r(C(W2x) C(WIL) Sh(WIL) + Ch(~lL) S(~2X) S(~lL))
(lOc) Again, several limiting cases can be investigated to check equations
(9) and (10):
Case 1. Let r approach infinity. Then h(x,t) vanishes as it
should for a "constant head" condition at x = O.
Case 2. Let r approach unity. Then h(x,t) again reduces to a
single left-traveling wave of amplitude soeW2 (X-L).
Case 5. Let r approach zero for x = O. Equations (8b, 8d) and
(lOb,lOc) yield consistent values for the amplitude and the phase
angles.
A second problem which can be analyzed is one where the permeability
varies linearly with position. If
K = KO + mIx = KO(l + mx) (11)
for a < x < 1 and if a new variable is defined by
t; = 1 + mx (12)
then equation (2) takes the form
~t; (t; dn ) icrD 0 dt; - m2K n =
0 (13)
7
Equation (13) is a modified Bessel's Equation and has a solution of the
form
(14)
where aD
a. = m2KO ' and Cl and C2 are complex constants.
The following boundary conditions
and
h(L,t) = ~osinot or net) = -i~o, t = 1 + mL (ISb)
require that a no-flow condition exist at the internal boundary (x = 0)
and that the piezometric surface vary as sinot at the "coastline" where
~o is the amplitude of the tidal change.
Imposing these conditions on equation (14) gives the following
expression for n:
n(x) = -i.i;;0[-Kl(i 1/
2v4a) Io(il/2~) + I1(i 1/
2/4(i) Ko(il/2~ )]
[II (i 1/ 2v'4a ) Ko(i 1
/ 2 /4a.l ) - Kl(i 1/
214a) IoCi l/ 2/4a.l )]
(16)
The functions Ko and 10 are complex quantities and may be expressed by
the equations:
IV Ci 1 /2X)
Ku(i 1 / 2X)
= Mu (x) e i8v (x)
= Nv(x) ei¢u(x) (l7a)
(17b)
Substituting equations (17) into equation (16), multiplying the results
by eiot and taking the real part of the product produces the desired
expression for the piezometric surface. That is,
h(~,t) = ~op(~,L) sin(ot + Bp) (l8a)
where
/
8
tan8p
(18c)
with
A = Mo Cl/J) No(ijid Ml(iji) Ndiji) B = M12(iji) NO(ijil) No(l/J)
c = MO(ijil) Mo(l/J) N12(ijj)
D = MO(ijil) Ml(iji) No(l/J) Nl(ijj) and l/J = ~,ijj= ~, ih = l4at
A constant head boundary condition at x = 0 (s = 1) requires that
n (1) = ClIo (i 1/2A'CX) + C2Ko (i 1/2~) = 0 (19)
Thus, for this boundary condition, the piezometric surface is expr(ssed
by equations (18), provided all the "III subscripts are replaced with
liD" subscripts.
In the preceding, h(x,t) is referenced from its equilibrium
surface.
The Electric Analog Model
The electric analog model is based on the analogy between electric
current flowing in a resistance-capacitance circuit and water flowing
through a resistive porous media. An analysis (see Williams" et at. ~ 1970 or Walton and Prickett, 1963) gives the following relation for the
time scale ratio between the two systems:
(20)
Here, K4 is the ratio of time in the hydraulic system to time in the
electrical system, S is the storage, T is the transmissability (ft 2 /sec),
R is the electrical resistance (ohms), C is the capacitance (farads)
and a is a length in the hydraulic model.
In the case of a linear variation of K with position the trans
missability becomes
T = zK = ZKO(l + mx), m = (KL/KO - l)/L (21)
9
where KO and KL are the permeabilities at the internal boundary and the
coastline l respectively, and z and L are the thickness and length of
the aquifer, respectively. Then, the value of T at anyone of N evenly
spaced points along the aquifer is
Tn = ZKo[ ] ; n = 1, , N (22)
Expressing zKO in terms of TNI equation (22) can be written
= [ 2N + mL(2n-l)]
TN 2N + mL(2N-l) ; n = 1, ••• , N
As R is inversely proportional to T, then
R [2N + mL(2N-l)] N 2N + mL(2n-l)
n = 1, .•• , N
(22a)
(23)
Therefore, the product RT in equation (20) is a constant and can be
taken as RNTN. Equation (23) may be used to calculate the Rn for a given
RN where Ru in the electric analog model corresponds to the resistance
encountered in a particular length LIN = a in the porous media.
For the discontinuous aquifer, the product RT is the same for both
regions 1 and 2; therefore
(24)
EXPERIMENTAL CONSIDERATIONS
Experimental Set-Up
The electric analog model of the aquifer of linearly varying
permeability consisted of a set of 20, single-turn, 0 to 300 ohm, 2 watt,
potentiometers mounted on a piece of 8 1/2" x 17" vector board. The
potentiometers were placed in series with a .Ol~f capacitor connecting
the junction between two adjacent potentiometers to ground, except at the
first and last (nineteenth) junction where .015~f capacitors were used.
The circuit was excited with a General Radio (#13l0-A) audio oscillator
and a Hewlett-Packard (#122a) dual-trace oscilloscope monitored the input
at the "coastline" and the response at a given point (xn = na). For a no
flow boundary condition at x = 0, the first half of the circuit (i.e., the
first ten potentiometers and corresponding capacitors) constituted the aqui
-------------- ...... --~~--
10
while the second half provided an "imagell aquifer in which that portion of the
input reflected from the no-flow boundary could be developed. For the
constant-head boundary condition, only the first half of the circuit was used.
The analog circuit for the discontinuous aquifer utilized the same
piece of hardware described above to model that portion of the aquifer
extending from x ;;; -L to the coastline at x ;;; L with the location of the
discontinuity at x ;;; O. This circuit was then extended to x ;;; -2L with
an array of components similar to those representing the aquifer on
-L < x < L and from -lOL < x < -2L with eight resistors and capacitors,
each representing the resistance and storage encountered in a length L
of region 2 in the aquifer. This circuit was also excited with the audio
oscillator and the input and output monitored on the dual-trace oscillo
scope.
Figure 1 presents photographs of the experimental apparatus. Figure
2 shows the circuit diagrams for the two electric analog models described
above.
Experimental Procedure
The experimental procedure for both models was essentially
identical. For the linearly varying permeability model, RN was
selected first, then the remainder of the variable resistances were
adjusted according to equation (23). Next, the time scale factor was
determined from equation (20) for a selected value of sYT and the
electrical frequencies calculated for a selected tidal period. The
final step involved adjusting the audio-oscillator to the calculated
frequency. and then observing on the oscilloscope the amplitude of the
input and the amplitude and the phase of the response at the desired
point (x ;;; an). n
For the discontinuous permeability model, Rz was fixed first and
Rl determined from equation (24). The remainder of the procedure was
identical to that described above.
For both models the values of N and a were 10 and 0.4 feet,
IThis method of simulating the no-flow boundary condition was used here for comparison with the simpler technique of leaving an open circuit at the interior end of the analog model (see Fig. 22). Both techniques give similar results, but the open circuit is more practical, especially when two or three dimensions are involved (see for example, Walton and Prickett, 1963).
FIGURE 1. PHOTOGRAPHS OF THE EXPERIMENTAL APPARATUS.
11
S2
~ S, ;: SINGLE POLE SWITCH
S2 i DOUBLE POLE SWITCH S3 i SINGLE POLE SWITCH
S4 1/ TWO POSITION SWITCH
CI " ~29 " 0.015 p.f.
C2 " ... " C8 " <;0 " ... " Cl8 " 0.01 p.f. C20 " ... " C28 " 0.01 p.f.
C30 " .. · " C37 " 0.1 p.f.
R21 " ... " R30" 200.n R31 " ..... R38" 2000.n
B~O : C
li
I _~~I ci ~I c'I ___ 3 VARIABLE PERMEABILITY MODEL, NO FLOW B. C. S, . CLOSED, S2 AND S3 OPEN.
Cg " O.QI p.f., C19 " 0.015 p.f.; Co OUT OF CIRCUIT R, TO R20 FROM EQUATION 23.
OSCILLOSCOPE i
o 0
X=-2L -3L -4L -5L -IOL
TO ANY NODAL POINT
EXTERNAL EXCITE
VARIABLE PERMEABILITY MODEL, CONSTANT HEAD B.C.
S, AND S2 OPEN, S3 CLOSED. Cg = 0.015 p.f., Co IN CIRCUIT. RI TO RIO FROM EQUATION 23.
DISCONTINUOUS PERMEABILITY MODEL
S, AND S3 OPEN, S2 CLOSED, Co OUT OF CIRCUIT. Cg " C'9 " 0.01 p.f.; RI TO RIO FROM EQUATION 24.
RII TO R20" 200.n.
FIGURE 2. CIRCUIT DIAGRAMS FOR THE DISCONTINUOUS PERMEABILITY AND THE LINEARLY VARYING PERMEABILITY ELECTRIC ANALOG MODELS.
I-' N
13
respectively, and the amplitude of the "tide" was set at 3.5 volts for
all tests.
PRESENTATION OF THE RESULTS
The results are presented as plots of the dimensionless ampli
tude p and the phase angle e versus the dimensionless position x/L. p
Experimental values of p were determined by taking the ratio of
the double amplitude of the voltage fluctuation at each x to the n
double amplitude of the input at the coastline. Both amplitudes were
scaled off the wave form traces on the oscilloscope. Similarly, the
phase angles were determined as the ratio of the distance between the
peaks of the input and the response fluctuations to the distance se
parating two successive peaks of the input. Distances on the oscillo
scope screen could be estimated to within ±l rom which corresponds to
an accuracy of ±O.l volts for the amplitudes and ±6 degrees for the
phase angles.
The mathematical models were evaluated on the IBM 360 digital
computer, which calculated p and e at predetermined values of x for p
each tidal period selected. Amplitudes and phase angles for the dis-
continuous aquifer, represented by equations (8b), (Sd) , (lOb), and
(lac) were calculated only to x = -L in region 2, and are plotted in
Figures 3 through 6 as the solid line curves. Figure 7 shows the effect
of extending the circuit to include the region x < -L and of the
"lumping" together of the individual resistors and capacitors corres
ponding to a length L of the media for r = 1 and a tidal period
to = 3 sec.
The results for the linearly varying permeability model repre
sented by equations (18b) , e18c), and (19) are presented as the solid
line curves in Figures 8 through 19. Figures 8 through 13 pertain to
the no-flow boundary condition at x = a and Figures 14 through 19 per
tain to the constant-head boundary condition at x = O. The correspond
ing electric analog results are indicated on these same plots by the
individual data points at each x. Since the amplitude curves for the n
constant-head boundary condition are so close together, only the electric
analog results for the two extreme periods used have been included in
Figures 14 to 19.
14
Figure 20 presents a comparison of the variation of the amplitude
and phase angle with x/L for KL/KO = 0.3/0.1, KL/KO = 0.1/0.1 and
KL/KO = 0.1/0.3 for to = 1.5 sec.
Figure 21 shows the effect on the phase angle of the value of r,
i.e., of the amount of positive or negative reflection at the section
where K changes discontinuously.
Finally. Figure 22 shows the effect of using both an open circuit
and an image circuit to simulate a no-flow boundary condition and com
pares these results with DAMP for r = 1. 0, a semi-infinite aquifer. and
for r = 0, an aquifer of length 2L with a no-flow boundary.
Summaries of the values of the significant parameters are given
in Tables 1 and 2. The following quantities and their indicated
TABLE 1. SUMMARY OF CONDITIONS:~ FOR DISCONTINUOUS PERMEABILITY MODEL.
Kl K2 Rl R2 to FT/SEC FT/SEC r2 OHMS OHMS K4 SEC FIG. NO.
0.1 0.05 0.5 100 200 16,000 0.75,1.5,3,6,9 3
0.1 0.08 0.8 160 200 10. 000 0.75,1.5,3,6,9 If
0.1 0.125 1.25 250 200 6,lfOO 0.75,1.5,3,6,9 5
0.1 0.150 1.50 300 200 5,333 0.75,1.5,3,6,9 6
0.1 0.1 1.0 200 200 8,000 3 7
.. THE INDICATED VALUES FOR THE FOLLOWING QUANTITIES WERE USED IN ALL CALCULATIONS: D E'/2 .01, a O.lf FT., L = 4.0 FT., C .01j.lf.
TABLE 2. SUMMARY OF CONDITIONS:: FOR LINEARLY CHANGING PERMEABILITY MODEL.
Ka KL m Ro R to B.C. AT X = 0 FT/SEC FT/SEC FT- 1 OHMS OH~S K4 SEC NO-FLOW CONST. HEAD
0.1 0.3 1/2 26lf 100 5517 1.5,3,6,9 FIG. 8 FIG. 14
0.3 0.1 -1/6 100 264 5517 1.5,3,6,9 II 9 II 15
0.1 0.4 3/4 268 80 5200 J..5,3,6,9 II 10 " 16
0.4 0.1 -3116 80 268 5200 1.5,3,6,9 " 11 II 17
0.1 0.5 1 280 70 lf762 1.5,3,6,9 II 12 " 18
0.5 0.1 -115 70 280 4762 1.5,3,6,9 " 13 " 19
0.1 0.1 0 264 26lf 6060 1.5,3,6 II 20
~~ THE INDICATED VALUES FOR THE FOLLOWING QUANTITIES WERE USED IN ALL CALCULATIONS: D = E~z = .01, a = 0.4 FT., L If.O FT., C = .01j.lf.
15
1.0
0.8
0.6
0.4
0.2
0.0 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
X/L
to MATH MODEL E.A. MODEL
360 (SEC.) DAMP 9 0
320 6 y
3 D
- 280 1.5 A
en 0.15 • I.LJ I.LJ IX 240 2 C) r = 0.5, KI = 0.1 FT./SEC. UJ A Q 200 -~ 160 ...J
Q. 120 1(3)
80
40
0 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0
X IL
FIGURE 3. AMPLITUDE AND PHASE ANGLE VS x/L FOR A COASTAL AQUIFER WITH A DISCONTINUOUS PERMEABILITY: r2 = 0.5.
16
1.0
0.8
0.6
0.4
0.2
360
320
C;; 280 IIJ IIJ ~ 240 IIJ 9 200 (.!)
j 160
a. 1(1) 120
80
40
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 X/L
!9 MATH MODEL E.A. MODEL (SEC.) DAMP
9 0
6 ~
3 a
1.5 C:.
0.75 •
2 r = 0.8, KI = 0.1 FT.I SEC.
0.6 0.4 0.2 0.0 -0.2 -0.4 -0.6 .. 0.8 -1.0
X IL
FIGURE 4. AMPLITUDE AND PHASE ANGLE VS x/L FOR A COASTAL AQUIFER WITH A DISCONTINUOUS PERMEABILITY: r2 = 0.8.
-en ft: a:: (!) IJJ 0 -(.!)
« ..J
Q. l(b
17
1.0
0.8
0.6
0.4
0.2
oL-~--~--L-~--~~~~~~~~~ 1.0 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1.0
320
280
240
200
160
120
80
40
0 1.0
t (SEC.)
9 6 :3 1.5
0.75
x/L
MATH MODEL E.A. MODEL DAMP
o
• 2
r = 1.25, K, =0.1 FT.lSEC.
0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1.0 x/L
FIGURE 5. AMPLITUDE AND PHASE ANGLE VB x/L FOR A COASTAL AQUIFER WITH A DISCONTINUOUS PERMEABILITY: r2 = 1.25.
18
-11.) lIJ lIJ 0:: (!) lIJ c -(!) <t -l
Q.
IQ)
1.0
0.6
0.4
0.2
0.8 0.6 0.4
to MATH MODEL
360 (SEC.) DAMP 9
320 6 3
280 1.5 0.75
240 r2 = 1.5, KI =0.1
200
160
120
80
40
0 1.0 0.8 0.6 0.4
0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 x/L
E.A. MODEL
0
y
CI
A
•
FT./SEC.
0.2 0 -0.2 -0.4 -0.6 -0.8 -1.0 X IL
FIGURE 6. AMPLITUDE AND PHASE ANGLE VB x/L FOR A COASTAL AQUIFER WITH A DISCONTINUOUS PERMEABILITY: r2 = 1.5,'
(/) !oJ !oJ 0:: (l) !oJ Q -(l) <t ..J
Q" Ill)
1 ~
160 . 0--
140 I- '\
120
100
80
60
40
20
MATH MODEL - DAMP: r 2 "1.0, Kit 1.0 FT ISEC, to" 3 SEC
CIRCUIT TO X -I., ai: 1.110
CIRCUIT TO X" -41., a" l ON -4l !i X !S. -I.
CIRCUIT TO
CIRCUIT TO
CIRCUIT TO
X .. -8l, a"l
X " -4l, a" I.
X " -IOL, a" l
ON -SI. :i X !S. -I.
ON -41. !i X .:s. -2l
ON -101. !i X :i -2l ! ! 1
o wkC~ ________ ~ ________ ~ ________ ~ ________ ~~ ________ L-________ ~ ________ ~ ________ ~ ________ ~~ ______ ~
ID O.S 0.6 0.4
1.0
0.9
0.8
0.7
0.6
0.2 o XiI.
-0.2 -0.4 -0.6 -0.8 -1.0
0... 0.5
0.4
0.3
0.2
0.1
! ! ! 1 ~~~r
o IL-______ ~----____ _L ________ ~ ________ ~ ______ ~ ________ _L ________ ~ ________ L_ ______ ~~ ______ _...I
1.0 0.8 0.6 0.4 0.2 o XiI.
-0.2 -0.4 -0.6
FIGURE 7. COMPARISON OF THE MATH MODEL FOR AN AQUIFER OF INFINITE EXTENT (EQUATIONS 8, WITH r = 1) WITH ELECTRIC ANALOG MODEL RESULTS FOR SEVERAL OPEN-END CIRCUIT CONFIGURATIONS.
-0.8 -1.0
""""co """""'""_"~" C"~~="," ."" .. _"."~'"="_" ~C·.d='""'"""'''''''' __ '''''r-__ _
'il>
0.6
0.4
0.2
o
100
-~ 80 LLI a: (,!) l&J o -(,!)
<t ...J
Q.
60
1(1) 40
20
1.0
21
0.8 0.6 0.4 0.2 o x/L
1 MATH MODEL E.A. MODEL (SEC.) VAMP- N F
9 0
6 y
3 a
1.5 A
K = 0.1 (1+ ~ X) FT./ SEC .• KL /KO. 3
o~~~~-L--~~--~~---L--L-~ 1.0 0.8 0.6 0.4 0.2 0
x/L
FIGURE 8. AMPLITUDE AND PHASE ANGLE VS x/L FOR A COASTAL AQUIFER WITH A NO-FLOW BOUNDARY CONDITION AT X = 0 AND A LINEARLY VARYING PERMEABILITY: K = O.1(1+(1/2)x) FT/SEC.
22
0 1.0 0.8 0.6 0.4 0.2 0
x/L
100 K = 0.3 (1- Y6 xl FT./SEC., KL 1Ko = I/S
to MATH MODEL EA. MODEL (SEC.) VAMP-NF
9 0 - 6 fa 80 LIJ 3 D: 1.5 (!)
~ - 60 (!) <C .J
I~ 40
20
o~--~----~--~~--~----~--~----~--~~--~--~ 1.0 0.8 0.6 0.4 0.2 0
x IL
FIGURE 9. AMPLITUDE AND PHASE ANGLE VB x/L FOR A COASTAL AQUIFER WITH A NO-FLOW BOUNDARY CONDITION AT x = 0 AND A LINEARLY VARYING PERMEABILITY: K = O.3(1-(1/6)x) FT/SEC.
23
0 1.0 0.8 0.6 0.4 0.2 0
JelL
100 MATH MODEL E.A. MODEL t
(SEC.) VAMP-NF 9 0 - 6 '" CJ) 80 LtJ :3 a
lIJ a:: 1.5 A (!) LtJ 0 - 60
K == 0.1 (1-1- \ X)FT./SEC. t KL 1KO == 4 (!) <[ -l
Q. 40 I~
-1:1
20
o~~~===== 1.0 0.8 0.6 0.4 0.2 o Jell
FIGURE 10. AMPLITUDE AND PHASE ANGLE VB x/L FOR A COASTAL AQUIFER WITh A NO-FLOW BOUNDARY CONDITION AT X = 0 AND A LINEARLY VARYING PERMEABILITY: K = O.1(1+C3/4)x) FT SEC.
24
0.4
0.2
0 1.0 0.8 0.6 0.4 0.2 0
x/L
K = 0.4 (1-316 x) FT. I SEC., KL. 1KO = 14
to MATH MODEL E.A. MODEL 80 (SEC.) VAMP-NF - 9 0
en UJ 6 UJ 3 0:: (!) 60 1.5 UJ Q -(!) a « a a
a ...J 40
a. 1(1)
20
o~---~----~------~---~------~--~----~-----~--~--~ 1.0 0.8 0.6 0.4 0.2 o
x/L
FIGURE 11. AMPLITUDE AND PHASE ANGLE VS x/L FOR A COASTAL AQUI FER WITH A NO-FLOW BOUNDARY CONDITION AT X = 0 AND A LINEARLY VARYING PERMEABILITY: K = O.4(l-(3/16)x) FT /SEC.
0.6
0.4
0.2
o 1.0
100
_ 80 fI) UJ UJ a:: (!) UJ 60 o -(!) « ..J
25
0.8 0.6 0.4 0.2 o X/L
to MATH MODEL E.A. MODEL (SEC.) VAMP - NF
9 0
6 • 3 a
1.5 A
K = 0.1 ( I -4- X) FT./SEC., KL 1KO= 5
X IL
FIGURE 12. AMPLITUDE AND PHASE ANGLE VB x/LFOR A COASTAL AQUIFER WITH A NO-FLOW BOUNDARY CONDITION AT X = 0 AND A LINEARLY VARYING PERMEABILITY: K - O.l(l+x) FT/SEC.
26
1.0 ~:::::--~i:-====:==:::===:==~e:==:a:===:e==~a~=:::::o ==~o
0.8
0.6
0.4
0.2
0.0 1.0 0.8 0.6 0.4 0.2 0.0
X IL
100 to MATH MODEL A.MODEL (SEC.) DAMP
9 0
80 6 ., 3 a - 1.5 A (f)
LLJ K = 0.5 (1- ~5X) FT. SEC., LLJ
0: 60 (!) LLJ 0 -(!)
« 40 a ...I
Q. IQ) a
• • ., 20
O~--~-----L-----~--~----~----~----~--~----~-----1.0 0.8 0.6 0.4 0.2 0.0
x/L
FIGURE 13. AMPLITUDE AND PHASE ANGLE VS x/L FOR A COASTAL AQUIFER WITH A NO-FLOW BOUNDARY CONDITION AT X = 0 ~~D A LINEARLY VARYING PERMEABILITY: K = O.5(1-(1/S)x) FT/SEC.
27
1.0
0.8
0.6 to = 6, 9 SEC.
0.4
0.2
0 1.0
X IL
K = 0.1 (I + 1/2 x) FT. SEC., KL IKO =3
40 to MATH MODEL E.A.MODEL - (SEC.) VAMP-CH CI) l1J 9 0 l1J
6 a: C!)
30 3 l1J 0 .1.5 D. -C!) « ..J
20
• &- to = 3 SEC .
10 to II 6 SEC.
o~~~~o~o:== 1.0 0.8 0.6 0.4 0.2 0
x/L
FIGURE 14. AMPLITUDE AND PHASE ANGLE VS x/L FOR A COASTAL AQUIFER WITH A CONSTANT HEAD BOUNDARY CONDITION AT x = 0 AND A LINEARLY VARYING PERMEABILITY: K = O.I(I+(1/2)x) FT/SEC.
28
-V) UJ LLl a:: (,!) LLJ C -~ ...J
~
1.0
0.8
0.6
to = 3 SEC.
0.4
0.2
o ~--~----~--~~--~----~--~----~----~--~--~ 1.0
40
30
20
10
0.8 0.6 0.4 x IL
K = 3.0 (I - 1/6 X ) FT ISEC t KL IKO = 1t3
to MATH MODEL E.A. MODEL (SEC.) VAMP-CH
9 0
6 3 1.5
9
0.2 o
to=3 SEC,_
to =6 SEC,
o e
O~~~-----~----L---~------~--~~--~-----~--~----~ 1.0 0.8 0.6 0.4 0.2 o
x/L
FIGURE 15. AMPLITUDE AND PHASE ANGLE VS x/L FOR A COASTAL AQUIFER WITH A CONSTANT-HEAD BOUNDARY COf\IDITION AT x = 0 AND A LINEARLY VARYING PERMEABILITY: K = O.3(1-(1/6)x) FT/SEC.
29
1.0
0.8
0.6
0.4
0.2
o ~ __ ~ ____ ~ __ ~ ____ -L ____ ~ __ ~ ____ ~ ____ ~ __ ~ __ ~
1.0 0.8 0.6 0.4 0.2 x/L
K = 0.1 (I +\ X) FT.lSEC., KL 1KO = 4
- 40 en lIJ to MATH MODEL E.A. MODEL lIJ D: (SEC.) VAMP-CH C!) 9 0 lIJ 0 30 6 - 3 C!) 1.5 <t ..J
20 Q.
1<1) to = 3 SEC.
10 to = 6 SEC.
0
0 1.0 0.8 0.6 0.4 0.2
X IL
FIGURE 16. AMPLITUDE AND PHASE ANGLE VS x/L FOR A COASTAL AQUIFER WITH A CONSTANT-HEAD BOUNDARY CONDITION AT x = 0 AND A LINEARLY VARYING PERMEABILITY: K = O.1(1+C3/4)x) FT/SEC.
o
0
30
1.0
0.8
0.6
0.4
to = 1.5 SEC.-~~
0.2
o ~ __ ~ ____ ~ __ ~~ __ ~ ____ ~ __ ~ ____ ~ ____ ~ __ ~ __ ~W
-(J) LLI LLI ct: (!) LLI o -Q.
'<1)
1.0
40
30
20
10
0.8 0.6 0.4 0.2 o x/L
K = 0.4 (1-~6 x) FT.lSEC., KL 1KO = 1/4
to MATH MODEL E.A. MODEL (SECr) VAMP-CH
9 0
6 3 1.5
to = 3 SEC.
to =6 SEC.
o~~======:===o == 1.0 0.8 0.6 0.4 0.2
x IL
FIGURE 17. AMPLITUDE AND PHASE ANGLE VB x/L FOR A COASTAL AQUIFER WITH A CONSTANT-HEAD BOUNDARY CONDITION AT x = a AND A LINEARLY VARYING PERMEABILITY: K = O.4(1-C3/16)x) FT/SEC.
o
31
1.0
0.8
0.6
0.4
0.2
o~--~----~----~--~----~--~----~----~--~--~ 1.0 0.8 0.6 0.4 0.2 o
x IL
K = 0.1 (I + x) FT./SEC., KL/KO = 5
- 40 CJ)
to MATH MODEL E.A. MODEL IJJ IIJ (SEC) VAMP-CH a:: (!) 9 0
IIJ 30 6 Q - 3 (!) 1.5 <[ ..J 20 a.
1(1) to = 3 SEC.
10
t = 6 SEC. 0
0 1.0 0.8 0.6 0.4 0.2 0
x/L
FIGURE 18. AMPLITUDE AND PHASE ANGLE VS x/L FOR A COASTAL AQUIFER WITH A CONSTANT-HEAD BOUI\IDARY CONDITION AT x = 0 AND A LINEARLY VARYING PERMEABILITY: K = O.1(1+x) FT/SEC.
32
0.8
to = 6,9 SEC. 0.6
0.4
0.2
o----~----~--~~--~--------~----~----~--~--~ 1.0 0.8 0.6 0.4 0.2
x IL
K = 0.5 (I - 'l5 xl FT.lSEC., KL 1Ko = ~5
40 to MATH MODEL E.A. MODEL - (SEC. 1 VAMP-CH
CJ) 9 0 LLJ LLJ 30 6 a::: 3 (!) LLJ 1.5 0 -(!) 20 c( ..J
Q. = 3 SEC. 1(1)
10
0 0
0 1.0 0.8 0.6 0.4 0.2
X IL
FIGURE 19. AMPLITUDE AND PHASE ANGLE VS x/L FOR A COASTAL AQUIFER WITH A CONSTANT-HEAD BOUNDARY CONDITION AT x = 0 AND A LINEARLY VARYING PERMEABI LITY: K O. 5Cl-Cl/5)x) FT /SEC.
o
0
1.0
0.8
0.6
0.4
33
" .... .... ..........
.......•.......
"", " ' ...... ............
...... ........
• It ........... " ..................... .
---- ........ --... ..... --------------_.
o----~~--~----~----~----~--~-----L----~----~--~
100
Cii 80 lLI lLI a::: ~ e 60
C)
« ..J
Ie:- 40
20
1.0 0.8 0.6 0.4
x/L
K = KO O-l-mX) FT/SEC t
to = 1.5 SECt NO-FLOW
~ .... /
/// ....... . // ..... .
// ....
,,'
, //':: ............. . , ,. .. -
", .. -"," ..... ". .. -
.. " ---
....... 'it';;' •••••
~ .... , . . ' '
.. .'
0.2 o
. . ..... ........... .... .' .'
K L 0.1 = KO 0.1
KL 0.3 - = ----KO 0.1
o ~ __ ~ ____ ~ ____ ~ ____ L-__ ~ ____ ~ ____ L-__ ~ ____ ~ __ ~
1.0 0.8 0.6 014 0.2 X IL
fIGURE 20. COMPARISON OF MATH MODEL RESULTS FOR AQUIFERS HAVING K = O.l(l+mx) FT/SEC: m = 1/2, 0, -1/6, FOR to = 1.5 SEC AND A NO-FLOW BOUNDARY CONDITIOI\!,
o
34
..... (() I.d I.d 0:: (,!) I.d .e (,!) c .J
a. Ictl
240
aI •
0 Q 220 II /I
aI aI ... ...
200
180
160
r2: 104
140
120
r2 = CD 100
0
80 II
t\I ...
60
I,
40
20
0~~~ __ L-~~ __ L-~-L __ L-~-L __ L-~-L~ 1.0 0.8 0.6 0.4 0.2 o -0.2 -0.4
X IL
FIGURE 21. A COMPARISON OF THE PHASE ANGLE VB POSITION RELATION FOR SEVERAL VALUES OF r AND to FOR THE DISCONTINUOUS PERMEABILITY MODEL.
-en IJJ IJJ 0:: (!) IJJ 0 -(!) « .J
Co 1(1)
35
1.0
0.8
0.6
0.4
0.2 -~~"----+-----t) --------
O~ __ -L----L---~----~--~----~--~----~--~----~ 1.0
180
160
140
120
100
80
60
40
20
0 1.0
0.8 0.6 0.4 0.2 0 -0.2 X IL
2 --- DAMP: r = 1.0, i .•. SEMI -INFINITE AQUIFER
2 --- DAMP: r = 0, i.e. FINITE AQUIFER, LENGTH 2L
t:;. E. A. MODEL: a = L 110, Rn = 200 .0., OPEN CIRCUIT
v" II Q = L/5, Rn= 200.0., IMAGE CIRCUIT t:;.
..Jr---
0.8
I( = 0.1 FT ISEC I to = 3 SEC t L = 4.0 FT ~ ............. ,,"
"
0.6 0.4
l:!."",""/">" -'J.}l
.&l
//
;,/ //
0.2 0 -0.2 -0.4 -0.6 -0.8 -1.0 x/L
FIGURE 22. COMPARISON OF ELECTRIC ANALOGS HAVING AN IMAGE CIRCUIT AND AN OPEN CIRCUIT TO SIMULATE NO-FLOW BOUNDARY CONDITIONS.
36
values were conunon to all calculations: D::::: sri = .01, a = 0.4 ft.,
L = 4.0 ft., C ::::: .Ol~f. The length L and the tidal periods were
selected to facilitate direct comparisons of the results obtained in
the reported study with those from future tests ~sing the Hele-Shaw
flow analog. However, the results can be projected to give the response
of an aquifer characteristic of real field conditions. That is, if
a = 750 ft., S = .002, Z ~ 200 ft., and K = 1000 ft/day for each 0.1
ft/sec used in the above calculations, then the amplitude and phase
angle curves for the 3 sec and 1.5 sec period tides will represent
the response of this characteristic aquifer to the diurnal and semi
diurnal tidal components, respectively.
The computer programs are included as Appendix B. DAMP (1 and
2), or discontinuous aquifer-model permeability, was the program used
to evaluate equations (Sb) , (Sd), and (lOb), (IDe) for regions I and
2, respectively, while VAMP, or variable aquifer-model permeability,
was the program used to evaluate equations (lSb) and (18c). The de
signations NF and CH indicate the no-flow and the constant-head
boundary conditions, respectively. For the latter boundary condition,
equations (lSb) and (18c) are valid provided all the "Ill subscripts
are replaced with "0" subscripts, i.e., only zero-order functions are
required.
The solutions to the modified Bessels equation can be expressed
as
Hence, for VAMP, sub-routines were written for ber x, bei x, ker x, v v v and keivx, using the first twenty terms of their respective series
expansions. The functions Mv' 6v ' and Nv ' ~v were subsequently eva
luated as the modulus and argument of the complex quantities Kv and
Iv' The series expansions failed to converge to the correct value
for arguments greater than about five,l as a result of the rapid in-
!This does not limit the use of the program for field data, since real tidal components have periods of sufficient length to keep the argument less than five.
crease in the size of each successive term in the series and the
limited capacity of the computer to represent such large numbers
(i.e.~ e 170 is the maximum for the IBM 360 computer).
DISCUSSION OF THE RESULTS AND THE CONCLUSIONS
There generally good agreement between the results of the
mathematical and the electric analog models shown in Figures 3 through
6 and 8 through 19. This good agreement verifies the computer pro
grams DAMP and VAMP. But more important, the good agreement verifies
the mathematical models as representing the response of the two given
non-homogeneous aquifers to tidal changes since such resistance
capacitance circuits have been shown, by Williams, et al. (1970), to
model periodic flows in a porous media, provided the appropriate
diffusion coefficient is used to determine the time scale factor.
An examination of the discontinuous aquifer model results shown
in Figures 3 through 6 indicates clearly the discontinuity in the
permeability since the slope of both the amplitude and the phase
37
angle curves exhibit a discontinuous change at x ; O. In comparing
results for the extreme values of r used (Figures 3 and 6), the steep
ness of both the amplitude and the phase angle curves is seen to increase
across the discontinuity when r2 = 0.5 because of the increased resis
tance to flow in region 2 of the aquifer. Also, the phase angle
curves become slightly concave downwards in region 1 as x = 0 is approach
ed from the coast, indicating that some positive reflection occurs as
the disturbance moves into the less permeable or shallower region 2.
However, when = 1.5, the steepness of both the amplitude and the
phase angle curve decreases discontinuously when passing from region
1 into region 2. The phase angle curve becomes slightly concave up-
wards in region 1 as x = 0 is approached, indicating a partial negative
reflection as the disturbance moves into the more permeable or deeper
region 2. The variation of phase angle with position in re-
gion 2 is for both cases, as it should be for a semi-infinite
aquifer.
The amplitude p, determined by the electric analog, is as much
as 10 to 15 percent greater than that predicted by the mathematical
model for the range -1 2 x/L 2 -0,3 for the longest tidal period of
· --.. ~ ---~ ........ ---
38
9 seconds. This error appears to increase as r increases, i.e.~ as
the resistance to flow in region 1 increases relative to that in region
2, and becomes evident in the amplitude VB position curves for a tidal
period of 6 seconds when r2 > 1. This error is most likely the result
of modeling an aquifer of infinite extent in the x-direction by a re
sistance-capacitance circuit of finite length. In particular, the
longer period fluctuations penetrate further into the aquifer model
and the end of the finite length analog circuit provides a no-flow
boundary and a subsequent partial positive reflection which increases
the amplitude of the free surface fluctuation above that predicted by
the mathematical model. When r2 > 1.0, region 2 offers relatively
less resistance to the flow than region 1, thereby permitting the
shorter period fluctuations to penetrate further into region 2 with
a resulting increase in the amount of energy reflected from the open
circuit end of the analog model.
Figure 7 indicates that the significant factor in the accuracy
of the electric analog model for the semi-infinite aquifer is the ex
tention, to x = -2L, of the fine grid spacing, a = Lila, when measure
ments are required for x > -L. Also, no significant difference in the
accuracy is observed on extending the circuit from x -4L to x = -8L
with "lumped" components, i.e • ., a = L, when a = L/lO is used only to
x = -L. However, when a = Lila is used to x = -2L, then ext~nding the
circuit to x = -4L with lumped parameters reduced the error between
prediction and measurement to approximately one-third of that observed
for the circuit of the same length but us x = L/lO to x = -L. The
extention of the circuit from x = -4L to x = -lOL, with lumped components
and where a Lila for x > -2L is seen to have relatively little in
fluence on the amplitudes but does noticeably improve the agreement
of the measured phase angles with the mathematical model predictions.
From the above discussion it is clear that the accuracy of the
electric analog for the semi-infinite aquifer depends on both the finite
difference grid spacing 1 and the length of the circuit. The rule for
lIt can be shown that errors caused by discreetizing the problem space depend upon the grid spacing and the fourth and higher order derivatives of the solution function and are zero if these higher derivatives are z~ro. Since hex, t) has non-vanishing higher derivatives these errors are not zero for the tidal response problem. See Karplus, 1953.
extending the circuit is simply to add resistors and capacitors out to
a point ~here changes in the voltage can no longer be detected. Since
the decay in amplitude at a distance of one penetration length, i.e."
39
, V4 Kzto f . . f' . . f . 1 2.11' b 99 8 A = £1 or a sem1-1n 1n1te aqu1 er 1S - e or a out • percent,
it would seem reasonable that a circuit of length equivalent to or great
er than one penetration length would be more than adequate. Penetration
lengths for the tidal periods tested varied from 7.0 1 to 41.5 1 and are
recorded in Table 3. The overall circuit length for region 2 was
Kz
TABLE 3. PENETRATION LENGTHS FOR SEMI-INFINITE AQUIFER IN REGION 2.
35.5 ~:: ,),.2 eFT) r2 FT/SEC FT/SEC1 / 2 to ;:: .75 SEC to ::: 1.5 SEC to 3 SEC to ::: 6 SEC to = 9 SEC
.5 .05 7.92 7.03 9.7 14.06 19.4 23.8 10%
.8 .08 10.02 8.9 12.3 17 .80 24.6 30.3 8% 13%
1.25 .125 12.58 11.18 15.3 22.36:::: 30.6 37.6 17% 16% 17%
1. 50 .150 13.80 12.22 16.85 24.44 33.7 41.5 17% 16% 17%
1.00 .100 11.20 9.94 13.75 19.88 27.5 33.6
"
~~~~
A2 :: y4'JTzzKzto/E:' C35.5(K;){tO)
IN THE COLUMNS RECORDING THE PENETRATION LENGTH, AJ THE UPPER NUMBER IS A AND THE LOWER NUMBER IS THE PERCENTAGE ERROR BETWEEN THE PREDICTED AND MEASURED AMPLITUDES.
equivalent to 10L = 40 1• Thus, it seems 1ike1y'that the errors in the
amplitude-position curves mentioned above are basically the result of
not using the spacing of a = LI10 = 0.41 for x < -2L. It is of interest
to note that when A > 20 1, measured amplitudes begin to eXCeed pre
dicted amplitudes in the neighborhood of x = -L. This implies that for
the semi-infinite aquifer, no significant error will result from the
circuit configuration if a ~A/50 and the circuit length is equivalent
to 2A; furthermore, "lumpedll components, i.e." a = A/5, may be used to
extend the circuit beyond x = .4A if measurements are restricted to
x > -.21...
Figures 8 through 13, representing the linearly varying permeabi
lity model with a no-flow boundary condition, reveal that a greater
40
attenuation of the tidal fluctuation is produced by an aquifer with
permeability that increases rather than decreases with distance from
the coast', i. e. J when KL/KO < 1 (see also Fig, 20). This is the re
sult of having a disproportionately greater portion of the energy
attenuated in the less permeable region near the coastline. The phase
angles at any given position are likewise greater when the permeability
increases rather than decreases with distance from the coast. The phase
angle curves for the case KL/KO < 1 are generally concave downwards,
while those for the case KL/KO > 1 are concave downwards over the in
terior half of the aquifer but have a point of inflection at about
x/L :::: 0.5. This point of inflection shifts towards the coastline as
the ratio KL/Ko becomes smaller for a given tidal period (e.g. compare
curves in Figs. 8, 10, and 12 for to :::: 1.5 seconds), as well as when
the tidal period becomes larger and the ratio KL/KO remains fixed.
This shift is the result of the reflected fluctuation being able to
penetrate more closely to the coastline because of the relatively less
resistive medium in the vicinity of the boundary for the former condi
tion and because of increased energy in the longer period fluctuation
for the latter condition.
The effect of the variable permeability when a constant-head
boundary condition is applied is Clearly visible in the amplitude
curves of Figures 14 through 19. When the permeability increases with
distance from the coast, these curves are concave upward, indicating
a decreasing rate of attenuation as the tidal-generated fluctuation
propagates into the interior of the aquifer. When the permeability
decreases with distance from the coast, the amplitude curves are
concave downward and the rate of attenuation increases as the fluctua
tion propagates into the interior.
The phase angle curves are generally concave downward, have no
points of inflection, and give phase angles which are smaller than those
for the no-flow boundary condition. The phase angles at the interior
boundary are essentially equal regardless of whether the permeability
increases or decreases, provided the change is linear and between the
same extreme values of K.
The constant-head boundary produces a negative reflection and
would represent a node in a standing-wave system, whereas the no-flow
41
boundary produces a positive reflection and would correspond to an
anti-node in the same standing-wave system. Equations (Sd) and (IDe)
have been used as bases for studying the effect on the phase angle of
the partial positive and negative reflections which exist between these
two limiting conditions of a positive reflection (r2 = 0) and a negative
reflection (r2 = 00), and the results are seen Figure 21. From this
figure, for to = 0.75 seconds, it is clear that partial positive and
negative reflections are characterized by phase-angle vs position
curves which are concave downward and concave upward, respectively, in
the neighborhood of the reflecting boundary (see also Fig. 3 to 6).
In the limiting case of a complpJ, ; negative reflection, the point of
inflection is located at the reflecting boundary and the phase angle
curve is entirely concave downward (see also Fig. 14 to 19). The
shift of the point of inflection toward the coastline with increasing
period for a positive reflection is evident on a comparison of the
curves for to = .75 sec and to = 9 sec but having the same values of
r2 < 1.
It is worth noting that in the variable permeability model the
incremental change used for K(x) was 10 percent, and that errors of
less than 10 percent between measured and predicted values of amplitude
and phase angle were observed.
In light of the results as discussed above, the following con
clusions are made:
1. The mathematical model developed for the one-dimensional
semi-infinite coastal aquifer having a discontinuous change in per
meability, or in both permeability and depth for the free surface aqui
fer,at a distance L from the coastline is valid provided that the
assumptions delineated for equation (2) are satisfied. The mathema
tical model expressed by equations (8) and (10) and the corresponding
computer program, DAMP land 2, may be used to study the response of
this aquifer to sinusoidal tides of a given period to' In particular:
i. The steepness of both the amplitude and the phase-angle
curves increase of decrease discontinuously at the section where the
permeability decreases or increases discontinuously.
A partial positive (r2 < 1) or negative Cr2 > 1) re
flection of energy takes place at the discontinuity. This partial
42
reflection is characterized by the curvature of the phase angle VB
position curve in the neighborhood (x/L 2 0) of the reflecting boundary:
concave downward for a positive reflection and concave upward for a
negative reflection. In the limiting case of a complete negative re
flection, the point of inflection coincides with the reflecting
boundary and the entire phase angle VB position curve is downward in
region 1.
2. The mathematical model developed for the one-dimensional,
finite coastal aquifer (confined or unconfined) of length L having a
permeability which varies linearly with distance from the coastline
is valid, provided the assumptions delineated for equation (2) are
satisfied. The mathematical model as expressed by equation (18) and
the corresponding computer program VAMP-CHI or VAMP-NF may be used to
study the response of this aquifer to sinusoidal tidal fluctuations of
period to. In particular:
i. The amplitude VB position curves for a constant
head boundary condition are concave downward for a permeability which
decreases with tance from the coast and are concave upward for a
permeability which increases with distance from the coast.
ii. The amplitude VB position curves for a no-flow
boundary condition flatten out more rapidly and indicate greater
attenuation when the permeability increases rather than decreases with
distance from the coast.
iii. The phase angles at a constant-head boundary are essen
tially the same regardless of whether the permeability increases or
decreases, provided the change is linear and between the same extreme
values of K.
iv. The phase angle VB position curves for the no-flow
boundary condition are concave downward near the reflecting boundary
with a point of inflection which shifts towards the coast when either
the tidal period increases or the ratio KL/Ko decreases. This effect
is most pronounced when KL/Ko > 1. When a constant-head boundary
condition is maintained these curves are concave downward over their
1If the constant-head boundary condition is used, all the first order functions must be replaced with zero order functions in equation (18).
entire range.
3. In the electric analog model of the semi-infinite aquifer,
no significant error will result from the circuit configuration if a
..:s AlSO and the circuit length is equivalent to 21.. where A is the tidal
penetration length; furthermore, "lumped" components J i.e' J a = 1../5,
may be used to extend the circuit beyond x = -.41.. if measurements are
restricted to x 2: -.21...
ACKNOWLEDGEMENTS
The authors would like to thank Mr. Ronald N. Wada for his
assistance in the initial stages of writing the computer program and
Drs. L. S. Lau and D. C. Cox for their helpful comments regarding the
preparation of the manuscript.
43
44
REFERENCES
Karplus, W. J. 1958. Analog 8imulation. McGraw-Hill Book Company, Inc., New York.
McCraken, D. A. 1967. Fortran IV manual. John Wiley & Sons, Inc., New York, 4th printing.
McLachlan, N. W. 1934. Be88el funation8 for engineers. Oxford University Press, London.
Sneddon, I. N. 1969. Speaial funations of mathematiaal phY8ias and ahemistry. Oliver and Boyd, Edinburgh and London.
Walton, W. C. and T. A. Prickett. 1963. "Hydrogeologic electric analog computers." ASCE Journal of Hydraulia Divi8ion. HY-6. pp. 67-91.
Williams, John A., Ronald N. Wada, and Ru-yih Wang. 1970. &odel studies of tidal effeats on ground water hydraulia8. Technical Report No. 39. Water Resources Research Center, University of Hawaii.
Young, Andrew· and Alan Kirk. 1964. Title of chapter - "Royal Society Mathematical Tables," In Bessel Funations" Part IV" Kelvin Funations. Royal Society, University Press, Cambridge. Vol. 10.
APPENDICES
a
all, ... , aZ2
A
Ao, ••• , A4
bIl, ... , bZ2
B
C
CI, Cz
Ch
D
D
e
h
i
Iv
K, K1 , K2
KV KO
Kit
Ku
Z.
L
47
APPENDIX A. LIST OF SYMBOLS.
grid spacing in finite difference representation of the equations
constants of integration
coefficient composed of a product of zero and first order Kelvin functions
coefficients composed of combinations of circular and hyperbolic functions
constants of integration
coefficient composed of a product of zero and first order Kelvin functions
cosine, capacitance, and a coefficient composed of a product of zero and first order Kelvin functions
complex constants of integration
hyperbolic cosine
coefficient composed of a product of zero and first order Kelvin functions
coefficient expressing storage per unit volume of aquifer
base of natural logarithms
piezometric head
Kelvin function of the first type of order u
Darcy coefficient of permeability; in region 1; in region 2
Darcy coefficient of permeability at the coast; at the interior boundary
time scale factor for electric analog model
Kelvin function of the second type of order u
aquifer length in transformed coordinate system
aquifer length
48
m' • m
M 1)
n
N
Nu
r
R, R 1, ••. ,
R
Sh
S, S s
t
to
T, Tl, T2
wo
X, X n
z
- - -z, Zl. Z2
Ct
£ I. £
n
RN
rate at which the permeability and the normalized permeability, respectively, change with position
modulus of Iu (i 1/ 2X)
integer
integer--total number of elements in finite difference representation of equations
modulus of Ku (i 1 / 2 X)
square root of ratio of transmissibility in region 2 to that in region 1
electrical resistance
real part of
hyperbolic sine
storage and specific storage
time variable
period of the sinusoidal tide
transmissibili ty
specific weight of fluid
horizontal position variable; continuous and discreetized
elevation of the free surface
average elevation of the free surface and aquifer thickness
compressibility of porous media, coefficient combining aquifer properties and tidal period
coefficients composed of combinations of circular and hyperbolic functions
coefficients composed of combinations of circular and hyperbolic functions
apparent porosity, porosity
amplitude of tide
space varying part of piezometric head
argument of Iu (i 1 / 2 X)
phase angle
l;
p
a
<Pu
49
penetration length--the distance, normal to the coast, in a semi-infinite aquifer, to the point where fluctuations in h are in phase with the tide at the coastline.
transformed space variable
dimensionless amplitude
angular frequency of tide
argument of Ku(i 1 / 2 X)
1/J 1 , 1/J2 coefficients containing aquifer properties and tidal period
1/J, iJ;, iJ;1
--<0
coefficients containing aquifer properties and tidal period
z
REGION 2 REGION I
: 0 ........ *' .......... If. •••• ........... .......... ••• ..... I
CHANGE AS WELL AS K.
UNCONFiNED COASTAL AQUIFER MODEL
REGiON 2 - (I) -::: x -:: 0
o
z
REGION OS x S L
z
CONFINED COASTAL AQUIFER MODEL
L
I
L
I
x e
x e
NOTE: FOR DISCONTINUOUS PERMEABILITY MODEL: -(I) <5x":: LAND r2. Kzz
2/K
1Z
1
FOR LI NEARLY VARYING PERMEABILITY MODEL: 0 -:;: x ~ L AN D K ( X) : Ko (I -I- m x I
WITH ah/ax = 0 OR h = CONSTANT AT X '0.
FIGURE A-I. SKETCH DEFINING THE AQUIFER MODELS ANALYZED.
-------------------------------
51
APPENDIX B
computer programs and sample output for:
DAMP (1) Kl 0.1 ft/sec, r2 ::;:; 1/2
DAMP (2) Kl ::;:; 0.1 ft/sec, r2 ::;:; 112
VAMP - NF - KL/KO 113, KO ::;:; 0.3 ftl sec
VAMP CH KL/KO ::;:; 113, KO = 0.3 ft/sec
•
C DAMP - REGICN 1 10 READ IS, 11 PERIOD, CKi, ALL, RZ
1 feRMAT (f5.2, FIO.7, f6.3, F9.3) IF (PERIGC .E~. 12.00) STOP
C ~RGI = (SIGMA*EPSI) I (CKl*lI) = (SIGMA*C) / CKI C ALPHAl = (SQRT(~RGl/2.0)J * ALL C EPSI = PORCSITY C SIGMA = FREQUENCY = (2.0*PI)/(PERIGD) RAD/SEC C CK = PER~EABILITY C Ll = ThE LENGT~ CF REGICN. 1 C R = «K2*12)/(Kl*11))**O.5 I = DEPTH
R = S'RTCRZ) SIGMA = 6.2831583/PERJOD C = 0.01 ARGl = (SIGMA*DJ / CKI ALPHAI = ISQRT(ARGl/2.0)) * All WRITE (6, 2) CKlf H2, PERIOD, ARGI
53
2 FORMAT (lHl,5HKl = ,f9.6.4H FPS,2X,SHR2 = ,F7.3/1HO,9HPERIOD ; t
1 F5.2,4H SfC,2X,7HARGl = ,FS.3/1HO,8HLOCATICN,2X, 2 9HA'PLITUDE,2X,llHPHASE ANGLE)
C x = LCCATICN (DI'E~SIO~LESS' C CEGREE = PHASE ANGLE C CALCULATE AND PRINT THE AMPLITUDES AND PHASE ANGLES FeR VALUES Of X
TENX ; 10.0 20 X = TENX/IO.O
CA = CO$(_LPHAl) CAX = C(S(~lPHAl*X) SA = SI~(ALfHAl) ~AX = SI~CAlPHAl*X) CHA : CC$~'AlPHAl) (HAX = CCS~(AlPHAl*XJ ShA = SINH(ALPHAl) SHAX = SI~t(AlPHA1*X) AOX = lCAX**2) + (SHAX**2) ACL = «(A**2) + (SH~**2) A4X = (SAX**2) + (SHAX**2) A4L = {S.**2J + (SHA**2) AO AOX*AOl Al = 12.0*CHAX*SHAX*AOL) + (2.0*CHA*SHA*AOX) A2 = (4.0* CHAX*CHA*SHAX*SHA) + (A4X*AOl) + (A4l * AOX) A3 = (2.0*CHAX*SHAX*A4l) + (2.0*CHA*SHA*A4X) A4 = A4X*A4l RHCl = (AC + Al*R + A2*(R**2) + A3*(R**3) + A4*(R**4)) RH02 = (AOL + 2.0*R*CHA*SHA + (R*.2)*A4l) ** 2 RHC = S'~T(RHOI/R~02) TAO = CAX*CHAX*SA*SHA - CA*CHA*SAX*SHAX TAl = CAX*SA*ShAX*ShA + CAX*CHAX*CHA*SA - CA*SAX*SHAX*SHA -
3 CA*CHAX*CHA*SAX TA2 CAX*CHA*SA*SHAX - CA*CHAX*SAX*SHA Teo SAX*SHAX*SA*SHA + CAX*CHAX*CA*CHA 181 = CAX*CA*CHAX*SHA + CAX*CA*CHA*SHAX + CHA*SAX*SA*SHAX +
4 CHA~*SAX*SA*SHA Tez = CAX*CA*SHAX*SHA + CHAX*CHA*SAX*SA TA~ = (lAC + TA1*R + TA2*IR**2.0» I (TBO + TBl*R + TB2*(R**2.01) T~ETA = ATANCTAM) DEGREE = ThETA * (360/6.2831853)
54
~RITE tt, 3) X, RHO, DEGREE 3 fORMAT (lhC,F4.2,lX,F7.5,4X,F10.5)
C X IS PRINTED UNDER lOCATION hEADING C RHO IS PRINTED UNDER AMPLITUDE HEADING C CEGREE IS PRINTED LNCER PHASE ANGLE hEADING
IF ex .EQ. 0.0) GO TO 10 lENX :;: TE~X - 1.0 GO Tn 20 END
OUTPUT - DAMP REGION 1
Kl == C. HeGeC FPS R2 :;: o. sec
PERICD = 3.00 SEC ARGI 0.2C9
LOCA lIeN A~PLI TUDE PHASE A f'..G LE
1.00 1.COOOO -c.ocoee
C.90 c.nEl7 7.72643
0.80 C .17250 15.53<;53
0.70 c .68136 23.42622
0.60 C.60322 31.35088
0.50 (.:3662 3C:>.24757
0.40 C.48010 47.01573
C.30 C.43216 54.52057
0.20 C.39128 61.60020
C.IO C.35sn 68.C7764
0.0 C.32463 73.17243
C DAMP - REGION 2 10 READ (5, II PERIOD, cn, All, R2
1 FORMAT (F5.2, FlO.7, F6.3, F9.3) IF (PERIOD .EQ. 12.00) STOP
C ARGl = (SIGMA*EPSI) I (CKl*ll) = (SIG~A*DJ I CKI C ARG2 = ARGl 1 R2 C ALPHA1 = (SQRT{ARGI/2.0» * ALI C ALPHA2 = (SQRT(ARG212.0)) * ALL C EPSI = POROSITY C SIGMA = FREQUENCY = (2.0*PI) ! PERIOD RAD/SEC C CK = PERMEABILITY CALI = THE LENGTH OF REGION I C R = «K2*Z2)/IKI*Zl».-C.5 Z = CEPTH C R2 = R**2
R = SQRTfR2) SIGMA = 6.2831583 / PERIOD D = 0.01 ARGI = (SIGMA*D) I CKl ARG2 = ARGl / R2 ALPHAl = (SQRT(ARGI/2.C) * ALI ALPHA2 = (SQRT'ARG212.0») * ALL WRITE 16, 2) CKI, ~2, PERIOD, ARG2
2 FORMAT (lHl,5HKl = ,F9.6,4H FPS,2X,5HRZ = ,F7.3/1HO,9HPERIOD = 1 F5.2,4H SEC,ZX,7HARG2 =: ,F5.3/1HO,8HlOCATION,2)(, 2 9HAMPLITUDE,2X,1IHPhASE ANGLE)
C X = LOCATIO~ (OIMEf\Sl(f\LESS) C DEGREE = PHASE ANGLE
55
C CALCULATE AND PRINT THE AMPLITUDES ANC PHASE ANGLES FOR VALUES OF X TENX = 0.0
2C X = TENXIlO.O CAL = COS(ALPHAl) SAL = SIN(ALPHAI) A2X = ALPHA2 * X CA2X = CCS(A2X) SA2 X = SIN ( A2 X) CHAI = COSH(ALPHAll SHAl - SINH{ALP~AI) RHOI = (2.7l83t**(A2X) RH02 ({CAl**2 + SHAI**2t + (2.C*R*CHA1*SHAI)
3 + (R**2t*{SAl**2 + SHAl**2») ** (0.51 RHO = (RHOl) ! (RH021 TAMI = (CA2X*SAI*SHAl - CA1*CHAl*SA2X)
4 + R*ICA2X*CHAl*SAl - CA1*SA2X*SrAl' TAM2 = (CA2X*CAI*CHAI + SA2X*SAl*SHAI)
5 + R*ICA2X*CAl*SHAl + CHAl*SA2X*SAll TAM = (TAMIl I (TAM2) THETA = ATAN(TAM) DEGREE = THETA * (360/6.2831853) WRI TE (6, 3) X, RHO, DEGREE
3 FORMAT CIHO,F5.2,6X,Fl.5,4X,FlO.51 C X IS PRINTED UNDER LOCATION hEADING C RHO IS PRINTED UNDER A~PLITUOE HEADING C DEGREE IS PRINTED UNDER PhASE ANGLE HEADING
IF (X .EQ. -1.0) GO TO II) TENX = TENX - 1.0 GO TO 20
56
END
OUTPUT - DAMP REGION 2
K1 = 0.100000 FPS R2 = 0.500
PER 100 -=
LOCATION
0.0
-0.10
-0.20
-0.30
-0.40
-0.50
-0.60
-0.10
-O.SO
-0.90
-1.00
3.00 SEC
AMPLITUDE
0.32463
0.27033
0.22511
0.18145
0.15609
0.12998
0.10824
0.09013
0.07506
0.06250
0.05204
ARG2 = 0.419
PHASE ANGLE
73.77243
84.26086
-85.25060
-74.76215
-64.27371
-53.78529
-43.29694
-32.80850
-22.32005
-11.83163
-1.34314
r'
I ~ I
57
C VAMP - NF C THIS PROGRAM IS USED TO CALCULATE T~E AMPLITUDE AND THE PHASE C ANGLE FOR COASTAL AQUIFER WITH LINEAR INCREASE OR DECREASE IN C PERMEABILITY, NO FLOW AT INTERNAL BQU~OARY
DOUBLE PRECISION Al, Cl, CKL, CKO, ~, CD, PERIOD, SIGMA, ALPHA, 1 TENX, X, XI, P, PS, PI, BEROP, BEIOP, Al2, AI, A2, CKROP. 2 CKIOP, A32, A3, A4, SERlPS, BEIIPB, B12, Bl, B2, CKRIP8, 3 CKIIPB, B32, 83, 84, SEROPl, BEIOPI, C12, Cl, e2, CKROPl, 4 CKIOP1, e32, e3, C4, At B, C, Df RI, RXI, RL. RCL, ~, ~HO,
5 SCA, SINA, AS, see, SINS, BS, SCC, SINe, CSt SCD, SIND, OS, 6 TS f COSA, AC, COSB, BC, cesc, ce, COso, DC, Te f TAM, THETAP 7 , DEGREE
C AL = THE LENGTH BETWEEN INTERIOR BOUN£ARY AND COASTLINE C CKL = THE PERMEABILITY AT (Xl:: L (FT/SEC) C CKO = THE PERMEABILITY AT (X) :: 0 (FT/SEC) C SIGMA:: FREI;UENCY C X = (Xl I L THE RELATIVE LaCATI(~ FPOM INTERIOR BOUNDARY C P = PS I C PB = PSI BAR C Pl = PSI BARI C BEROP:: BERO FUNCTION ~ITH ARGUMENT P C BEIOP = BEIO FUNCTION ~ITH ~RGUMENT P C CKROP :: KERO FUNCTION hITH ARGUMENT P C CKIOP = KEIO fUNCTION kITH ARGUMENT P C RHO = AMPLITUDE C DEGREE = PHASE ANGLE
10 READ (5, 2o, CKO, CKL 20 FORMAT (2015.7)
IF (CKl .EQ. 0.00000000 00) STOP AL = 4.0 CL = CKl I CKO M .: (Cl - 1.0) I AL CD= 0.01 PER 100 '" 3.0
1 SIGMA:: 6.2831853 I PERIOD ALPHA = (SIGMA * CO) I (M * M * eKOI WRITE (6, 30) tKO, CKL, PERIOD, ALPHA
30 FORMAT {lHl, 5HKO '" , 013.7, 4X, .5HKl = t D13.711HO, 9HPERIOO :::: , 8 09.3, 4X, 8HALPHA '" , DI0.4/1~Ot 3HLOCATION, 4X, 9HAMPlITUDE, 9 7X, IlHPHASE ANGLEt
TENX = 20.0 2 X = TENX I 20.0
XI = 1.0 + (M * At * XI P = DSQRT(4.0 * ALPHA. XI) PB = DSQRT(4.0 * ALPHA) PI = DSQRT{4.0 • ALPHA * Cl) CALL CALBER (SEROP, P) CALL CALOE I (BE lOP, P. A12 : (SEROP * BERO?) + (BElOP * BEIOPI Al = DSQRTt 112) 12 = OATAN{BEIOP / RERCP) IF (REROP .LT. 0.0 .A~O. BEIOP .GT. 0.0) A2 = A2 + 3.1415921 IF (BEROP .LT. 0.0 .ANC. BEIOP .LT. 0.0) A2 = 12 + 3.1415q21 If (BEROP .GT. 0.0 .ANC. BEIOP .LT. 0.0) A2 = 12 + 6.2831853 CALL CALKER (CKROP, PI CALL CALKEI (CKIOP, PI
•
58
A32 = (CKROP * CKROP) + (CKIOP * CKIOP) A3 = DSQRT (A3 2) A4 = DATAN(CKIOP I CKROP) IF (CKROP .LT. 0.0 .ANO. CKIOP .Gl. 0.0) IF (CKROP .IT. 0.0 .ANO. CKIO? .IT. 0.0) IF (CKROP .GT. 0.0 .AND. (KIO? .LT. O.OJ CALL CABERI (BERlPS, PB)
A4 == A4 .=
A4 ::
A4 + 3.1415927 A4 + 3.1415921 A4 + 6.2831853
CAll CABEll (SEIlPB, pel B12 :: (BERlPS • BERIPS) + (BEIIPB * BEIlPS) Bl :: OSQRT{Bl2) B2 :: DATAN(BEIIPB I BERIPBJ IF (BERI?S .LT. 0.0 .AND. BEII?S .GT. 0.0) IF (BERI?S .IT. 0.0 .ANO. BEl IPS .LT. 0.0) IF (BERlPS .GT. 0.0 .ANC. BEI1?S .LT. 0.0) CALL CAKERI (CKRIPB~ PB) CALL CAKEIl (CKIIPS, PS. 832 = (CKRIPB * CKRlPS) + (CKIlPS * CKIIPB) 63 :: DSQRT{B32) 64 :: DATAN(CKI1PB I CKRIPS) IF (CKR1?S .IT. 0.0 .ANt. CKIIP8 .GT. 0.0) IF (CKRIPB .LT. 0.0 .ANO. CKIl?B .IT. O.OJ IF (CKR1PB .GT. 0.0 .ANC. CKIIPS .IT. o.Ot CALL CALBER (BEROPl, PI) CALL CALBEr (BEIOP1, PI) tl2 = (BERO?l * BEROP1) + (BEIOPl * BEIOP1) C 1 = DSQRT( e12) C2 = DATAN(BEIOPl I BERe?l) If (BEROPI .LT. 0.0 .ANC. BEIOPl .GT. o.ot IF (BERO?1 .LT. 0.0 .AND. BEIOPI .IT. 0.0) IF {BEROP! .GT. 0.0 .ANC. BE10?1 .LT. 0.01 CALL CALKER (CKROPl, PI) CALL CALKEI (CKIDPl, PI) (32 = (CKROP1 * CKROPI) + (CKIO?l * (KIO?I) C3 '" DSQRT(C32) C4 = OATANtCKIOPI I CKROPIJ IF { CKROPI .IT. 0.0 .AND. CKID?1 IF ( CKROPI .LT. 0.0 .A~D. CKIO?1 IF ( CKROPI .GT. 0.0 .ANO. CKIOPI A = Al * C3 * 61 * 83 8 :: B12 * C3 * A3 C = Cl * Al * 832 D = Cl * Bl * 43 * 83 RI = ({A2 - B21 - (44 - B4)) RXI = (B32 * A12) + (B12 * A32.
.GT.O.O)
.IT. 0.0' .LT. 0.0)
$ - (2.0 * B1 • 83 * '1 * 43 * OCCS(RI)) RL = «C2 - 82) - (C4 - 84') RCL = (B32 * C12' + (812 * (32)
$ - (2.0 * Bl * 83 * Cl * C3 * OC(S(RLI) R :: RXI I RCL RHO = DSQRT(OABS(Rl) SCA = (A2 - B21 - (C4 - B4)) SINA = OSIN(SCA) AS :: A * 51 NA seA == (A4 - C4) SINS = OSIN(SCB' BS :: B * SINB
B2 = S2 '" B2 :::
B2 + 3.1415921 B2 + 3.141592? B2 + 6.2831853
B4 ::: B4 + 3.1415927 B4 ::: 84 + 3.1415927 B4 ::: B4 + 6.2831853
C2 == C2 = C2 =
C2 + 3.1415927 (2 + 3.1415927 C2 + 6.2831853
C4 ::: C4 + 3.1415927 C4 ::: C4 + 3.1415927 C4 ::: C4 + 6.2831853
sec = (A2 - e2) SINC = OSIN{SCCJ CS=C*SINC SCO = «A4 - 641 - (C2 - 82») SIND = OSIN(SCO) OS :: 0 * SI NO TS = -AS • as + cs - as eOSA = OCOS(SCA) AC = A * COSA cosa = OCOSCSCBt BC = B * eOSB case:: OCOS{SCC) ec = C * cnse coso = OC05(SCO) DC ,: 0 * COSD Te = -AC + BC + CC - DC TAM = TS / TC THETAP = DATAN(TAMI IF (TS .GT. 0.0 .AND. Te .GT. 0.0) lHETAP = THETAP - 6.2831853 IF (TS .GT. 0.0 .AND. TC .LT. 0.0) T~ETAP :: THETAP - 3.1415921 IF (TS.LT. 0.0 .AND. Te .LT. 0.0) THETAP :: THETAP - 3.1415q27 DEGREE:: THETAP * (360.0 I 6.2831853. WRI TE (6, 4;) t X, RHO, C EGRE E
40 fORMAT tlHO, D9.3, DI5.6, 016.6' IF (TENX .EQ. 0.0) GO Te 3 TENX = TENX - 1.0 GO TO 2
3 PERIOD = PERIOD. 3.0 IF (PERIOD .EQ. 6.0) GC TO 10 GO TO 1 END
SUBROUTINE CALKER IBKER, Z) C THIS PROGRAM CALCULATES THE KERO FUhCTION
59
DOUBLE PRECISION Z, RF., RFMl, S, SI, 'i, SUM, BERt BEl, R, R2, X, 1 TERM, k, BKER, XX, TI2,TIP12, TIM12
RF ;: 1.0 RFM 1 ;: 1.0 S = 0.0 51 = 0.0 y ;: 1..0 SUM = 0.0 CALL CALBER (BER, Z) CALL CALBEI (BEl, Z) W = «0.1159 - DLOG{Z») '* BERt + {(I.e I 4.01 * 3.1415927 * BEll DO 3 I ; 2, 30, 2 R = I RF = RF '* R kFMl = RFMl * (R - 1.0) P2 = (RF * qFMl) ** 2 S = S + (1.0 I Rl S1 = SI + (1.0 I (R - 1.0) x = 10.500 * ZI *'* (2.0 '* R) y = v '* (-1.0)
60
TERM = (X '" Y '" (S + Sl») I R2 SUM ::: SUM + TERM BKER = W + SUM IF (DABStTERM) .LE. 1.0[-10) GO TO 4
3 CONTINUE 4 RETURN
ENO
SUBROUTINE CALKEI fBKE!, zt C THIS PROGRAM CALCULATES THE KEIO FUNCTION
DOUBLE PRECISION RF, RF'l, $, SI, Y, SUM, BERt BEl, R, R2, X, 1 TERM, W. BKEI, XX, l
RF '" 1.0 RFMl:; 1.0 S = 1.0 Sl = 0.0 y ::: 1.0 SUM = ll/Z)**2 CALL (ALBER leER, Z) CALL CALBEI (BEl, l) W '" (0.1159 - OLOGfZ)) * BEl) - «(3.1415927 '" BER) I 4.0) DO 3 I = 3 t 3 1 t 2 P :. I RF :; RF '" R RFM1 = RFMl '" (R - 1.0) R2 = (Rf '" RFMll*"'2 S ::: S + Cl.0 I R) S 1 ::: S 1 + (1.0 I «R - 1.0}t X :; tZ/2)**(Z.O"'R) y '= y '" (-I.O) TERM::: ex '" y '" (S + 51)) I R2 SUM ::: SUM + TERM BKE I :; W + SUM IF (DABS(TERMI .LE. 1.or-l0) GO TO 4
"3 CONTINUE 4 RETURN
END
SUBROUTINE CAKERI tBKER1, ZI ( THIS PROGRAM CALCULATES T~E KERI FUNCTION
DOUBLE PRECISION Zt RF, RP1F, $, 51, SUM, BER1, BEll, W, V, R, 1 THETA, X, TERM, ThETAl, SUM1, PTS, BKERI
RF = 1.0 RP1F = 1.0 S = 0.0 S 1= 1.0 SUM:; 0.0 CAU CABERI (BERl, Z) CALL CABEll (BEll, Z) W ;: f(O.1l59 - DlOG(l) I * BERU + «(1.0 I 4.0) '" 3.1415921 * BElli DO 3 I = 1, 20 y = ( -1 • 0) ** (1 + I.
R = I THETA = 0.250 • (1.0 + (2.0. RJ) - 3.1415921 RF :: RF • R RPIF:: RPIF • (R + 1.0) S = S + tl.O I RI Sl = 51 + H.O I «R + 1.0 II X = (0.5 * I) •• (1.0 • (2.0 • RI' TERM = 0.500 • (ex. y ~ (5 + SI) * OCOS(THETA») I (RF * RPIF)) SUM ;; SUM + TERM THETAI = 0.250 * 3.1415~27 SUMl = ((-l.Ot I Z) • CCOS{THETAl) PTS : (-0.250 * ZI * DCCS(THETAl) BKERI = W + SUMl + PTS • SUM If (DABS{TERM) .lE. I.CD-IO' GO TO 4
3 CONTINUE 4- RETURN
END
SUBROUTINE CAKEIl (BKEIl, ZI C THIS PROGRAM CALCULATES THE KEIl FUNCTION
DOUBLE PRECISION 1, RF, RPIF, 5, 51, SUM, BERI, BEll, W, y, R. 1 THETA, X, TERM, THETAl, SUMl. PTS, BKEIl
RF = 1.0 RPIF :: 1.0 S :: 0.0 Sl "" 1.0 SUM'" 0.0 CALL CABERI (fERl, zt CAtL CABEll (BEll, Z)
61
W = «0.1159 - OLOG(zt) * BEll) - ((1.0 I 4.0) • 3.1415921 * BERl' DO 3 I = 1 t 20 Y :: (-1.0) •• (1 + II R :: I THETA:: 0.250 * (1.0 + (2.0 * R) t * 3.1415921 RF ::: RF • R RPlF = RPIF * (R + 1.0) S = S + (1.0 I R) SI ; 51 + (1.0 I fR + 1.0») x = (0.5 * IJ .* (1.0 + (2.0 • R) TERM = 0.500 * (X * V * (S + 51) * DSIN(THETA» I IRF * RPIF» SUM = SUM + TERM THETAI = 0.250 * 3.1415q27 SUMl = «-1.0) I Zt * D5INlTHETAl) PTS = (-0.250 * ZI * DSIN(T~ETAl) BKEll :: W + SUMl - PTS - SUM IF (OABS{TERM) .LE. 1.OC-IO) GO TO 4
3 CONTINUE 4 RETURN
END
SUBROUTINE CALBER (BER, Xl C THIS PROGRAM CALCULATES T~E BERO FUNCTION
62
DOUBLE PRECISION TJP12. TIZ, XX, V. x. TERM, BER BER = 1.0 TERM = 1.0 DO 3 I :: 1 t 100 TI2 = (2 * II ** 2 TIP12 = ((2 * II - U ** 2 XX : «(0.25) * (X ** 2)1 ** 2 V = ((-1) * XX. / (TIZ ,. TIP12) TERM = TERM * V BER :: BER + TERM IF (DA8S(TERM) .LE. 1.0C-071 GO TO 4
3 CONTINUE 4 RETURN
END
SUBROUTINE CALBE! (8El, X) C THIS PROGRAM CALCULATES THE BEIO FUNCTION
DOUBLE PRECISION TI2. TIMI2, XX, V. Xt TERM, BEl TERM = 0.25 * ex ** 2) BEl = 0.25 * (X ** 2) 00 3 I ::: 1, 100 TI2 :: (2 * II ** 2 TIM 12 = {{ 2 * I I + 1 J ,.,. 2 xx = ((0.25) * (X ** 2). ** 2 Y ::: «-11 * XX) I (TI2 ,. TIM12' TERM- TERM * V BEl:: BEl + TERM If lOABSITERM) .LE. I.OC-071 GO TO 4
3 CONTINU.E 4 RETURN
END
SUBROUTINE CABERl (BERS}, Xl C THIS PROGRAM CALCULATES THE BERl FUNCllON
DOUBLE PRECISION BERI, TERMl, BER2, TERM2, Xt FTI2, FTIMl, XX, 1 VI, STI, STIP2, STIPl2, Y2, BERSl
BERI ::: 1.0 TERMI :: 1.0 BER2 = ({0.5 * Xl ** 2) I 2.0 TERM2::: «0.5 * Xl ** 2) I 2.0 00 3 I = 1, 100 FTI2 = (2 * I) ** 2 FTIPl = (2 * I) + 1 FT I Ml = «2 * I) - 1 XX = (0.5 * Xl ** 4 Vi:: «(-1.0) * XX) I (FTl2 * FlIPI '" HIMl) TERMl :: TERMI * VI BERI ::: BERl + TERMl STI ::: (2 * II STIP12:: «2 * I) + 11 ** 2 STIP2 = (2 * II + 2 Y2 =«-1.0) * XX) I (511 * STIPl2 * STIP2)
1
TERM2 = TERM2 * YZ SER2 = BERl + TERMl BERSl:: (BERI + BERZ) * ((-X) I (2 * SQRT(2.0»)) IF (OABStTERM1) .LE. 1.00-01 .OR. DA8S(TERMZ) .LE. 1.00-07)
ZGO TO 4 3 CONTINUE 4 RETURN
END
SUBROUTINE CABEll (BERSl, X) C THIS PROGRAM CALCULATES THE BEll FUNCTION
63
DOUBLE PRECISION BERI, TERM1, Xt SER2, TERM2t FTIl, fTIPl, FTIM1. 1 XX, Ylt STI , STIP12., STIP2, Y2, BERSl
BERI = 1.0 TERMI = 1.0 BER2: «-1.0) * (IC.S * Xl ** 2)) I 2.0 TERM2 ~ ({-I.O) * (0.5 * XI ** 2)1 I 2.0 DO 3 I = 1, 100 FTl2 = (2 * II ** 2 FTIPl : (Z * I) + 1 FTIMl = (2 * It - 1 XX : (0.5 * X) ** 4 VI = «(-1.01 * XX) I (FTI2 * FTIPl • FTIMI) TERMI : TERMI * VI BERI = BERI + TERM1 STI = (2 * I) STI Pl2 :: C (2 * I) + 11 ** 2 ST I P2 = (2 * 1) + 2 Y2 = «-1.0) * XX) I (STI * STIP12 * STIPZ) TERM2 = TERM2 * Y2 SER2 = SER2 + TERM2 BERSl = (BERI + BERZ) * (X I (2 * SQRT(2.0)1) IF WAB$(TERMl) .LE. 1.00-07 .OR. DA8SHERM2) .LE. 1.00-071
2(;0 TO 4 3 CONTINUE 4 RETURN
END
OUTPUT - VAMP NF
KO = 0.30000000 00 Kl ::; /).10000000 00
PERIOD = 0.3000 01 ALPHA = 0.25130 01
LOCATlON AMPl I TUDE PHASE ANGLE
0.1000 Cl 0.1000000 01 -0.810611D-15
0.9500 00 0.9354860 00 -0.5732450 01
0.9000 00 0.885169D 00 -0.1120 laD 02
0.8500 00 0.846213D 00 -0.163584D 02
64
0.8000 00 0.8163270 00 -0.2116570 02
0.7500 DO 0.1936380 00 -0.2559550 02
0.1000 00 0.776618C 00 -0.2963340 02
0.6500 00 0.7640200 00 -0.3327670 02
" 0.6000 00 0.754837D on -0 .365~ 24D 02 j
O.550D 00 0.748261D 00 -0.394145D 02
0.5000 00 0.7436480 00 -0.4194170 02
0.450D OC O.740493D 00 -0.441353D 02
0.400D 00 0.7384020 00 -0.460176D 02
0.3500 00 0.7370680 00 -0.4761060 02
0.3000 00 0.7362600 00 -0.4893560 02
0.2500 00 0.7358030 00 -0.5(101260 02
0.2000 00 0.735569[; 00 -0.5085990 02
0.1500 00 0.7354640 00 -0.514943D 02
0.1000 00 0.7354280 00 -0.51931.30 02
0.5000-01 0.7354200 00 -0.521844D 02
0.0 0.1354200 00 -0.522662D 02
65
C vAMP-C~ C THIS PRCGRAM IS USED TO CALCUlAT'E HE AMPLITUDE AND THE PHASE C ANGLE FCR lO.STAL A'UlfER WITH LINEAR INCREASE OR DECREASE IN C PERMEABILITY, CONSTANT hEAD AT I~TE~NAL BOUNCARY
COUBlE PRECISION Al, Cl, CKl, CKO, ~, CO, PERIOD, SIGMA, ALPHA, 1 TENX, X, Xl, f, PB, PI, BEROF, BEIOP, A12, AI, A2, CKROP, 2 (KIep, '32, A3,. '4, SEROPE, BEIOPB, B12, fj' I"l(o nPB, 3 CKIOPS, 832, 83, H4, BEROFl, BEIOPl, e12, Cl, C2, CKKurL' 4 CKIOFl, C32, C3, 04, At Sf e, 0, ~[, RXI, RL, RCL, R, RHO, 5 seA, SINA, AS, SCB, SINB, BS, sec, SING, es, SCD, SIND, DS, 6 IS, COS,, AC, eOSB, BC, cose, ce, eeso, DC, Te, T'~, THE TAP 7 , CEGREE
C At = IHE LENGTH BET~EEN INTERIOR BCUNOARY AhD COASTLINE C CKL - THE PERMEABILllY AT (X) = L (FI/SEC) C CKO = l~E PERMEABILITY Al (X) = 0 {fT/SECa c SIGMA = FREQUENCY £ X = (Xl J L ThE RELATIVE LtCATICN FROM INTERIOR BOUNDARY C P ::: PSI C pa = FSI e,R C PI = PSI EARl C BEROP = BERO fUNCTION WITH ARGUMENT P C BE lOP = BEIO FUNCTION WITH ARGUMENT P C CKROP = KERO fUNCIICN WITH ARGUMENT P C CKIOP = KEIO FUNCTICN WITH ARGUMENT P C RHO -= A~FlIIUCE C CEGREE = PHASE ANtlE
10 RE'D (5, to) CKO, CKL 20 fOR~AT (2C15./)
IF (CKL .EQ. o.oecoceOD 00) STOP AL :; 4.e Cl -= CKl , CKe ~ :: lel - 1.C} I AL CD :: C.Cl PER IOC :: 3.C
1 SIGMA:: 6.28318:3 I PERIOD ALPHA = (SIGMA * CD) I (M • M * CKO) ~RITE (6, 30) CKG, CKlt PERICD, 'lP~A
30 FORMAT (lfl, 5~KO = , 013.7, 4X, 5HKL = , Dl3.7/1HO, 9HPERIOD = t
8 09.3, ~Xt 8HALPHA = , DIO.4/1HO, 8HLOCATIDN f 4X, ~HAMPlITUDE, 9 7X, IlHPhASE A~GlEl
TEIliX = 20.C 2 X = TENX , 20.0
Xl = 1.C + (~ * AL * XJ P = DSQ~T(4.0 * ALPHA * XIJ fB = CSGRT(4.C $ ALPhA) Pi = CSQ~1(4.C * ALPHA * CLI CAll CAlBE!< (BEROP, P) CALL (ALBH (BElOP, PI Al2 = (EEROP * BEROPI + (BEIOP * BEIOP) Ai ..; CSCRIU12. A2 = CAIA~(BEICP J EEROP) IF (SEROP .LT. o.c .A~D. BEIOP .GT. 0.0) A2 = A2 + 3.1415927 If (BERCP .IT. 0.0 .A~D. BEIOP .LT. 0.0) A2 = '2 + 3.1415927 IF (BERCP .GT. 0.0 .A~D. tiEIOf .LT. O.OJ A2 = A2 + 6.2831853 (ALL CAlKEx ((KROPf PJ CALL CAlKEI (CKIOP, Pl
66
A32 : C(KROP • CKROF) • «(KIOP * CKICP) A3 =: CSQR1(A32) A4 = CATA~(CKICP I CKROPJ If (CKROP .IT. 0.0 .AND. (KIOP .Gl. C.O) If (CKRCP .LT. 0.0 .AND. (KIOP .LT. 0.0) IF (CKRCP .GT. 0.0 .AND. CKIDP .LT. 0.0) CAll (ALBER (eE~OFB, PBl
A4 = A4 + 3.1415921 A4 = A4 + 3.1415927 A4 = A4 • 6.2831853
(All (ALBEI (BEIOF8, PH) e12 = (SEROPH * BEROPB) + (REIOPB * BEIOPB) Bl = [S<;RT(E1lJ 82 = [ATA~{BEIOPB I BEROPS) IF (BERepS .LT. 0.0 .AND. BEIDPB .GT. 0.0) IF (SEROPH .LT. 0.0 ,",AND. BEIOPS .LT. 0.0) IF (BERCPS .Gl. 0.0 .AND. BEIDPS .IT. O.C) CAll CALKfR (CKROPS, PB) CALL CALKEI ((KIOPB, FB. 832 = «(KROPS • CKRepS) + (CKIDPS • CKIDPS) S3 ;: CS'FHE321 84 = CATA~(CKICPB I (KROPS) If ((KROPS .LT. 0.0 .AND. CKIDPS .GT. o.o) IF {(KReFS .LT. 0.0 .AND. CKIOPS .LT. 0.0) IF (C K Rep B • G T. C. 0 • A N C .C K lOP B .1I. O. 0 J {ALL {AL8ER (BEROPl, PI) CALL CAlBEI (BEIOfl, P1) (12 : {BERCPl * BERCP11 + (BEIOPI * BEroPlt Cl ;: CSQRlICl2j (2 = CATA~(BEICPl I BEROP1) IF (BEROPI .IT. 0.0 .AND. BEla?l IF (BERDPl .LT. 0.0 .AND. BEIDPl IF [BEROPl .GT. c.e .AND. BEIOPI CALL CALKER «(KROFl, Pl) (ALL {AlKEI «(KIOPl, PI) (32 = {CKRCFl • CKROFI. + (CKIDPl C3 = CS'R Tl (32) C4 = [A1A~(CKICPI I CKROPl) If I CKRCPl .LT. c.e .AND. (KIOPl IF ( CKROPI .IT. 0.0 .AND. CKIOPI IF ( (KROFl .GT. O.C .A~D. CKIO?1 A = Al * C3 • B1 • 83 E = E12 * C3 • A3 C = Cl • Al * 832 C = (1 • el * A3 • B3 RI = ((.42 - 82)- (P4 - 54)} RXI = (832 * A12) + (B12 * A32)
.GT. O.C)
.l T • 0.0.
.IT. 0.0)
* (KIOPU
.GT. c.ot
.LT. o.O)
.IT. C.O)
$ - (2.e * 61 • 83 • Al * A3 * CCOS{RI)t RL = ({C2 - B2) - {C4 - 64») RCL = (E32 * Cl21 + 1812 * (32)
$ - (2.0 * 81 * 83 * CI * C3 * (COS(Rl)) R ::: R)! I I RC l R~G • CSQRlIDABS(R») SeA = ((AL= - (2) - (C4 - B4)) SINA = [SHdSCi.lJ /.IS = A '" SIM SCE = 04 - C41 S I1'<B = [S 11\( SCBl es = e '4 SI~8
. ( ;;x .•
B2 :: 62 • 3.1415921 82 :: B2 + 3.1415921 62 = 82 + 6.2831853
84 = B4 :: B4 ::
C2 ::
C2 C2 -=
(4 C4 C4
-= = ::
B4 + 3.1415921 B4 + 3.1415927 84 • 6.2831853
C2 + 3.1415921 C2 + 3.1415927 C2 + 6.2831853
C4 + 3.1415927 e4 .. 3.1415n7 C4 + 6.2831853
1
sec;: (A2 - C2) SINe = CSIf'.(SCC) es '" e ,. S I I\C seD -= «(jl4 - 84. - (C2 - B2)} SIND;:: DSHd sec) [$ ~ C ,. SIt;o IS = -AS + as + cs - CS CCSA'" [C(S(SCId AC ": A '" C(SA cose :: CCCS(SCB) EC =: E! * C( 5B cose ;:: tees(sec) ce = c * cesc cos c "" eccs (seC) cc = c *' ceso Te = -At + BC + ec - DC lAM = T 5 I Te l~ETAP ;:: (ATA"(TA~. If (IS .GT. o.c .A~D. Te .GT. O.OJ T~ETAP ;:: T~ETAP - 6.2831853 IF (15 .GT. 0.0 .A~D. Te .LT. O.O) TrETAP '" ThETAP - 3.141SQ27 If (15 .Ll. 0.0 _'"c. TC .LT. 0.0) TrETAP = THETAP - 3.1~15927 [EGREE • IhETAP * (36C.0 I 6.2B318~31 wRITE If, 4C) X, ~HC, DEGREE
40 FeRMAT (IrO, C~.3f CI5.6, 016.6) IF (TENX .EQ. C.O) GC TO 3 lE~X • lE~X - 1.0 GO TO 2
3 PERlec ;:: PERIOD + 3.0 IF (PERleD .EQ. 6.0) GO TO 10 GO TO 1 END
SLBRClTINE CAL~ER ISKER, If C THIS PRCG~A~ CALCLLATES THE KERO fL~CTI(~
67
CCUBlE FRECISICN Z, RF, RFMl, Sf SI, V, su~, BER, BEl, R, R2, X, 1 TER~, w, BKER, XX, TI2,TIP12, TI~12
RF = I.e RFJI(l ::: 1.0 S :: c.o 51 = c.c ¥= 1.0 SLfi = C.O CALL (AlbER (BER, l) (ALL (ALBEI (BEl, Z) ~ = {(O.1159 - DLCGIZI) * BERI + ({I.O I 4.0) * 3.1415927 * BEll C[ :! I ::: 2, 3e, 2 R = I Elf :: Rf * R RF~l • PFM1 • fR - 1.0) R2 :: (Rf * RF~lJ .* 2 S :: S + (I.e I RI S1 :: Sl + (1.0 I (R - 1.0)) X (C.50C * Z) ** (2.0 * RI 'y '" Y * (-l.C)
a
68
HRM:: ()( » Y * (5 .. 51) I R2 SL~ :: S~M .. TE~M
8KER :;: lit .. 5UM ]f (OAES(TERM) .LE. 1.00-101 GO Te 4
3 CCI\TII\UE 4 f<EHRf>;
EI\ C
SLBROlTII\E CAlKEI (eKEI, Z) THIS FRCGRA~ CALCLLATES THE KEtO fLI\Cll(~ DOUBLE PRECISICN RF, RFMl, S, 51, Y, SUM, BER, BEl, R, R2, X,
1 lER~, w, BKEl, XX, Z RF = 1.0 RFf"1 :: 1.0 S :: 1.0 51 :: C.C y= 1.0 SUM.;: (l/2JH2 CALL (ALBER (BER, 1) CALL CAlBE) (BEl, Z) ~ :: (0.1159 - OLCG(Z» * BEl) - '(3.1415927 * BER' I 4.01 CC 3 I :;: 3, 31, 2 R :: I Rf :: Rf * R HI"1 :: RHI * (R - 1.C) RZ :: (Rf * RFMIJ**2 S :: S ~ (l.C J AI 51 .;: 51 .. (1.0 / (R - 1.0)>> X :: IZ/21**(Z.C*RJ y= y * (-l.C) lERf" :: Ix * y * (S .. 511) I R2 SL~ :: SLM .. TE~M
EKEI = k .. 5UjII If (OAES(TER~) .LE. 1.OG-I0) GO Te 4
3 CCNTII\UE 4 I'ETlJR/\
He
5LBRCLTINE CALCER (EER, X) C THIS PRCGRA~ (ALC~LATES THE BERe fLNCTI(~
COLBLE FRE(ISI(~ TIF12. TI2. XX, Y, X, TERM, BER HR = 1.0 lER~ ::: I.e co 3 I ;:: 1, HC T I 2 -: (2 '* [I '** 2 TIP12 :: el2 '* l) - 1) ** 2
XX ;::: ((C .. 2:) 11< 0 ** 211 ** 2 Y ;::: ((-11 '* XX) I (T12 * TIP12) H~~ = TEIdI * Y eER ;::: BEl< + TERM If IDAES(TER~I .LE. 1.00-071 GO Te -4
3 C (I'd 11\ l;E
%,?L_Al\! it
c
4 IiETURI\ END
SleR(LTl~E C_leEI tBEI, Xl T~lS PRC'~_M CAlC~lATES THE EEIC FLI\CT(CN e(UBLE P~ECISICN T12, TIMI2, )X, V, X, TERM, BEl TER~ : C.~5 * {X ** 2. eEl = G.25 * () ** 21 co :! I ;;: 1 t lCO 112 ;;: (2 * I) ** 2 TI~12 -= (2 * II + 1) ** 2 XX = (Ce.25) * (X ** 2») ** 2 Y ;;: (-1) * XX) I (112 * TIMI2. lER~ -= TERt< * V EEl;;: BEl + TERM IF (OABSITERM) .lE. 1.00-071 GO TG 4
3 CCf\TII\.UE 4 fiEn.R~
Ef\C
OUTPUT - VAMP CH
KC = (.3CCCCCCIJ 00 I<l :: C decocceD
PERIOD = C.3CCIJ 01 PLPHA = C.2513D
00
01
letA TlCI\. AlfH ITUDE PHtlSE tll\GLE
'.ICOD C 1 C.J.COCOOD (1 c.e
(.95eo cc C.9093290 00 -C.~194040 (1
(.9000 OC 0.8216990 CO -0.415398D 01
C.S50D CC 0.1534420 CC -C.5~c!:SI0 (1
0.8000 CC C.685302C CO -0.7475350 C1
C.7500 CC 0.6223120 CC -C.EE1181D (1
(;.1COO ac C.5637120 00 -( .le 13020 02
0.6500 CC C.5G89C1C OC -C.11245'10 02
C .6(, CC cc C .4513 9lC CO -c .12 2300 02
0.5500 C( (.4C87850 CO -C.1311260 C2
(.5000 CC (.3(:21560 CC -C.13882CO 02
0.4500 CC C.319C29C 00 -C.1455450 C2
C.4000 DC 0.217315D co -0.1.513510 C2
69
10
(.35CC C( C.2375S60 00 -C .15t3CtD C2
C.300D ac C.1S9526D 0,) -C.16C4t5D 02
C.2500 C( C.H:3(U~C CC -C.H:387ED (2
l (.2000 OC G.127~4dC OC -c. lU: 58';;D C2 , C.150D OC C • <14 2 C 28 C - 0 1 -(.H:E63S0 C2 ..,
: C.ICGD CC C.61b8t3C-Cl -(.17(((;30 02
C.5ceO-C1 C.3031121::-01 -(.17C8~6D 02