AN ABSTRACT OF THE THESIS OF
WILLIAM EDGAR AVERA
prese9d on
GEOPHYSICS
in
Title:
ITERATIVE TECHN
Abstract approved
for the degree of
MASTER OF SCIENCE
April 30, 1981
LIEARiZEDfREE SURFACE FLOW
Redacted for Privacy
Gunnar Bodvarsson
The displacement of the free liquid surface in geothermal and
hydrologic reservoirs is an important capacitance factor. An
iterative approach to determining the drawdown of the free liquid
surface for a single sink region in a homogeneous, isotropic, Darcytype porous mediums is discussed. The iterative approach involves a
stepwise adjustment of the pressure, on the reference surface which
replaces the time-dependent free surface condition by a fixed plane
Dirichlet type condition so that readily availiable, standard
techniques can be applied. Grouping of producing wells into a single
analogous well may be used to treat multiple well cases with the
iterative approach.
An analytic solution for the infinite half space situation is
used to compare solutions with the iterative technique. The analytic
solution is derived for a point sink within an infinite, homogeneous,
isotropic, Darcy-type porous half space. It is obtained by
linearizing the free liquid boundary condition provided that the
free surface deviates from its equilibrium reference position by
only a small slowly undulating displacement
h
.
The flow pressure
at the equilibrium surface is then approximated by the hydrostatic
pressure for a column of height
h
A standard model is designed to be analogous to the analytic
solution. Testing the iterative-procedure calculations for this
model against the derived analytic solution produces very
satisfactory results provided that the numerical grid spacing is
adequately chosen for the problem. Calculations of the linear and
quadratic terms of the free surface condition indicate that the
neglected quadratic terms are in general small, and the
approximation is reasonable.
Iterative Techniques in Linearized
Free Surface Flow
by
William Edgar Avera
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Master of Science
Completed April 1981
Commencement June 1981
APPROVED:
Redacted for Privacy
Professor of Geophysics and Mathematics
in charge of major
Redacted for Privacy
Dean 'ôf School of Oceanography
Redacted for Privacy
Dean o
Graduate Sdhool
Date thesis is presented
Typed by William Avera for
April 30, 1981
William Edgar Avera
ACKNOWL EDGMENTS
My most sincere appreciation goes to my major professor,
Dr. Gunnar Bodvarsson, for his guidance to me as a student, and for
the opportunity to become involved with his work and ideas.
During
my studies here I have gained an insight to his philosophy of science
which will benefit me throughout my career. In addition, I will
always be indebted to him for his advice and critical reviewing of
this manuscript.
Dr. Elliot Zais has also been a good influence on my character
and science as a student at OSU. His encouragement and helpful
discussions will always be appreciated. I am grateful to
Dr. Richard Couch for his guidance and for entrusting me with the
use of his HP-41C which has contributed a valuable part in this
research. Also, the assistance of Dr. Jonathan M. Hanson in attaining
the group of subroutines PWSCYL and of Dr. F.T. Lindstrom in getting
it operative is greatly appreciated.
The encouragement and discussion from all of my fellow students
is deeply appreciated; in particular, G. Stephen Pitts and
Osvaldo Sanchez Zamora have been particuarly helpful. Also the
assistance in drafting from Paula Pitts is very much appreciated.
The encouragement and love from my parents, Joseph A. Avera
and Jane Y. Avera, is the source of my strength. Their guidance plus
that of Jesus has brought me through the difficulties.
I am also grateful to the Milne Computer Center at OSU for a
grant to fund the use of the CYBER 7300 computer.
TABLE OF CONTENTS
Chapter
Page
I. INTRODUCTION
1
Relative Importance of the Free Liquid Surface
Capacitance
II. CONSIDERATIONS WITH AN ANALYTIC SOLUTION
2
6
The Concept of Drawdown
7
Analytic Solution for a Point Sink with a Linearized
Free Surface Condition
7
Discussion of Assumptions
15
Other Reservoir Models and the Motivation of the
Present Investigation
17
The Reference Numerical Model (RM1)
18
Borehole Grouping
20
III. THE TIME ITERATION TECHNIQUE
25
Technique
25
The Analytic Time Step Approximation
28
Computer Programs
31
IV. COMPARISON BETWEEN ANALYTIC AND NUMERICAL MODELS
34
Relationship Between the Analytic and Numerical
Models
34
Comparison of the Models
36
The Homogeneous Boundary Value Solution
37
Approximation of
dp/dz
by a Polynomial
42
The Validity of the Linearized Free Surface
Approximation
43
The Time Iterative Solution
45
BIBLIOGRAPHY
55
APPENDICES
56
Appendix A:
Appendix B:
Development of a Fourth Order Polynomial
Derivative
56
Program Listings
59
LIST OF FIGURES
Figure
Page
1.
Schematic cartoon of the time step process.
28
2.
Profile of R values for the homogeneous boundary
value solution for three different boundary distances
relative to a constant point sink location.
39
Profile of R values for the homogeneous boundary
value solution along the radial axis for three
different boundary distances relative to a constant
point sink location.
40
Profile of R values for the homogeneous boundary
value solution radially just below the free surface
for three different boundary distances relative to
a constant point sink location.
41
Profile of the free liquid surface radially outward
from the central axis for several different times
of the model RH1 with t= 4XlO7sec.
46
R values plotted radially out from the point sink
at several different times for RM1 with t 4XlO7sec.
48
3.
4.
5.
6.
7.
values plotted radially out along the reference
surface at several different times for RM1 with
R
t= 4XlO7.
8.
49
Maximum time steps before reversal of pressure
gradient as a function of the characteristic
interval for RM1
.
54
LIST OF TABLES
Table
Page
1.
Selected parameters for model RM1.
19
2.
Brief discussion of numerical parameters used in
the time iterative technique.
33
3.
Selected examples of the maximum QR for different
times using a time step of 4X107 sec with model
RM1
4.
5.
45
.
Comparison between analytic and RM1 polynomial
derivative values for t= 4XIO7sec at two
selected points on the reference surface.
The maximum flow rate
om
50
kg/sec) possible for RM1
to maintain the established limits on QR through
successive time iterations. Based on the largest
QR value for each iteration (Time is sec X107)
52
Input for subroutines DIFTBL and DERVZO to
calculate the polynomial derivative fo the
free surface
57
.
6.
ITERATIVE TECHNIQUES IN LINEARIZED FREE SURFACE FLOW
I.
INTRODUCTION
The production of fluid from a liquid dominated geothermal
reservoir is generally associated with changes in the free liquid
surface position. Changes in the liquid level depend largely on how
the production has affected the reservoir pressure field. The lowering
of the pressure field releases stored fluid for production. The three
main storage capacitances within the reservoir are the compressibility
of the reservoir rock and fluid, the displacement of the liquid
surface, and the vaporization of liquid (Zais and Bodvarsson, 1980).
The free surface capacitance is of particular importance, and the
mechanism of the liquid surface response is very important in
estimating the production capabilities of a reservoir.
The purpose of this study is to develop an iterative technique
which can be used with existing numerical potential field programs to
model the changes in the free liquid surface. A linearized form of
the free-surface boundary condition is used in an iterative technique
that calculates the pressure field solution at finite time increments
incorporating an existing program obtained from the Lawrence Livermore
Laboratory. We compare the iterative solution to an availiable
analytic form for a particular case, and illustrate how the iterative
technique can be used.
Relative Importance of the Free Liquid Surface Capacitance
We want to compare the relative importance of the free liquid
surface with that of compressibility and liquid vaporization which
can occur within a reservoir. Each of these physical processes
releases fluid upon a decrease in pressure. Knowing the relative
amount of fluid released for each will give us an idea as to which
process is most itiportant for the production capacity of the reser-
voir. Zais and Bodvarsson (1980) have carried out an investigation
of this type.
Starting with a reservoir of thickness
in equilibrium, the
H
production of fluid results in a change in pressure that propagates
through the system. The amount of fluid that can be produced upon a
given unit reduction in pressure is the fluid capacitance of the
system.
First, compare the relative fluid storage or release by the
compressibility of the reservoir formation with that of the level
of the free liquid surface. The reservoir is assumed to be an isotropic and homogeneous slab with a porosity
S
,
and fluid density
p
.
,
storage capacitivity
The amount of fluid mass released due
to compressibility for a small change in pressure
of unit area and height
H
p
on a column
is
= 2pH
.
The specific capacitance per unit area is defined as
(1.1)
3
dq/dP = pSH
(1.2)
.
Secondly, the displacement of the liquid surface by
= ip/pg
(1.3)
,
corresponding to a pressure reduction of
qf = (iip/pg)(p4) =
p
releases
(1.4)
g
and hence the specific capacitance of the free surface
dqf/dp = p/g
(1.5)
.
The relative amounts of fluid released for these two mechanisms
can be compared by the ratio
dqf/dp
dq/dp
4/psHg
(1.6)
Common reservoir field cases may involve a thickness H= 1 X103m,
capacitivity S= 2Xl0h1Pa
0.2
.
,
and a porosity in the range 4
0.01 to
Substituting into the ratio (eq. 1.6) we find that the
displacement of the free surface releases about
50
to
1000 times as much liquid mass as the compressibility effect.
Finally we will want to examine the significance of the fluid
release due to liquid vaporization within the reservoir. The relationship for the change in vapor pressure
PS
with temperature
degrees Kelvin along the saturation line denoted by the subscript
v
is given by the Clausius-Clapeyron equation that we can
T
4
approximate by
dp
F
where
p5
S\
-
(1.7)
is the vapor density, and
is the latent heat of
L
vaporization for the liquid. Rearranging and assuming saturation
conditions the temperature change is
= TAp/p5L
for a change in the pressure
or specific heat and
r
ip
.
(1.8)
If
is the heat capacitivity
Cf
the density of the wet formation then the
heat released per unit volume of wet formation is
= PrCfTsL
(1.9)
.
The mass of vapor released by a unit volume upon a change in
pressure
tp
at saturation conditions is
= zh/L = (pprCfT)/psL2
The specific capacitance for a column of height
dq/dp
=
(PrCfTH)/Ps2
(1.10)
.
H
is then
.
(1.11)
We compare the vaporization with the compressibility
effects on the basis of
dq/dP
(dq/dp) =
rCfTs
(1.12)
5
Assuming some typical values of 1=200°C (473°K),
Cf TX1O3J/kg°K , s= 2Xl0hiPa, p5=7
kg/rn3
,
r=2500
kg/rn3
L= 2X106J/kg
a value of about 2000 is obtained for the ratio.
Comparing this ratio to the values obtained previously for
the free surface and the compressibility this result indicates that
unlike the compressibility effects, total vaporization within the
reservoir material could theoretically release as much fluid mass
as the free surface effect. In most cases, however, in liquid
dominated systems vaporization is confined to the immediate vicinity
of the producing boreholes. The relative effects of the free surface
would then be about
10
times that of vaporization.
In the following discussion we will restrict ourselves to
cases in which vaporization is not significant and compression will
be negligible. This will involve not only geothermal situations
but also many hydrologic reservoirs as well. The iterative technique
we describe is equally suited for both situations provided that
a free liquid surface effect remains dominant.
II.
CONSIDERATIONS WITH AN ANALYTIC SOLUTION
In the most general sense, a reservoir is a collection or
storage place for anything in quantity. The particular situation
which we will consider involves the collection of a liquid within a
large volume of porous and permeable rock that extends to the
surface. Small interconnecting cavities or openings between grains
of the rock provide the space for the fluid storage and movement.
In some instances a reservoir may be bounded by fault planes,
lithologic changes, or layers of low permeability which restrict
fluid flow into or out of the reservoir area. The interconnection
of pore space within the rock allows for fluid movement and the
formation of a fluid level or liquid surface. Particuarly in our
case, the fluid surface is a liquid interface free to move and
respond to pressure changes exerted on it.
Basically the reservoir to be considered here is composed of
a porous, isotropic, homogeneous rock volume within the earth,
containing a liquid (water) with a freely moving liquid-air
interface and possibly bounded from above by some impermeable layer.
We assume that the fluid motion within the reservoir obeys Darcy's
law, that is, the fluid mass flow density is proportional to the
local pressure gradient induced. For the analytic model used, we
will consider the reservoir as infinite in extent.
7
The Concept of Drawdown
The free liquid surface of the reservoir responds to changes
in the pressure field. The gas phase above the liquid in our model
can be at atmospheric pressure or be confined by impermeable
overlying material. A homogeneous static pressure field acts on the
liquid surface, and by the principle of superposition we can
subtract the gas pressure from the pressure field within the fluid.
The ambient pressure field around a borehole withdrawing liquid
from the reservoir will be depressed. The decrease in local pressure
results in an observable lowering of the liquid surface in the
vicinity of the borehole. The drawdown is defined as the lowering
of the free liquid surface below the equilibrium position.
Analytic Solution for a Point Sink with a
Linearized Free Surface Condition
The following development is in part an overview of a paper
written by Bodvarsson (1977) with an emphasis on particular items
that are essential to the understanding of the material in the next
chapters. Although our notation will be adapted for the specific
case of radial symetry, we will maintain similar notational symbols.
In all cases the z-axis is defined as positive downward using
cylindrical coordinates.
Consider an isotropic, incompressible, homogeneous half space
of porous material with an area porosity
.
The area porosity is
defined as the fractional area of fluid conductive pores in a given
[:3
cross section surface. The material is saturated with a liquid of
density
p
.
Choose an equilibrium reference surface
which
corresponds to the initial liquid surface position at z=O and let
c
represent the liquid surface at a later time. The points on
are
S=(r,O)
E
and let P=(r,z) be the general field point within
,
the half space z>O
Our basic assumption is that flow within the porous medium
obeys Darcy's law
- p)
(P,t) = -C(v
where
C=K/y
conductivity
(P,t) = total fluid pressure
;
K= permeability
;
-= kinematic viscosity
;
(2.1)
= mass flow vector
g = acceleration of gravity
The second term on the right of equation 2.1 accounts for the
gravitational pull on a fluid flowing in the z-direction. If there
is given a mass flow source density as a function of position and
time
f(P,t)
then,
+
v.q = f
(2.2)
We now assume that the total pressure is the superposition of the
fluid hydrostatic pressure
pressure
p
h
and the flow or perturbation
due to the source density, that is,
(2.3)
where initially
h
= pgz
.
The external pressure on
c
zero. Since the second partial derivative with respect to
will be
z
of
is zero, equation 2.2 reduces in a homogeneous isotropic space
to
-v2p = f/C
(2.4)
The boundary condition on the free surface is
=0
Ot
where
D/Dt
(2.5)
is the material (or total) derivative. This is a non-
linear condition which results in the loss of the principle of
superposition. Consequently we are interested in a method of
linearizing this condition without placing too rigid a restriction
on the reservoir model.
If the position of the free surface
only a small, slowly undulating amplitude
c
deviates from
h(S,t)
by
z
that is positive
up, we can modify the free surface condition by moving the fluid
half space boundary to
pressure on
c2
z
and replacing the condition of zero
by the condition
p = pgh
on
plane
.
(2.6)
In other words, we replace the undulating surface
c
by the
and assume that to the first order the pressure difference
between the two surfaces is only hydrostatic. Moreover, assuming
that
h(S,t)
is very small compared to the source depth, we can
take that the flow pressure
p
at
E
is a small perturbation of
the total pressure and that to the first order
p
on
c2
is equal
10
to
on
p
E
In addition, the motion of the fluid at the free surface can
be approximated as strictly vertical. Then the kinematic condition
on the boundary represents the strictly vertical flow in a column
just below the free surface. Hence, the mass flow to raise the free
surface is
pth
t(Ph) =
(2.7)
and since by equation 2.1
= C3P
(2.8)
we obtain the relation
pth = CP
- aap = 0
or
a = Cg/q
where
(2.9)
z=0
at
(2.10)
is a characteristic fluid velocity for a porous
medium.
Equation 2.10 represents the linearization of equation 2.5
where the only major restrictions imposed have been that the free
surface
c
deviate only by a small amplitude from the horizontal
reference surface
z
,
and that the hydrostatic reservoir pressure
be much larger than the flow pressure in the neighborhood of the
free surface. A peculiarity about these conditions is that
takes on negative values when the
ci
surface is below
total pressure on the reference surface
E
E
h(P,t)
,
and the
is then negative.
However, negative pressures in this situation represent a
mathematical abstraction and do not present any physical problems.
Now that we have gained a physical feeling for how the free
surface condition (eq. 2.5) is approximated by equation 2.10 , it
will be helpful to look at the form of the terms neglected by the
linearization. The magnitude of the neglected terms will be useful
later to determine the error involved in using this approximation.
Starting with the original boundary condition (eq. 2.5) for the
free surface and the definition of the total pressure (eq. 2.3) we
expand the material derivative into its component terms. In our case
of axial symmetry
where
r
= ..2
r
+ wa
+
Dt
.
'
(2.11)
--)
= (pg
z
(2.12)
.
=(s,w) is the velocity vector of the pore fluid within the
reservoir. The pore fluid velocity can be expressed in terms of the
mass flow
flow
/p
by first dividing by the density to obtain a volume
, then by dividing by the area porosity
we get an
expression which represents the velocity of the fluid elements.
=
(2.13)
/pt
Rewriting the mass flow due to the flow pressure as
=
-Gyp = -C[
r
+
z
]
(2.14)
12
the pore velocity is then
=
where
S
5r
(2.15)
z ]
+
W_2.
p ar
p
Sz
Sz
/
(2.16)
The free surface condition then becomes
Dt
+
5t
St
o =
p
Sr
p
at
-
Sr
Sr
p
(2.17)
(2.18)
az
p4)
a[ ()2
a()]
pg
Sr
()2
az
]
(2.19)
Neglected Quadratic
Linearized Free
Term
Surface
Approximation
Thus in cases where the quadratic derivative term is small compared
to the linear derivative term
- a
(2.20)
we may neglect the quadratic and obtain the linearized free-surface
condition just as before.
Using the basic equation 2.4 and the linearized free-surface
condition (eq. 2.10) we can solve for the pressure field within the
half space. Consider first the solution obtained for the source-free
case where f0=0
13
The basic equations are
where
c
for
p = pgh0
and
(2.21)
z > 0
v2p = 0
t
0
,
z = 0
.
(2.22)
is the initial vertical amplitude for the free surface
h0(S)
relative to
z
and
,
is some general point on the
S
surface.
Bodvarsson (1977) observes that the solution for the pressure field
will be of the form
p = p(r,z+at)
(2.23)
which satisfies the boundary condition (eq. 2.10) for all time.
Since
p
as expressed by equation 2.23 is for t>O a potential
(r,z+at) in z>O , we can continue the function into
function of
z>0 by standard potential theoretical formulas (Duff and Naylor
1966) and hence
p(P,t) = [pg(z+at)/2ir]
for
with
and
t > 0
,
z > 0
,
rp
I
(l/r)
h0(u) d:
(2.24)
U(r' ,0)
= [ (r-r')2 + (z+at)2
dz = 2irr'dr'
(2.25)
(2.26)
.
The corresponding motion of the fluid surface
setting z=0 in equation 2.24 and 2.25 giving
c
is found by
14
h(S,t) = (at/2r) J (1/rt)h0(U) dE
= [ (r-r')2 + (at)2
and
]
(2.27)
; t>O
(2.28)
.
Next we will examine the pressure field due to a concentrated
point sink of strength
at a position Q=(0,d) within the half
f0
space. At t0 the fluid is in a static equilibrium with the fluid
surface corresponding to the reference surface
begins withdrawing fluid at
t=O+
z
The point sink
.
with a constant rate
f0
.
The
basic equation to be solved then takes the form
-v2p =
where
I(0)=0
1(t)
(P-Q) 1(t)
-fe/C)
(
(2.29)
is the causal unit step function that takes the value
for t=0 and
for t>0
I(t)=l
.
Equation 2.10 represents the
boundary condition placed on the free surface, and there is now an
initial condition of p=0 in z>0 , at t=0
.
Bodvarsson (1977) solves
this problem by applying the method of images. To obtain the
stationary pressure field as t
an image of strength f0 is placed
at Q'=(O,-d) giving the Neumann type solution of no flow through E
= (- f0/4irC)[ (1/rQ) +
(l/rQI)
1
,
t
(2.30)
and on the basis of equation 2.6 , the surface amplitude is
h5(S) = - fQ/2TrCP9rSQ
where
rpQ
[
r2 + (z-d)2
]
(2.31)
(2.32)
15
rpQI= [ r
+ (z+d)2
rSQ = [ r
+ d2
(2.33)
]
(2.34)
J
The general solution for t>0 is obtained by adding to the
stationary pressure field (eq. 2.30) a time varying component which
is initially equal and opposite to the stationary field thus
satisfying the initial condition of p=0 at t=0
.
The time varying
component is given by equation 2.24 with h0(U)= -.h5(S)
(eq. 2.31)
which upon adding to that of 2.30 results in the solution
p (P,t) = (-f0/41TC)[ (l/rQ) + (1/rQI)
where
rpQIt= [ r
(2.35)
(2/rpQIt) ]
+ (z+at+d)2
(2.36)
.
By setting z0 we obtain the flow pressure at the reference surface
and using our approximation (eq. 2.6) the vertical amplitude of the
free surface
relative to the reference surface
h(S,t) = (-f0I4TrCpg)[ (l/rSQ)
rSQIt- [ r2 + (at-Fd)2
:
(l/r$Q.t) ]
]
is
(237)
(2.38)
for S=(r,0) on the surface. Equations 2.35 and 2.36 represent the
analytic solution which we will refer to later.
Discussion of Assumptions
In the previous developement we made several approximations.
iI
Some of these have already been pointed out. One is using the
hydrostatic approximation to derive the pressure on the reference
surface, and another is assuming a constant liquid velocity in the
z-direction during each incremental time. In addition to these, we
have neglected capillary pressure as a force on the free liquid
surface. This is probably quite reasonable provided that we restrict
the applications to porous materials such as sandstone etc. in which
capillary forces are known to be small.
The flow pressure within the reservoir is being calculated as
a pure potential field neglecting compressibility and the resulting
pressure diffusion. In other words, we have assumed that the
pressure diffusion through the reservoir requires much less time
than the response of the free liquid surface. This is a reasonable
assumption because the time required for the pressure signal to
diffuse to one half of its full value over a distance
d
within
the reservoir is on the order of (Carsiaw and Jaeger, 1959)
(2.39)
tD
where
K
is the diffusivity of the reservoir formation. Conversely
equations 2.31 and 2.37 indicate that the time required for the free
surface to reach one half of its stationary drawdown is on the order
tn
where
a
d
(2 40)
.
is the characteristic velocity of the reservoir. The ratio
of the two times is
17
=
tFi/tD
where
s
/gpsd
(2.41)
is the capacitivity or storage coefficient that has values
of a few times 10_li Pa
values as
(d/Cg)(C/psd2)
q=0.l and
(Bodvarsson, 1970). Hence for such common
d=l03 m , the ratio is a few times io2
Other Reservoir Models and the Motivation
of the Present Investigation
The above results are quite simple and have been obtained by an
elementary method. Several alternative procedures including HankelLaplace transform techniques are also applicable for the same
purpose. The transform techniques are of a more general scope and
can also be applied to models of other geometries, in particular, to
the case of a reservoir of a finite thickness with the same free
surface boundary condition as above. This case is of considerable
practical interest. Bodvarsson (personal communication, 1981) has
investigated such cases and shown that a solution is readily
availiable in the transform space, but the Hankel-inversion is not
elementary and can not be expressed in a closed analytic form. It
is a complex double series where convergence poses a non-trivial
problem.
It is interesting that because of the parculiarities of the
free surface condition a simple method of images technique breaks
down in the case of the finite thickness reservoir. To cope with
such problems, it may in many cases appear to be of interest to
modify the free surface condition such that the Hankel inversion
iE3
becomes elementary. As will be elaborated on below, we will present
one such technique that replaces the free surface condition by an
iterative approach and has the following two advantages.
(1)
The time-dependent free-surface condition is replaced by a
stationary fixed plane Dirichiet type condition that moves the
problem back to conventional potential theory where availiable
standard solution techniques and numerical procedures can be applied.
(2)
The iterative approach is equally applicable to models with
more complex boundary conditions.
Our principle goal is to test the iterative approach on the
infinite reservoir case where the computational results can be
compared with a simple analytic solution.
The Reference Numerical Model
(RM1)
In this section a model will be developed that will approximate
the analytic situation for the infinite reservoir with the iterative
technique. The magnitudes of the parameters selected for the
numerical model are strictly relative and represent a reference case
called Reference Model
1
(RM1) upon which other models can be based.
Parameter values are chosen as reasonable values for a particular
case of a Darcy flow type sandstone reservoir with very distant
lateral boundaries from a borehole penetrating below a free liquid
surface within the reservoir. The multi-hole case will be discussed
later. The selected parameters are listed in Table 1
Table 1.
Selected parameters for model RM1
Parameter
Area Porosity
0.2
1 Xl07 sec
Conductivity
C
Radius (to boundary)
r
14.4
km
z
14.4
km
Borehole Flow Strength
f0
50.0
kg/sec
Depth of Borehole Flow
d
1.4
km
Flow Across Lower Boundaries
-
0.0
kg/km2sec
Pressure at Free Liquid Surface
-
0.0
kg/kin sec2
Depth
(to bottom)
The radius and depth of the reservoir are selected as
convienient approximations to the infinite half space of the
analytic model. They are convienient because they give a grid
spacing of 0.1 km for the numerical model when 144 numerical
divisions are used to calculate the solution values. The borehole
flow strength chosen is a reasonable figure for the rate of
production by a single well within a geothermal field.
In the numerical case the allowed drawdown is limited by the
requirement that it be small compared to the borehole depth and
the radial scale of the free surface response. Preferably, the
total drawdown of the free fluid surface not exceed
borehole depth. The
1,'5
1/5
of the
limitation is based on experience with
the perturbation techniques (Dr. G. Bodvarsson, personal communication, 1981).
The depth of the borehole was chosen to be approximately
1/10
the distance to the bottom and side boundaries. This puts the bottorn and sides sufficiently far away from the borehole that their
20
overall influence on the pressure solution values will be small.
The flow across the bottom and side boundaries is set to zero.
Since the boundaries are relatively far from the borehole, the
effects of this assumption are small within the borehole vicinity.
Borehole Grouping
A practical problem which may arrise is to determine at what
distance R0 can a particular group of boreholes be replaced by a
single borehole having a flow strength equal to the combined strength
of the group of boreholes and yielding nearly the same pressure
field. In particular, we want to know the distance beyond which the
pressure field due to the single borehole will be approximately
equal (within 10% for example) to the value from the borehole group.
Take
Mf
mf
as the flow strength of each of the grouped boreholes and
as the flow strength for the single borehole representing the
group. Assume for example that we have a distribution of boreholes
such that the flow points are placed at the vertices of a cube. At
some distance R0 the pressure field due to this group of wells will
be within a given fraction of that due to a single well with a flow
point located at the center of the grouping. The distance at which
the group of wells can be replaced by a single well will depend on
the group spacing.
Solving for the whole-space, homogeneous, isotropic, Darcy-type
porous medium pressure solution with the origin at the center of
the cubic grouping
21
MfS(r)
2
-7 p
(2.42)
41TCr
we obtain the pressure field for the single grouped well
(2.43)
PS = Mf/4TrCrS
where
r5
(x2 + y2 + z2)
(2.44)
.
Likewise the solution for each of the distributed boreholes is
p.
where
mf/4JrCr
(2.45)
+ (y-y')2 + (z-z')2
= [ (x-x
]
(2.46)
.
Summing up the pressure due to the distributed boreholes we have
Pg = (mfI4C)
(lIr)
The spacing of the cubic group will be
2L
(2.47)
.
on edges. Consider a
point P(x,y,z) directly out from the center of one of the faces of
the cubic group and at a distance
from the center of the cube.
Pg can be rewritten as
mf
Pg =
([[
3L2-2R
1+
2
R0
0
L
+ [1+ (
3L2+2R L
20 ) ]. (2.48)
R0
J
Using an expansion for (1+X), JX<1 and neglecting third order
and higher terms we obtain
22
Pg
rrCR0
2-]
[
.
(2.49)
The pressure at P(x,y,z) due to a single well at the center of
the cubic array with flow strength
P5 =
Mf
is then
M/4CR
(2.50)
The difference between the pressure field at P(x,y,z) from substituting a single well at the center of the cube compared with that of
the field from the cubic group of wells will be within 10% provided
R0 > L(2.34763)
where P(x,y,z) is at a distance
(2.51)
from the center of the cubic
group. If the spacing is L=lOO m then R0235 m
For a point on the free surface the drawdown from a cubic borehole group will appear nearly the same as that of a single well of
equal strength if the distance to the surface is larger than that
given by equation 2.51
.
In cases in which a group of boreholes are
producing from the same vicinity the group can be approximated by
a single well which can be examined using the iteration technique
to be discussed in the next chapter.
We can do a similar analysis for the fluid flow field in a whole
space situation. Consider a homogeneous, isotropic, whole space Darcy
type porous medium such that the fluid flow field is given by
vq5
M
f+ (r S )
4wr
The flow field for a single well at the origin is then
(2.52)
23
+
N r
fs
q5
where
r5
(2.53)
4'rrr
is defined by equation 2.44
.
The coordinate axis
x,y,z
are oriented perpendicular to the faces of the cube with the origin
at the center of the cubic grouping.
Consider a point on the z-axis P(O,O,z) and at a distance R0
from the origin. The fluid flow field along the z-axis from each of
the boreholes in the cubic configuration of strength
mf
is
mf (z-z.)
(2.54)
.
4-:;:
The total flow field in the z-direction at P(O,O,R0) from the
cubic configuration will be
(z_z)
8
m
For a cubic spacing of
irR
equation 2.55 can be rewritten as
(R0-L)
m
=
2L
(R0+L)
3L2-2R
o
[
R
(
(2.55)
.
3
0
L
2'
R0
+
3L2+2R
L
Ro1l+(_-2
Using a second order Taylor series expansion for (lxy
.
)J
(2.56)
1
J
xkl
and neglecting third order and higher terms we obtain
m
_i.2. [
rrR0
4
2+ 78.75(i_)
(2.57)
24
The fluid flow field at P(0,0,R0) for a single well of flow
strength
Mf
at the origin will be
(2.58)
2
4 R0
Taking the difference between
and
we find that if the
group of wells were replaced by a single well at the origin, then
the distance at which the two flow fields differ by no more than
10% is related to the group spacing by
R0 > L(4.33876)
(2.59)
If the error in determining the flow field is at least 10%
then for some region of interest at a distance R0434 m from the
origin a cubic group of producing wells with a spacing L=lOO m
could be replaced by a single well without any significant effect
on the flow field results at R0
25
III. THE TIME ITERATION TECHNIQUE
Time dependent effects of the free liquid surface can be
approximated using an incremental or iterative proceedure. This
involves replacing the free liquid surface by a stationary reference
surface at
z=O
and stepwise adjusting the pressure on the
reference surface such that the free surface condition is approximated as closely as possible.
Technique
The proceedure begins by solving for the pressure field due to
a point sink (source) placed on the symmetry axis of the reservoir
and assuming the initial free surface condition
t=O
.
p=O at z=O and
We then obtain a fluid velocity at the surface and use this
to determine an incremental displacement of the surface during a
sufficiently small time interval. Next a hydrostatic approximation
is used to adjust the pressure on the reference surface for the new
position of the free surface. These pressure values form a new
boundary condition for recalculating the pressure field within the
reservoir during an additional small time increment. The process
can then be repeated stepwise.
Considering the process in more detail, after the pressure
field is found for the initial free surface condition we can calculate the fluid flow at the free surface and subsequently the free
surface velocity. Since the free surface is sufficiently far from
the sink and the displacement sufficiently small then the fluid
flow in this region can be approximated as strictly vertical. Thus
only the z-component of the mass flow vector need to be considered
in order to obtain an expression for the velocity of the free
surface.
In chapter (2) we obtained a relation for the pore liquid
velocity
w0
-(C/p) ap/z
;
z = 0
(3.1)
.
If we assume that the calculated velocity does not change signifi-
cantly during the selected time increment then multiplication of
by the incremental time gives the displacement
h
w0
of the liquid
surface in the z-direction. Due to the fact that we have defined
h
as positive up
= (C/p)(p/z)t
.
(3.2)
In order to complete the time step procedure, the boundary
values for the pressure of the reference surface at z=0 must be
calculated. As discussed in chapter (2) we can take that if
h
is
small in comparison to the scale of the undulation of the liquid
surface, the difference in the total pressure for the reference
surface and the free liquid surface is essentially due to the hydrostatic pressure of the fluid column between them. Thus we can
approximate the flow pressure on the reference surface by the hydrostatic pressure of a fluid column of height
p = pgh =
h
t
(3.3)
Using this relation a new set of boundary values is generated
for the reference surface, and the pressure field within the
27
reservoir can then be calculated during the following time step.
Repeating this process allows us to derive an approximation of the
pressure field as a function of time. The schematic cartoon of
figure 1 illustrates several such time steps showing the relative
positions of the reference surface
c
E
and the free liquid surface
from both an analytical and a numerical point of view.
The Analytic Time Step Approximation
The analytic solution can be used to examine the criteria under
which the time step technique approximation is a valid representation
for the solution. We begin with a developement that is analogous to
the developement of the computer technique given above with the
analytic pressure field at t=0+
Upon completing a derivation of the
.
first time step flow pressure for a point on the reference surface we
show how this result is related to a first order Taylor series
expansion of equation 2.37 at (0,0). From the Taylor series
expansion we gain some insight as to the restrictions which must be
placed on the iteration technique.
Equation 2.35 for t=0+ is
- fo
p(P,0+)
[ l/r
-. l/rpQI
]
.
(3.4)
Taking the gradient with respect to z
f
az4
-
o
r
(z+d)
3
rpQI
(z-d)
3
J
(35)
rpQ
the vertical flow of liquid can be obtained from equation 2.8
.
A
relation for the initial vertical pore liquid velocity (eq. 2.16) is
h+
2+
point source at depth
Analytical Situation
d
Numerical Situation
p=O
- z=O
p=O
F(1,1)
d
::::
F(1,15)
Initial Conditions
Sclve for pressure field numerically
then increment the time to obtain
fluid displacement.
h
[Begin Time Step 1
z=O
E
h1(r) =
]
h1(r)
\4p1ogh1
og(0) + p1
Approximate flow pressure on z=O (z)
as the hydrostatic pressure of the
p=O
overlying water. This gives new
(
boundary values for the reference
surface (z). Substitute in
p1pgh1
r(1,15)
:
Solve for pressure field numerically
[End Time Step 1
Then increment the time to obtain
fluid displacement.
[Begin Time Step 2 ]
--- Z=
d
h2(r)=h1(r) +
p2=pgh2
\
h2(r)
N
1=g(0) + p2
Approximate flow pressure on z=O (z)
as the hydrostatic pressure of the
overlying water. We get a new boundary
condition for
.
Substitute in
p145)
:::
Solve for pressure field numerically
Continue incremental displacement
in time.
Figure 1.
1$
with new boundary condition of
time step 2
[End Time Step 2
Schematic cartoon of time step process.
3
then
=
4
r
(z+d)
I.
3
(z-d)
]
3
rpQ
rpQI
,
t=O
.
(3.6)
Assuming that the velocity is approximately constant over some
small time interval
by a distance Ah
.
t the free surface will be displaced vertically
Consider a point (0,0) on the reference surface
directly above the point sink, the displacement will be
=
w0(0,0)t1
which is analogous to equation 3.2
.
(3.7)
Substituting in gives
at
f
(3.8)
for
t1
=
t
and
r = 0
. The iterative flow pressure at (0,0) is
obtained using a hydrostatic approximation on the reference surface
as
p1
= pgh
=-
f
0,'
at
)
(3.9)
.
From the complete analytic solution the vertical amplitude of
the free surface
c
relative to the reference surface
z
is given
by equations 2.37 and 2.38 as
h(S,t) = -(f0/2rrCpg)
[
lirSQ
lircQlt
]
.
(3.10)
For the point (0,0) on the reference surface we have
h(S,t) = -(f0/2TrCpg) [lid
which can be rewritten as
- l/(atd) ]
(3.11)
30
h(0,t) = -(f0/27rCpgd)[ 1
!)
- 11(1+
The first order Taylor series expansion for
11(1+ -)
-
1
II << 1
provided that
11(1 + -)
is
at
(3.13)
(3.14)
.
Then substituting into equation 3.12 for the time
t = t1
we get
at
f
h(0,t1)
(3.12)
]
2pgC
)
(3.15)
and using the hydrostatic approximation we obtain the flow pressure
on the surface at (0,0)
at
f
p
C
_-.1-
,
=1
(3.16)
which is identical to equation 3.9 that we obtained using the
iteration technique.
In making the Taylor series approximation we found that it
is necessary to have the restriction stated in equation 3.14 to
obtain a solution equivalent to equation 3.9
.
Equation 3.14
gives a measure of the magnitude of the time step that can
be allowed.
31
Computer Programs
The iteration technique described here is implimented by a
main program supplemented by a group of subroutines. The basic
iterative subroutines are VELOC, DIFTBL, and the subroutine package
PWSCYL along with the function DERVZO. These subroutines with the
assistance of the calling program POSGO perform the iteration
technique under the direction of the user. Additional subroutines
RADF, QUADR, and RZDIFT along with the function DERVR assist the
user in obtaining information about the iterative solutions
calculated.
Before getting into a more detailed discussion of the time
step technique used. it will be helpful for later reference to
briefly explain the notation and subroutines which are directly
involved with the iteration technique. A more thourough discussion
is supplied in Appendix B along with program documentation.
The main program and all subroutines except the subroutine
group PWSCYL were written specifically for use in studying the
time iterative technique. PWSCYL is a group of subroutines which
was obtained through Dr. Jonathan Hanson from Lawrence Livermore
Laboratory as part of their mathematical software library. Its
original authors are Paul Swarztrauber and Roland Sweet (Technical
note TN/IA-109
Research
,
July 1975) of the National Center for Atmospheric
Boulder Colorado 80307
Initially, the subroutine PWSCYL calculates the pressure field
solution for the case of zero pressure on the free surface z0
32
PWSCYL solves a finite difference approximation to the Poisson
equation in cylindrical coordinates. Solution values are then
stored in a matrix array set up by the main program. Once this is
accomplished the user then prompts the program to calculate a new
set of boundary conditions using the subroutines VELOC and DIFTBL
along with the function DERVZO
VELOC sets up the conditions whereby DIFTBL and DERVZO can
calculate a fourth order polynomial derivative at each point along
the free surface. VELOC then calculates a new pressure boundary
value by equation 3.3
,
and stores the value in the corresponding
matrix locations.
The main program then resets the input matrix array with the
new free surface boundary values. A prompt from the user calls
PWSCYL again to recalculate the pressure field solution values using
the new boundary conditions. This process may be repeated until the
job is completed.
Other features within the main program allow the user to output or view solution values at any stage in the procedure. Some
additional features also allow the user to output a comparison of
solution values at any stage of the process with the analytic solution given by equations 2.35 and 2.36
.
The user is also able to
obtain information on both the radial and z-derivative values for
points along the free surface.
In the next chapter we will be discussing some equations which
use and relate the notation of both the programs discussed here and
the analytic solution derived in chapter (2)
.
Table 2 is set up to
33
relate these two notational schemes and briefly explain specific
parameters utilized in the iteration process.
Table 2.
Brief discussion of numerical parameters used in the
time iterative technique.
Parameters
Numerical
Analytic
B
r
-
The radial range of the reservoir
0< r <B [A=0 always]
M
-
-
Number of radial subdivisions for the
interval [A,B]
D
z
-
The range of z (depth.) for the reservoir
0< z <0
N
-
-
Number of vertical subdivisions for the
interval [C,D]
-
Specified values for the normal derivative
on the boundaries A,B,C,D respectively
BDA,BDB
BDC,BDD
*
F
-
Input - matrix which specifies the source!
sink information along with
specified boundary data
Output - matrix of pressure solution value
-
-
Specifies the boundary condition
(see Appendix B )
ELMBDA
-
-
Specifies the calculation of the Poisson
equation by PWSCYL (ELMBDA = 0 always)
TINC
t
-
Time step increment
TIM
t
-
Time (sum of time step increments plus
initial time)
COND
C
-
Fluid conductivity of medium
-
Porosity of medium
-
Density of liquid
PHI
RHO
*
p
Relationship between
F
and
f0
given later in chapter (4).
34
IV. COMPARISON BETWEEN ANALYTIC AND NUMERICAL MODELS
Basically the analytic model and the time step reference model
(RM1) are very similar. Both use a linearized form of the free
surface condition, and both approximate the flow pressure on the
reference surface z by the hydrostatic approximation in the space
between z and c
.
The major differences between them are due to the
inherent problems of the finite difference computer techniques. Some
of these problems, like the infinite reservoir in the analytic case
can only be approximated by putting the sides and bottom boundaries
very far from the region of interest. Also the analytic point source
has to be approximated by a small source region in the numerical
situation. Yet the similarity of the calculated solutions for each
case indicates that the approximation techniques are quite adequate.
Relationship Between the Analytic and Numerical iodels
The source (sink) term of the analytic model (f0) can be
related to the source term of the numerical model (F) by integrating
over the space surrounding the point source in the analytic case and
suming over all of the grid points for the numerical case.
Analytic
:
-v2p = (-f0/27rrC)
(r)
(d-z)
Jv2p dv = J[(foI2rC) ç(r) 5(d-z) ] dv
dv =
Jv2p dv =
(4.1)
(4.2)
rdodrdz
(4.3)
f0/C
(4.4)
35
Numerical
v2p =
:
F(r,z)
(4.5)
10< r)
[ constant = F0
)< z <(d+
(d-
)
I
F(r,z)
Jv2p dv =
(4.6)
; Otherwise
0
1
JF(rz) dv
(47)
Id4
B
p dv =
-F0rrr
2t
(4.8)
z
J
Jv2
d-
'0
Equating equations 4.4 and 4.8 we obtain a relationship between
f0
and
F0
-
)2
D
(4.9)
)
2M
which allows us to determine an input value
F0
for the numerical
model that corresponds to some selected mass flow rate
f0
.
The
minus sign in equation 4.9 results from an interchange of sources
and sinks.
One other peculiarity of the computer technique is that we
have expressed the input reservoir dimensions in kilometers (km)
rather than meters (m)
of the parameters
.
This is done so as to reduce the magnitude
B (radius) and
D (depth)
.
It has been suggested
that the finite difference technique used to calculate the solution
for Poisson's equation will give better results if the magnitude
of these parameters is small (Dr. F.T.
Lindstrom, personal communi-
cation,1980). However, this implies an unusual set of units where
36
all length dimensions are in km (mass in kg and time in sec remain
the same).
F0
used as an input then has dimensions of
and the output values of pressure have dimensions
kg/km3sec2,
kg/kmsec2
Translating the pressure values into the standard MKS system is
fortunately a simple task. Simply by dividing
p
by l0
we obtain
the equivalent MKS value.
Comparison of the Models
Comparison between the numerical and analytic solutions can
be made using several techniques. The most obvious and quite satisfactory procedure is to plot the position of the free surface (or
some surface of interest) for both models in the same scale directly
over one another.
Another way to compare the two situations would be to consider
the ratio of the pressure values. On this procedure we operate with
the Ratio (R)
R
Numerical Solution
Analytic Solution
(4.10)
at selected locations within the reservoir.
In most of our comparisons we consider specific profiles which
are comon to both situations. Particuarly three specific profiles
are chosen to illustrate essential changes within the reservoir.
One of these is obviously along the reference surface (or slightly
below the surface for the t=O case) starting at the borehole and
extending radially out. This profile will be of special interest
because the boundary conditions along this surface define
37
the pressure values in the reservoir during the succeeding
iteration. Another profile of interest is along the central axis
extending from the reference surface down. A third profile can be
located below the reference surface and extend radially out. In
general we have chosen to look at a profile beginning somewhere near
the point sink. These three areas give a fairly good coverage of
the changes taking place in the reservoir. In the particular case
below we have extended these profiles out to a radius or depth of
about 3 km for a reservoir with an outer radius of over 14 km
.
At
this distance from the numerical boundaries any boundary effects
will clearly be negligible, and yet a good understanding of the
significant characteristics of the pressure field can be obtained.
The Homogeneous Boundary Value Solution
Initially we begin our iteration procedure by setting the
pressure on the free surface to zero. The solution to this situation
corresponds to the half space homogeneous boundary value case p=O
at t0 and z0 as we discussed in chapter (2) for the analytic
solution. However with the side and bottom boundaries having
a
no-flow condition, derivative normal to boundary equal zero, the
numerical solution represents a deviation from the half space case.
The homogeneous boundary value solution is a simple case and
will provide a good situation to obtain a measure of how well the
numerical solution matches the analytic solution. One thing to
consider is the effect of the lower reservoir boundary on the
solution values. By keeping the grid spacing constant and varying
the number of grid points between the sink point and the boundaries
we can examine the effect of the no-flow condition on solution
values. This changes the ratio
to the boundary distance
L
a reasonable value would be
d/L (ratio of point sink depth
d
) for which we concluded in chapter (2)
1/10
or less.
Looking at the plots generated in figures 2, 3, and 4
the most
apparent effect of fewer grid points is a poorer fit between the two
models. In particular, the R value increases at a greater rate for
larger radii. For M=N=144 or 100 the similarity of the R curve
shapes and their closeness to unity indicate that the boundary
effects are minor for these values. Considering that using M=N=lOO
rather than 144 requires about 30% less computational time it is
reasonable to consider the lower value as an adequate choice for
many cases. However, the choice of the M,N values should depend on
the specific requirements of the situation.
One other feature which we should point out is the way in
which the numerical and analytic solutions close to the borehole
sink match. Since the analytic point sink is of infinite density
we really can't compare the two situations at the sink position,
but the solution values are finite elsewhere. Within about two grid
spacings of the borehole sink, the two models become strikingly
different. However, beyond this distance the solution values compare
much better. This effect is observed in each of the graphs in
figures 2 and 3 regardless of the M and N values.
In the graphs shown we do not compare the two models out past
3 km (30 grid points) although they compare relatively well at
39
A
1.080
rr.i.
L020
1.1.1.]
0.1
0.5
2.1
1.3
r '
Z
Figure 2.
Profile of
R
2.9
1.4
values for the homogeneous boundary value
solution for three different boundary distances relative
to a constant point sink location.
Ei]
tE'IsI
1.0200
r
Sink at (0,1.4)
IA
0.4
2.4
1.2
2.8
z-.,
r=O
60
,
B=D=
B - M=N= 100
,
B=D= 10.0
- M=N= 144
,
B=D= 14.4
A - M=N=
C
Figure 3.
Profile of
R
6.0
values for the homogeneous boundary value
solution along the radial axis for three different
boundary distances relative to a constant point sink
location.
41
Jill
JóI:1
I.I
1.04
I.02
0
0.4
.2
r b
2.0
2.8
Z=O.2
A - M=N=
60, B=D=
6.0
B - M=N= 100, B=D= 10.0
C - M=N= 144, B=D= 14.4
)
Figure 4.
Profile of
R
values for the homogeneous boundary value
solution radially just below the free surface for three
different boundary distances relative to a constant
point sink location.
42
larger distances up to the vicinity of the boundaries where the fit
becomes poorer again. From an examination of R values at constant
depth radially out from the point sink we find that the pressure
values of the two models are within 10% of each other as far out
as 5.1 km
Because of linearity these results are independent of the sink
strength. When the strength is multiplied by some constant, the
solution values are uniformly also multiplied by some constant.
The analytic solution most clearly exemplifies this situation
because the strength of the sink is a simple multiple in the
pressure solution (see eq. 2.35), and any multiple of this value
then merely multiplies the solution values likewise. However, the
validity of the linearization depends on the drawdown amplitude.
Approximation of
dp/dz
by a Polynomial
In order to find the liquid velocity at the free surface in
the iteration technique we have to calculate the vertical pressure
gradient at the reference surface. An analytic expression for the
pressure as a function of depth at z=O is obtained by fitting a
polynomial curve to points there. The positions used consist of a
point on the reference surface and adjacent points along the zdirection. By taking the derivative of this polynomial at the z=O
position we obtain an expression which approximates
dp/dz
at z=O.
Appendix A describes the developement of the fourth order polynomial
derivative using Newton's Divided-Difference Method. A brief
discussion is also included on the subroutines used to calculate
43
the polynomial derivative.
The selection of a fourth order polynomial was based mainly on
a trial and error procedure in which a higher order polynomial was
found to offer no significant improvements. For example, in a
particular situation we calculated the incremental reference surface
pressure to be p
-9.9822Xl06at z=O using a fourth order polynomial
and p= -9.9949Xl06at z=O using a fifth order polynomial. The change
in the two surface values for an increase in polynomial order is
less than one percent.
The Validity of the Linearized Free Surface Approximation
There are at least two major factors to consider in deciding on
the validity of the time step technique. First we need to know how
well the linearization approximates the true situation, and second,
how well the time step technique approximates the analytic form of
the linearization. The second factor can be evaluated relatively
easily by simply selecting the model parameters such that they fit
the analytic model. Then by calculating some values and comparing
them we obtain a measure of how closely the two models match.
However, no analytic solution is known which satisfies the
complete non-linear free surface condition in equation 2.5
.
Thus
we really have no good analytical procedure at our disposal to test
how well the linearization would fit an exact solution. Nevertheless
by explicitly calculating the quadratic and linear boundary terms in
given cases it is possible to obtain a measure of their relative
magnitudes.
44
A convienient way to compare these two terms is by taking the
ratio which we call Quadratie
QR
Quadratic Term
Linear Term
QR
(4.11)
where the quadratic and linear terms are give by equations 2.19 and
2.20 respectively. QR is convieniently expressed as a percentage
value. If this value does not exceed 10% then we consider that a
valid justification for excluding the quadratic term since its
contribution to the overall free surface condition is small. The 10%
limitation is based on common experience with perturbation techniques.
Testing out this technique with the reference model RM1
discussed in chapter (2), we find that, in fact, the quadratic terms
are small in comparison to the linear term provided that the sink
strength and the total time of drawdown are within reasonable bounds.
Table 3 gives some selected examples that have been calculated using
this model with a time step of 4XlO7sec (463 days). We find that QR
tends to limit the admissible sink strength (f0) in this particular
model. In addition, if we know QR for some specific f0 value and some
specified model parameters, then within the present approximation the
QR' for some other f
value with the same model parameters is simply
fi
QR'
) QR
(
.
(4.12)
It is also of interest that the surface point having the highest QR
value tends to change with time and approaches in the case of RM1 a
stable position near (r,z)=(l.0,0)
.
Calculation of QR for the
45
analytic solution also indicates a similar change of position for
the maximum QR with time. The position it migrates to depends on
the depth of the point sink.
Table 3.
Selected examples of the maximum QR for different times
using a time step of 4XlO7sec with model RM1.
Time X107 sec
Position (r,z) km
QR (f0=50)
QR (f0=l000)
0
(0.0,0)
0.00416
0.0832
4
(0.0,0)
0.00299
0.0598
12
(0.7,0)
0.00221
0.0442
20
(0.9,0)
0.00244
0.0488
40
(1.0,0)
0.00405
0.0810
60
(1.0,0)
0.00634
0.1268
80
(1.0,0)
0.00902
0.1804
The Time-Iterative Solution
The time-iterative solution is the next step beyond the
homogeneous boundary value case since it begins with a zero pressure
on the free surface boundary then perturbs the situation and
recalculates a new solution. We still must consider the fact that the
time step procedure depends on a pressure field solution which
contains boundaries at distance. However, comparing this to the
analytic solution through many time steps will be a good test of the
iteration method.
Figure 5 shows a profile of the free liquid surface for RM1 as
set up in chapter (2) over a time period of 8OXIO7sec
.
Here we have
selected a time step of 4XlO7sec as an example. The surface drawdown
is in a conical shape around the borehole with the maximum at the
(Km) r-'
[I]
0.2
0.4
I
0.6
I
0.8
I
1.0
I
OXIO7sec
1.2
I
1.4
I
16
1.8
2.0
2.2
4
2.6
_I
0.5
(m)
Z=h
20X 101 sec
2°H
40 X
3.5
60 X IO7sec
4.0
8OXIO7sec
4.5-F
Figure 5.
Profile of the free liquid surface radially outward from the central axis for
several different times of the model RM1 with At= 4XlO7sec
.
2.8
47
central axis and smoothly becomes smaller radially outward. The
initial drawdown rate is large, but rapidly slows down with
increasing time as indicated by the time lines being closer together
for large
t
The profiles shown here are scaled with a rather large vertical
exaggeration of 1000 to emphasize the drawdown features. As noted
earlier for the homogeneous boundary value case, the magnitude of
the pressure field within the reservoir is directly proportional to
the value of the sink strength (f0)
.
This effect holds also through
the time step procedure. Consequently if a larger withdrawal
(e.g. 500 kg/sec) were considered the curves would still be the same
only the scale on the z-axis would be multiplied by a constant
factor (equivalent to a vertical exaggeration of 100).
The next problem is to find out how well figure 5 represents
the analytic situation which RM1 is designed to approximate. Figures
6 and 7 show the ratio R for two of the selected profiles (radially
out from the sink and along the reference surface, respectively)
over the time interval of the drawdown. After the first time step
(time = 4XlO7sec) the free surface has significantly overshot the
drawdown of the analytic solution near the borehole. However, it
quickly settles down and achieves a fairly close approximation to
the analytic solution after several time steps. Farther out from the
borehole the R values indicate that the difference between the
analytic and numerical solution becomes notably larger, and in
addition the trend over time is for this situation to become worse.
Yet figures 6 and 7 indicate that the divergence of the analytic
Time
III.'.'.]
A -
8(lO7sec
B - 6OXlO7sec
C
- 4OXlO7sec
D - 2OXlO7sec
1.0800
E
-
4XlO7sec
11
t.j
R11L.I.]
C
n
1.0200
E
'SI.'.'.]
[.WI;I.II
0.1
0.5
13
r '
2.1
Z=I.4
Figure 6.
R
values plotted radially out from the point sink
at several different times for RM1 with it= 4XlO7sec.
49
I.
I.
I.
I.0
R
I.0
1.0
I.0
I.0(
o
0.4
1.2
r '
2.0
2.8
z=o
Figure 7.
values plotted radially out along the reference
surface at several different times for RM1 with t=4XlO7.
R
50
and numerical models with time is not very rapid.
We also are interested in how well the polynomial derivatives
of the numerical solutions compare with the analytically derived
derivatives. To illustrate the similarity over time for several
surface points Table 4 list some calculated values for a particular
case (it= 4XlO7sec). Comparison indicates that the numerical
polynomial derivative values are fairly good approximations to the
analytic solutions.
Table 4.
Comparison between analytic and RM1 polynomial derivative
values for
t= 4XlO7sec at two selected points on the
reference surface.
dp/dr
Time
Analytic
Numerical
(r,z)=(0.4,0)
%Diff
Analytic
dp/dz
Numerical
%Diff
0
0
0
-36.09X106 -36.19X106
<1
20Xl07
8.05X106
8.60Xl06
7
-13.47X106 -l2.27X106
9
40X107
9.49X106
9.76Xl06
3
-6.90Xl06
-6.38X106
8
60X107
9.93Xl06
l0.11XlO6
2
-4.l7XlO6
-3.99X106
4
0
(r,z)=(2.4,0)
0
0
0
0
-5.l9Xl06
-5.26X106
1
20X107
3.96Xl06
4.l5X106
5
-4.90X106
-5.08X106
4
40Xl07
6.l9Xl06
6.41X106
4
-3.80Xl06
-3.89X106
2
60Xl07
7.34Xl06
7.49X106
2
-2.83X106
-2.92Xl06
3
In Chapter (2) we established that a drawdown of
1/5
the
borehole depth would be an upper bound for the applicability of the
linearized free surface approximation. Calculating out the maximum
drawdown at the borehole for RM1 after a time of 8OXlO7sec
51
(t= 4XlO'7sec) we get 4.42 m for a flow strength f0=50 kg/sec
.
This
compares well with the analytically determined value of 4.27 m for
the same time, and it is well within the established limit. Another
way to view this situation is to use the maximum drawdown limit for
this model and calculate the maximum flow strength required to
produce that drawdown over the time interval. Since the pressure
values calculated for a particular flow strength f0 can be used to
determine the pressure values for any other flow strength f
(as a
simple multiple f/f0 ), this task becomes an easy problem. We find
that a flow rate of 3168 kg/sec would produce the drawdown limit
for RM1 (t= 4X1O7sec) after 80X1O7sec
However, the maximum drawdown is not the only limiting factor
for the flow rate. The Quadratio (QR) also changes as approximately
a simple multiple (f/f0) of the flow rate. Taking the maximum QR
value calculated along the reference surface of the numerical model,
we use this to set up a proportionality and find the limiting f0
value. Table 5 list the maximum f0 values allowed based on the QR=O.l
limit defined earlier. The limitations placed on the quadratic term
specify an upper limit to the flow rate of 554 kg/sec for this model.
Thus the QR value becomes the limiting factor at least in this case.
The iterative technique
we have discussed works adequately for
most physical models. It is however limited by an inherent
complication which may arrise in certain cases. Generally these
computational artifacts are easily distinquished. In the cases we
have dealt with, a sudden reversal in the pressure gradient caused
anomalous values to occur at the point (0,0) (or nearby points on
52
Table 5.
The maximum flow rate
kg/sec) possible for RM1 to
f
maintain the established limits on QR through successive
time iterations. Based on the largest QR value for each
iteration ( Time in Sec. X107 ).
=
Itera.#
4X107
Time
Time
om
Time
lOXlO7
9X107
8X107
5X107
om
Time
om
Time
om
1
4
1672
5
1849
8
2108
9
2060
10
1960
2
8
2116
10
2234
16
2048
18
1905
20
1735
3
12
2267
15
2204
24
1735
27
1600
30
1501
4
16
2202
20
2009
32
1345
36
1132
5
20
2048
25
1778
40
1112
45
949
6
24
1863
30
1563
48
946
7
28
1684
35
1368
56
712
8
32
1516
40
1204
64
62
9
36
1365
45
1067
10
40
1233
50
951
11
44
1118
55
855
12
48
1018
60
771
13
52
931
65
702
14
56
855
15
60
788
16
64
729
17
68
677
18
72
632
19
76
591
20
80
554
53
the reference surface) after several time steps. Continuing the time
step procedure resulted in similar discrepancies at other points
along the surface. A possible cause for this condition may be that
the numerical polynomial derivative values become small after some
given number of time steps and the calculated value then becomes
subjected to irregularities of the polynomial approximation.
Consequently when the polynomial derivative becomes small it may
begin taking on incorrect negative values which result in the
observed pressure gradient reversals. Thea priori predicting of such
gradient reversals would require a rather involved analysis.
The parameter
b
(4.13)
b =
has been found to be related to the maximum number of time steps
taken before an anomalous pressure gradient reversal occurs. It is
a factor which appears in the calculation of the free surface
pressure values for a time step (see eq. 3.3), and we have called it
the characteristic interval. Its name is chosen because
Cg/q
is
refered to as the characteristic fluid velocity (Bodvarsson,1977).
Figure 8 shows how this is related to the number of time steps taken
for a point sink. The values of porosity, conductivity, and the time
increment can be varied for convenience without affecting the program
operation provided that
b
remains the same. We can also change the
magnitude of the point sink (source) strength f0 without affecting
the number of time steps taken before the pressure gradient reversal
occurs.
54
2(
Cl)
cI
E
a)
0
C
5
U.1
U.
Q.
0.4
0.5
0.6
0.7
0.8
0.9
1.0
b= 2-9-t
0
Figure 8.
Maximum time steps before reversal of pressure gradient
as a function of the characteristic interval for RM1
4
BIBLIOGRAPHY
Bodvarsson, G.,1970, Confined Fluids as Strain Meters; J. Geophys.
Res., 75(14), 2711-2718
Bodvarsson, G.,l977, Unconfined Aquifer FLow with a Linearized
Free Surface Condition; Jku1l, 27
Carnahan, B.,
H.A. Luther, and J.O. Wilkes. 1969. Applied Numerical
Methods. John Wiley and Sons, New York, 604 pp.
Carslaw, H.S. and J.C. Jaeger. 1959. Conduction of Heat in Solids.
2nd ed., Oxford University Press, London.
Duff, G.F.D. and D. Naylor. 1966. Differential Equations of Applied
Mathematics. John Wiley and Sons, New York, 423 pp.
Swarztrauber P. and R. Sweet, 1975, Efficient Fortran Subroutines
for the Solution of Elliptic Partial Differential Equations.
NCAR, Tech. Note , TN/IA-109
,
Boulder Colorado.
Zais, E.J. and G. Bodvarsson, 1980, Analysis of Production Decline
in Geothermal Reservoirs, LBL-112l5, GREMP-lO.
APPENDICES
56
APPENDIX
A
Development of a Fourth Order Polynomial Derivative
Development of an interpolating polynomial using Newton's
Divided-Difference Method is given by Carnahan, etal. (1969).
Adapting a similar notation we can write out a fourth order interpolating polynomial without the remainder term as:
y(x) = a4x4 + a3x3 + a2x2
a1x + a0
(A.l)
where,
a4 = y[x4,x3,x2,x1,x0]
a3 = y[x3,x2,x1,x0]
-
(A.2)
(x0+x1+x2+x3)y[x4,x3,x2,x1,x0J
(A.3)
a2 = [(x0x1+x0x2+x0x3+x1x2+x1x3x2x3)y[x4x3x2x1x0]
(x0+x1+x2)y[x3,x2,x1 ,x0] +
a1
=
(A.4)
[(x0x1+x0x2+x1x2)y[x3x2x1x0] - (x0x1x2+x0x1x3+x0x2x3+
x1x2x3)y[x4,x3,x2,x1,x0]
-
(x0+x1)y[x2,x1,x0] +
y[x1 x0]J
a0 =
-
(A.5)
{(x0x1x2x3)y[x4x3x2x1x0] - (x0x1x2)y[x3,x2,x1,x0] +
(x0x1)y[x2,x1,x03
-
(x0)y[x1,x0] +
y[xx1 ...,x0]=
.
Xl] Y[Xn1Xn_2
xn
-
xO
(A.6)
X0]
(A.7)
Taking the derivative we have,
y'(x) = 4a4x3 + 3a3x2 + 2a2x + a1
(A.8)
57
for the sample values y4, y3, y2, y,, y0 corresponding to the
positions x4, x3, x2, x1, x0
y
.
In this situation the spacing of the
values determines the denominator of the difference thus only
relative magnitudes of the
x
positions are significant.
We need to know the derivative at the reference surface.
Consequently, for y0 at the surface z=O we choose x0=O for convenience
such that y1,y2,y3,y4 and x1,x2,x3,x4 are successive points below the
surface. The derivative is then taken at x=O which helps simplify the
form of equation A.8 to
y'(0) = a1
(A.9)
To calculate the derivative values on the reference surface with
the time step technique a subroutine called DIFTBL is used to set up
the Newton's divided-difference table for the selected values. Input
parameters are listed in Table 6
Table 6.
Input for subroutines DIFTBL and DERVZO to calculate the
polynomial derivative of the free surface.
Parameter
X
-
Array containing position values x0,x1,x2,x3,x4
with x0x(1)
F
-
Matrix array of solution values for the reservoir
IC
-
Radial position F(IC,l) of first polynomial point
x0
TABLE
-
(x1,x2,x3,x4 are along the z-axis from x0)
Matrix array of the divided difference table values
(output)
MP
-
Polynomial order
NP
-
Number of points used
K
-
Dimension of arrays
OK
-
Check for polynomial order errors
X
and
TABLE
Arrays
F
and
X
must be set up prior to use of the subroutine.
Output values in matrix array
TABLE
are used with Function
to calculate equation A.9 for the point
fit to points
F(IC,J)
J=i,5
F(IC,l)
DERVZO
using a polynomial
APPENDIX
B
Program Listings
The interactive calling program for the iteration technique is
POSGO
.
Using a set of command prompts from a terminal the user
directs the computational actions and checks progress or results.
The program documentation contains a complete list of these commands
with explanation. In addition, all parameters required are explained
in the program documentation for each section or subroutine used.
Some commands require additional information from the user before
the job can be completed. In this case, the program prompts the
user with appropriate questions or format required for input.
The program operation of POSGO begins by a prompt for a command.
Entering a command from the terminal results in the program operation
being shifted to the specific command section. If additional
information is required (e.g. input of a parameter value, profile
position, etc.) a prompt will appear on the screen indicating the
type of information required and/or the format specification expected
on input. When sufficient information is obtained the appropriate
subroutines are called and jobs performed. After completion of the
calculations or task for a command section the program returns to
the command selection and prompts for another request.
POSGO and associated subroutines are written in single precision
FORTRAN IV and designed to be used on a CYBER computer. To use on
some other computer the array allocations and input/output sections
must be checked for compatability.
?ROGRAP4 POSGO(tNPUT,OtJTpUT,TAeQ,TApEj2,TApEjj5,T4pEj,54,J)
C
C
C
C
C
C
C
CC CCCCC CCCCCCCCCCCCCCCC CCCCCCCC CCCCC CCCCCCCCCC C 0000CCCCC coco cc c CCCC CCCC
POSGO IS AM INTERACTIVE CALLING PROGRAM FOR THE SUBROUTINES
PWSCYL,...
IT GIVES THE USER A WAY OF CHANGING PARAMETERS
AND RECALULATtNG THE SOLUTION FOR THE 2-OIM P0155CM EQ. (OR THE
MORE GENERAL MEL PHOLTZ EQ.)
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
WRITTEN BY
S
VERSION
*
WILLrAM E. IVERA
TIME ITERATIVE DROCEEDU
DEVISED BY *
DR. GUNNAR BOOVARSSCN
C
3
MARCH 191
1
INPUT I CUTPUT SPECIFICATIONS
TAPEINTIAL PARAMETERS INPUT TO P05CC
TAPEI1=OUTPUT FILE USED TO MAKE A SECOND RUM CR IS REALLY
JUST A COPY OF (I,J) AS
ALLEO ON INPUT IC PWSCYL.
TAPE12OUI°UT FILE CONTAINING SOME PARAMETERS USED ALONG
WITH THE OUTPUT VALUES
(AS FOLLOWS)
PARAMETERS
LINE 1
A,B,M,IBOCNO,CONOIFLG1,IFLG2 (ZX,2F15.4,2110,E15.8,jx,2Ij)
LINE 2
C,O,N,HBOCNO,ONI,FLG3,IFLG4 (2X,2F15.'4,2110,E15.8,1X,211)
(2X,L.Fj5.)
LINE 3
ELMBDA,RHO,TINC,T!M
LINE . THROUGH LIME (M+1)+3 OR LINE (M+j)+3 WHICHEVER LARGEST
BDA(t)
908(1), 30C(I), 300(I)
(2X,'.F15.4)
FROM THIS POINT TILL END OF FILE
OUTPUT FQOM PWSCYL CF(11J)]
(2X,E15.8,I1X,Ej5.8,11X,Ej5.8,I1X,E15.8,lj.X,E15.8,jjX)
TAPE1(.CCPY OF THE 4PAMETERS WHICH CAN BE USED LATER AS
A PARAMETER INPUT FILE
OUTPUT FILES
PRSENTLY
TAPEj.
AND
10IHF155 ALWAYS
TAPEL2
(THE
INCLUDE LINE PRINTER CARRAGE CONTROL
DIMENSION CF THE F ARRAY)
I
15 IN THE RADIAL DIRECTION. N IS IN THE Z DIRECTION
(USING CYLINDRICAL CURD.)
P
P
Q
N
MUST BE OF THE FORM (2) (3) (5) , P,Q,P ARE MON-NEGATIVE INTEGERS
LIMITS ON N AND P4 ARE SET At 150
H
CUTPUT TO TERMINAL MAY BE OBTAINED FOR AMY SIZE BLOCK OF THE
MATRIX BY SPECIFYING THE BEGINNING AND ENCING
AND
(1)5
(J)5.
(I3,4X,13, 1X,I3,'.X,I3)
F(I,J) OUTPUT TO TAPEI2 AND TAPE1
IS IN 5
COLUMNS WHERE THE
(I) DIMENSION OF THE BLOCK REQUESTED HAS BEEN BROKEN UP INTO
A MULTIPLE OF 5
INPUT INFORMATION FROM THE TERMINAL IS GENERALLY PROMPTED BY A GROUP
CF LETTERS ABOVE THE INPUT LINE tHAT INDICATE THE FCRMAT,POSITION,
AND TYPE OF INFORMATION REQUESTED.
C
NOTE 3 CAUTION IS PEQUTED WHEN INPUTTING VALUES TO PROGRAM
INTEGERS MUST BE RIGHT SHIFTED AND NON-INTEGERS MUST
INCLUDE A PERIOD.
FLG3 - FLAG TO ALLOW THE USER TO DECIDE IF THE DERIVATIVE VALUES
30A,BOB,BDC,900 WILL BE OUTPUT WITH THE SOLUTICH VALUES(POSLST)
OUTPUT OF THESE VALUES IS SURPRESSED UNLESS SPECIFIED EACH TIME
P
COMMAND
IS USED.
FLG3 = 0 DERIVATIVE OUTPUT SUPPRESSED FROM POSLST
FLG3
DERIVATIVE VALUES OUTPUT WITH POSLST
1
C MOTE 3 EXPLANATION OF THE MEANING AND USAGE OF PARAMETERS IS
GIVEN IN THE PWSCYL DOCUMENTATION.
A PCLYNCMIAL FIT TO THE FIRST FIVE POINTS ALONG THE
DIRECTION
Z
OF EACH COLUMN FOP CI,J) IS USED TO CALCULATE THE DERIVATIVE
CF THE PRESSURE FIELD AT !0. A TIME INCREMENT IS THEM USED TO
DETERMINE THE MOVEMENT OF THE LUIO SURFACE.
USING A HYDROSTATIC
P°ESSUE WE THEN CALCULATE A MEW PRESSURE VALUE FOR Z0
AN ITERATIVE PROCEDURE ALLOWS THIS TO BE CONE FOR AS LONG AS
NECCESARY.
PARAMETERS WITH SPECIFIC USAGES
(
NOT FOUND IN °WSCYL DOCUMENTATION
61
TINC - TIME INCEMEHT /ALJE USED TO EVALUATE THE INCREMENTAL
PRESSURE CHANGE OP THE FREE SURFACE ( F(I,j) 11,M+tl
DURING ONE TIME STE.
UNITS SECCNOS
RHO
- DENSITY OF FLUID IN cYLINDRICAL RESERVOIR
UNITS MASS/VOLUME
- EFFECTIVE
OROSITY OF °ESERVO!R MATERIAL
- TIME OF WHICH THE CURRENT SOLUTICNS HAVE BEEN
INCREMENTED 10.
TIw
SUM OF TIME INCREMENTS USED
UNITS SECONDS
CONO - FLUID CONDUCTIVITY OF RESERVOIR
CONG
PERMEABILITY /
KINEMATIC VISCOSITY
UNITS SECONDS
PH!
TIM
(
EXPLANATION OF COMMANDS
R - (RUN) CAUSES EXECUTION CF SUBROUTINE GROUP
PWSCY!..
TO
CALCULATE THE NUMERICAL SOLUTION 8ASEC ON THE INPUT VALUES
F
IN ARRAY
ALOPIG WITH THE BOUNDARY CONDITIONS (A,3,C,3,MBDCNO,
NeDCND,9O4,BQ8,BOC,3DO), AND PARAMETERS (M,N,ELMBOA)
I
- (INCREMENT) DIRECTS THE SYSTEM 3 INCREMENT 'RESSURE VALUES
ALCMG THE UPPE° SURFACE (REFERENCE SURFACE) OF THE CYLINDRICAL
It,M+t I. PRESSURE INCREMENTS ARE OBTAINED
EGION (I.E. F(I,1)
BY TAKING THE SOLUTION VALUES OF A COLUMN IN THE pMATRIX (J1,5)
AND CALCULATING A FOURTH ORDER POLYNOMIAL APPROXIMATION 10 THE
GRADIENT (DERIVATIVE) AT TIE REFERENCE SURFACE (Jt). USING THIS
A VERTICAL VELOCITY IS CALCULATED FOR THE FREE SURFACE AT EACH
CF THE COLUMNS BY
COND'(PRESSURE GADIENT) / (RHO
PHI)
UNITS
LENGTH / TIME
VEL
IT HAS BEEN SUGGESTED THAT REDUCING THE MAGNITUDE OF
A,B,C.D PARAMETERS WILL PRODUCE A MORE ACCURATE
THE
SOLUTION. FOR THIS REASON WE HAVE CHOOSEN TO REPRESENT
A,B.C,O IN <ILOMETERS RATHER THAN METERS. THIS RESULTS
IN SOMEWHAT STQANGE UNITS FOR PRESSURE OF
KG./t(.3)(SECZ)1
SINCE ALL LENGTH DIMENSIONS
M.
ARE THEM INPUT AS
INSTEAD OF METERS. LIKE4ISE THE
VELOCITY IS THEN KM./SEC
WE HAVE ALSO CHOOSEM A
CYLINDRICAL COORDINATE SYSTEM WITH OROGIN (1,O) AT F(t.1)
THE VELOCITY IS MtJLTIPLIED BY THE TIME INCREMENT TO OBTAIN AN
INCREMENTAL LOWERING DELTA (H)
OF THE FREE SURFACE.
NOTE
I
OELTA(H)
VEL
TINC
A NEW PRESSURE VALUE IS THEN CALCULATED FOP EACH POINT ON THE
SURFACE AT F(I,1) It,M+1
USING THE HYDROSTATIC ELATION
NEW PRESSURE = (OLD DRESSURE) s OELTA(H)GRAVRHO
GRAy = 9.8OS5E-3 KM/SEC
THE REMAINING FCI,J) I1,1+1 , J=a,N+I.
ARE RESET TO ZERO FOR
THE NEXT ITERATION.
NOTE I FOP TWE SPECIFIC
URPOSE OF DEALING WITH DCTNT SOURCE(SINK)
SOLUTIONS THE LAST Ftt,J)
CHANGED WITH THE CHANGE COMMAND IS
INSERTED INTO THE SAME
(!,J) LOCATION WITH THIS COMMAND TO
PRESERVE THE SOURCE VALUE DURING AN ITERATICN.
L - (LIST) ALLOWS THE USER 10 VIEW ANY PORTION CF THE
F(I,.J)
MATRIX
DESIRED Al' THE TERMINAL BY SIMPLY RESPONDING TO A OROMPT WITH
THE BEGINNING I VALUE FOLLOWED BY THE ENDING I VALUE FOLLCWEO
BY THE BEGINNING AND ENDING J VALUES. USE FORMAT
(I3,'.X,I3,1X,I3,.X,I3).
TAPEI2 CF SOME ARAMETERS
P - (PRINT SOLUTION) CREATES * FILE THROU(H
VALUES USED AND THE SOLUTION VALUES F(I,J)
CREATED BY PWSCYL.
ANY
CRTION OF THE F(I,J)
MATRIX MAY BE OUTPUT AS SPECIFIED
BY A QESPONC! TO A PROMPT 4ITH THE BEGINNING AND ENCI.'IG I AND
ALSO AS SPECIFIED BY THE SETTING OF FLG3
VALUES.
J
T - (TIEORETICAL VALUES) CDPUTES THE ANALYTICAL SOLUTTCN TO OISSONS
EQUATION FOR A P)INT SOtJRCE(SINK) IN AN INFINITE HALF SPACE
BOUNDED BY A FREE SURFACE ALONG A SELECTED FROFILE. THE PROFILE
VALUES (F(I,J)]
U
IS SELECTED BY SUPPLYING THE BEGINNING I AND
SUBSCRIPT FOLLOWED BY
I
FOLLOWED BY A VALUE TO INCREMENT THE
A VALUE TO INCREMENT THE U SU8CRIPT FOLLOWED BY THE TOTAL
NUMBER OF POINTS (INCREMENTS) TO BE USED IN THE PROFILE
1.55 3. U3,IX I3,LX 13,1x,13,1X,13)
(THIS CAN NOT EXCEED
TAPi2
THE PROFILE VALUES ARE THEN OUTPUT T
A RATIO OF THE NUMERICAL RESULTS WITH THE THEORETICAL VALUES
IS OUTPUT FOR COMPARISON PURPOSES.
RATIO = NUMERICAL / THEORETICAL
Q - (QUADRATIC RATIO COMPUTES THE FCURTH ORDER POLYNOMIAL DERIVATIVES
DP/OZ 3, AMO ALCULATES BOTH THE LINEAR AND QUADRATIC
OP/DR
t
TERMS. THEN FORMS THE RATIC
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
QR = QUADRATIC / LINEAR
ALCNG THE REFERENCE SURFACE ARE
COMPUTATIONAL POINTS F(I,i)
SELECTED BY INPUT OF BEGINNING I FOLLOWED BY ENDING I FOLLOWED
BY INTERVAL SPACING OF CALCULATEC F(I,1) POINTS.
(13,LX,13,1X,13)
WHICH CAN BE
lAPEL'.
0 - (DATA FILE) OUTPUTS A PARAMETER FILE TO
USED AS AN INPUT FILE. THE PWSCYL INPUT F-MATRIX MUST HAVE
TAPELL) IN ORDER
(TO CREATE FILE
BEEN SAVED WITH COMMAND SF
FOR THE ARAMETER FILE TO BE OUTPUT. USE OF THIS COMMAND DOES
NOT ALTER THE CONTENTS OF THE F-MATRIX.
C
C
C
C
C
C
C
C
C
C
C
E - (END) ALLOWS THE USER TO EXIT THE PROGRAM OPERATIONS WHEN WORK
IS COMPLETED.
PLACED BEFORE ANY OF THE PARAMETER NAMES
C - (CHANGE) A I C
C F,M,N,A,8,C.O, TINC,COMO,TIM,PHI,RHO,BOA,80B,BDC,800,
ALLOWS THE USER TO CHANGE THE
MBQCNO,NBDCNO,ELMBOA,FLG3
(EXAMPLE I CTINC
VALUE OF THAT PARAMETER.
THE USER IS PROMPTED FOR THE INPUT WITH AN INTEGER OR NON-INTEGER
DESIGNATING THE LOCATION OF INPUT FIELD.
OF THE CURRENT F-MATRIX
CREATES A FILE (TAPEIL)
SF - (SAVE F
(USUALLY USED TO SAVE THE F-MATRIX INPUT TO WSCYL BEFORE A
PASS THROUGH PWSCYL 14 THE EVENT THAT THOSE PARAMETER VALUES
WOULD NEED TO BE USED LATER)
UTILIZES THE FILE OF INPUT F VALUES CREATED
RF - (RESET
WITH THE SF COMMAND TO RELOAD THE F-MATRIX WITH THESE VALUES
EW VALUES ARE CHANGED IN THE INPUT F-MATRIX
(USEFUL IF ONLY A
FOR SUCCESSIVE RUNS WITH PWSCYL)
C
C
C
I
C
C
I
C
C
C
C
C
3
C
C
C
C
C
C
3
C
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCC
C
C
DIMENSION BOA(155),BDB(t55),9DC(155) ,800(155)
DIMENSION F(155,155),W(21?0),RATIO(155)
DIMENSION ORVR(155),)RVZtL55)
INTLO
ZERO ALL ARRAYS EXCEPT
00 21 I1,t55
BOA(I)=0.
908(1)20.
9DC( 1)213.
800(1)20.
RATIO(I)20.IE-19
00 22 J1,.31
JC1=J+31
JC2J+62
JC3J+ 93
JC.,J+t24
F (I,J)2O.
F(I,JCI) 0.
F(I,JC2) =0.
F(I,JC3)20.
F(I,JCk) 213
22 CONTINUE
21 CONTINUE
C
C
C
C
FSAVEQ.
IFLG1O
1F1G2 0
IFLG3O
IFLG'.0
IFLGO
SET 101MF155
10IMF155
W
63
C
READ IN THE INITIAL PARAMETER LIST
READ(9,103) A,B,M,M9DCNO.CONO1IFLGItIFLG2
READ(9,133J C,O,N,N8DCNOtPHI,LFLG3,LFLG4
READ(9,109) ELM9D,RH0,TLNC,TIM
MIPi+j
M1N+t
C
C
C
DETERMINE THE OPTIMUM F(I,J) LIST LENGTH
RRMM1
RRM(RRMI5.1 +0.9
0.9 ADDED TO ROUND-OFF TO NEXT INTEGER
MRR RRPf
MRR22MRR
MR23 3MRR
MRRI. 4MRR
C
CONTINUE READING PARAMETERS
IF(M1.GE.N1) 11M1
IF(N1.GZ.M1) ITN1
00 23 11,IT
REAO(9,109) BDA(I),808(I),8DC(I).800(I)
23 CONTINUE
00 2. Il,MRR
1C1 I+MRR
IC2I+MRR2
1C31+PIRR3
IC41+NRR'.
00 25 J1,Nt
REAO(9,126) F(I,J),F(ICI,J),F(1C2,J),F(IC3,,J),FtIC4,J)
25 CONTINUE
2'. CONTINUE
C
C
C
C
C
EFCEFCEFCEFCEFCEFCEFCEFCEFCEFCE FCEFCEFCEFCE FCEFCEFCEFCEFCEFCEFCEFCE FCEF EFC
COMMAND 4N0 EDITOR FUNCTIOMS
(THE ORDER OF COMMAND CMECS IS BASED ON EXPECTED MOST USED
COMMANDS FIRST)
26 WRITE tOl
C
C
C
C
READ 10?,RESPI.
ICHNG1
IF(RESP1.EQ.THR
)
IF(RESPI.EQ.7H1
)
IF(RESPI.EQ.7H1
)
)
IF(RESP1.EQ.?HP
P
IF(RESPI.EQ.THT
IF(RESPt.EQ.THQ
)
IF(RESPI.EQ.7H0
IF(RESPt.EQ.7HE
P
IF(RESPI.EQ.7HCTINC )
)
IF(RESP1.EQ.7HCF
IF(RESPI.EO.?HCM
IF(RESPt.EO.7HCN
P
IF(RESP1.EQ.7HCCONO )
IF(RESP1.EQ.7HCA
IF(RESP1.EQ.7HCB
)
IF(RESPI.EQ.7HCC
IF(RESP1.EQ.7HCD
P
IF(RESP1.EQ.THCPHI
P
IF(RESP1.EQ.7HCTIM
P
IF(RESP1.EQ.THCRHO
)
IF(RESPI.EQ.THC8OA
P
IFCRESPI.EO.7HCBDB
P
P
IF(RESPI.EQ.?HCBOC
IF(RESPI.EQ.7HCBOO
)
IF(RESP1.EQ.7HCMBDCNO)
IF(RESPI.EQ.THCNBOCIO)
IF(RESPI.EQ.7HCELMBOA)
IF(RESPI.EQ.?HSF
P
IF(RESP1.EQ.7HRF
P
IF(RESPt.EQ.7HCFLG3 P
WRITE 115,RESP1
P
P
P
P
C
C
C
GO
GO
GO
GO
GO
GO
GO
GO
GD
GO
CD
GD
GO
GD
GO
53
CD
GO
50
GD
GO
G0
CD
CO
GO
GO
GO
CD
GO
GD
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
TO
53
83
66
62
C
C
C
C
C
C
9'.
C
80
C
C
C
98
60
'.4
32
30
31
'.6
38
39
'.0
kt
'.7
45
'.8
33
3'.
35
36
42
43
37
50
TO 75
TO '.9
501026
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
CEFCEFCFECEFCEFCEFCEFCEFCEFCEFCEFCEFCEFCEFCEFCEFCEFCEFCEFCEFCEFCEFCEFCEFCEFC
C
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
COMMAND
S
CHANGE
=
C
C
*
C
64
ARAMETER EDITOR SECTION
(EACH GRCUP OF STATEMENTS RE
C
C
E3ENTS THE REFERENCING
CHANGE
COMMAND)
30 WRITE 103
READ 105,XN
M XM
Ml V4+ 1.
R R M Nj
RM=(RRM/5.)+0.q
MQRRRP
MQR22MRR
4RQ3=3MQR
MR P
RR
GO TO 26
31 WRITE 106
READ j05,XN
NXN
NIN+1
GO TO 26
32 WRITE 110
READ lj1,ICHNG
00 27 1t,ICHNG
WRITE 112
EAD 113,IVALF,JVALF
WRITE 108
READ 105,F(IVALF,JVALF)
27 CONTINUE
tcHN1
C
C
C
C
FOR SPECIAL ITERATIVE USES THE LAST F(I,J) VALUE CHANGED IS SAVED
FSAVEF(IVALF,JVALF)
IFLGkI
GO TO 25
33 WRITE 110
READ 111.ICHP4G
00 23 It,ICHNG
WRITE 11'
READ 111,IVAL
WRITE 108
READ 105,8D4(IVAL)
28 CONTINUE
ICHNGI
3
GO TO 26
WRITE 110
READ 111,ICI4NG
DO 29 Il.ICHNG
WRITE 11.
READ t11,IVAL
WRITE 108
READ 105,8D8(IVAL)
29 CONTINUE
ICHNGI
GO TO 26
35 WRITE 110
READ j11,ICHNG
DO 92 I,IcHNG
WRITE 11+
READ ill,IVAL
WRITE 108
READ 105,30C(IVAL)
92 CONTINUE
I)4NG1
GO TO 25
36 WRITE 110
READ 111,ICM'4G
DO 93 11,ICHMG
WRITE 11e
READ 111,IVAL
WRITE 108
READ 105,300CIVAL)
93 CONTINUE
ICNNG1
GO TO 26
37 WRITE 108
READ 105,ELMBDA
GO TO 26
38 WRITE 108
READ 105,A
GO TO 26
39 WRITE 103
65
A9 10 .8
'0 wRtT
C
108
C
A0 i05,C
C
C
C
C
C
GO TO 2
1 WRITE 108
READ 105,0
GOTO2S
2 WRITE 108
READ 105,XMBCD
C
C
C
IBOCNDXMBCO
0OT026
C
C
L3 WRITE 108
READ 105,XN8CO
M9DCNO=XNBCC
C
C
INTLO
t.
G0T026
C
C
WRITE 108
REAO 105,TINC
C
C
C
G0T026
5 WRITE 108
READ j05,TIM
C
001026
C
.6 WRITE 108
EAO 105,CONO
C
e7 WRITE 108
READ i05,PHI
C
8 WRtTE 108
READ 105,RHO
C
C
001026
C
c
G01026
C
c
001026
C
C
.9 WRITE 108
READ 105,FLG3
C
C
C
1FL03F103
C
001026
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
CSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSS
C
SF
SAVE F
CCMMANO I
C
S
*
C
C
C
C
C
C
CREATE FILE OF CURRENT
F(I,J)
FOR LATER USE
S
S
5
S
50 REWIND it
IFLG2I
00 51 11,MPR
1C11+MRR
IC2I+P'R2
1C31+MRR3
IC=I+MRR
00 52 .J=1,N1
WRITE(11,12) Fr,J),F(IC1,J),F(IC2,.J),FUC3,J),F(IC,J)
52 CONTINUE
St CONTINUE
C
S
001026
S
S
S
S
S
S
S
S
S
CSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFSFS
C
CRPRRRRRRPRRPRQRRRRRRRRRRRRRRRRRRRRRRRPQRRRRRRRRRRRRRRRRRRRRRRRPRRRRRR
C
C
C
C
C
C
C
C
C
C
C
C
C
RUN = P 1
COMMAND I
HE
CURENT
PARAMETERS
TO O9TAIN A SOLUTION
USING
53 CONTINUE
CALL PWSC'fL(INTL,a,9,M,NBDCNO,804,8C8,C,0,N,N8OCNO,BDC,800,
I ELP'80A,F,ICIMF,PERTR8,IERROR,W)
RESETTING INTL FOP MOPE RUMS WITH NEW PARAMETERS
INTL1
TERROR AND PERTR8
NCW WRITING OUT RRp INFORMATION
OF MATRIX (GRID) POINTS
ALONG WITH INFORMATION ON THE
AND THE TIME INCREMENT
WRITE 116, TERROR, PERTRB,Mt,Nt,TIMC,TIM
IFLGIQ
G0T026
R
P
R
p
P
P
P
p
C RRRRRRRPRRRRRRRRRRRRRRRRRRRRORRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
CLLLLLLLLLLLLLLLLLLLLLL.LLLLLLLLLLLLLLLL. LLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLL
C
L
C
CCMMANO I LIST = I I
L
C
C
C
C
C
C
C
C
c
C
L
OUTPUTTING SOLUTION VALUES TO TERMINAL
OUTPUT MAY BE OBTAINED IN BLOCKS OF SIZE DEPENDING ON THE INPUT
GIVEN AS FOLLOWS: BEGINNING (I)
VALUE FOLLOWED BY THE E'OING
(I)
VALUE FOLLOWED BY THE BEGINNING
U) VALUE AND ENDING
(U)
L
L
L
L
VALUE.
L
FURTHER BLOCKS MAY THEN BE INSPECTED
(THE PROCESS MAY THEN BE REPEATED)
66 WRITE 127
READ 12,I8,IE,.Ja,JE
00 63 IIB.IE
00 6'. JJB,JE
WRITE 122,F(t,J),I,J
6'. CONTINUE
63 CONTINUE
WRITE 129
READ t02,RESP6
tF(RESP6.EO.IHY) GO TO 6
I
L
1
I
I
I
L
L
L
1
L
I
I
I
G01026
CLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL
C
CPPPPOPPPPPPPPPPPPPP°PPPPPPPP°PPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP
C
C
C
C
C
C
C
C
C
COMMAND
PRINT SOLUTIONS = P
PRODUCING A LISTING OF SOLUTION VALUES AND SOME PARAMETERS USED
OUTPUT OF FUI,.j) IS IN BLOCKS WHICH APE ARRANGED IN 5 COLUMNS
WHERE THE
(I)
DIMENSION HAS BEEN BROKEN UP INTO THE 5 DIVISIONS
62 WRITE(12,123) A,8,M,MBDCNO,00NO,IFLGI,IFLG2
WRITEt12.iO3) C,O,N,NBOCNO,PHI,IFLG3,IFLGt.
WRITE(12,109) ELMBOA,RHO,TINC,TIM
CJJJJJ.JJ.JJJJJJJJJUJ.JJJJJJJJJJJJJJJJJJJJJJJJJJJ
C
C
STATEMENT INSERTED TO JUMP CURRENTLY UNNECESSARY
C
P
RIE(RIE/5.)+O.9
IIE1RIE
IIEIIEttB-t
t1E22'IIEl
tIE3=3IIE1
11E4k11E1
00 71 119,IIE
1C11+IIEI
1C2I+11E2
1C3214I1E3
tCkt+IIEL.
WRITE(127122) FU,J),I,J,F(ICI.,J),ICi,J.,F(IC2,J),ICZ,J,
1 F(IC3,J,,IC3,J,F(IC'.,J),t'.,J
72 CONTINUE
71 CONTINUE
WRITE 129
READ t02,RESP7
IF(RESP7.EO.IIIY) GO TO 73
C
G0T026
P
P
OUTPUT
WRITE(12,iOB) BDA(I),BOR(I) ,80C(I) ,BOOUI)
70 CONTINUE
IFLGIO
P
P
P
CJJJJJJJ.JJJJJJJJ JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ
C
73 WRITE 127
READ IZA,1B,IE,JB,JE
RIE=IE-!8+1
C
p
P
tF(IFLG3.NE.1) GO TO 73
00 72 JJB.JE
P
P
IF(M1.GE.N1) ITMI
IF(N1.GE..M1) ITNI
30 70 1t,IT
C
p
P
*
p
o
P
p
p
P
P
P
P
9
p
9
p
P
P
P
P
P
P
P
P
P
P
P
P
p
P
P
p
[*A
CIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
C
CCMMANO
C
C
C
C
C
C
C
C
C
C
t
INCREMENT (TIME)
= I
I
I
I
I
I
THIS SECTION FITS DATA TO A °OLYMCMIAL,CALCULATES THE DERIVATIVE
AND THE FLOW VELOCITY. THEM IT ASSIGNS NEW PRESSURE VALUES
AT
Z) (F(I,j)] BASED ON THE TIME INCREMENT INPUT. F(I,J) J>t
IS THEN RESET TO ZERO FOR THE NEXT ITERATION.
83 CALL VELOC(F,M1,O,M,CONO,TINC,TIM,pH1,RHO)
RESET
I
I
I
I
F(t,J) J>1
00 81. J2,t55
00 82 1=1,31
IC1I3t
1C21+62
IC31+93
ICI+126
F(I,J)=0.0
F(1C2,J)0.
F(1C3,J)=O.
F(IC'.,.J)=O.
82 CONTINUE
81 CONTINUE
F(IVALF,JvALF)=SAVE
IFLG1I
60T026
I
I
1
I
I
I
I
I
r
I
I
I
I
I
I
I
I
CIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII
C
C
I
CTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTT
I
COMMAND : THEORETICAL SOLUTION = I
C
I
C
C
C
C
C
C
C
C
C
C
C
THIS SECTION POOUCES A PROFILE OF PATiO VALUES BY COMPARING THE
NUMERICAL SOLUTION FOR A POINT S0UCE (SINK) TO THAT OF AN ANALYTICAL
SOLUTION GIVEN IN
(REF. UNCOMFINED AOUIFER FLOW WITH A LINEARIZED FREE SURFACE
CONDITION I BY DR. G. BODVARSSON 1 1977 )
(IHOTEJI
C
C
C
THE LAST F(I,J) VALUE
CHANGED
IS TAKEN AS THE VALUE AND
LOCATION OF THE O1MT SOURCE.
SEE FSAVE COMMAND STATEMENT #32
SELECT PROFILE
90 WRITE 106
READ 117,I3,J8,IINC,JINC,NNP
9'. IF(IFLG'..NE.l) GO TO 95
XM=P$
c
C
C
I
1
I
I
1
1
T
RSP(3-A)/XM
ZSP(D-C)/XN
R=XIBRSP
ZXJBZSP
RINC=PSPIINC
ZINC=ZSPJtNC
DOVALJVALF-t
I
DO0QVALZSP
I
I
CALL RACFCCOND1 PHI,FSAVE,RIMC,ZINC,R,Z,NNP,PATIC,OQ,TIM,
I
IIB'.z1133+IINC
JJBJB+((I-t)JIMC)
JJ91JJ4JINC
JJBZJJBI+JP4C
JJ83JJB2+JINC
JJ94JJ83.JINC
C
T
T
T
T
1 B4O,XM,XN,IFL)
IF(IFLG.NE.0) GO 10 96
WRITE OUT THEORETICAL SOLUTION VALUES FOR PROFILES
00 91 11,NNP,5
IIB!B,((I-1)IINC)
1191119+IINC
1I92=IIBDIINC
11B31192+IIMC
1.
I
I
I
I
XNN
XI9=I9-1.
C
T
I
I
WRITE(12,11.3) RATIO(I),tIB,JJ9,RATIO(141) ,II81,.JJBI,RATIO(I+2)
,IIBZ,JJB2,RATIO(T+3),I133,JJ83,RATIO(1+4),IIB'.,JJg'.
T
I
r
T
T
I
I
1
T
T
I
T
T
r
I
I
r
r
I
T
I
I
T
I
I
COMPUTE THE
C
C
RATIO
(NUMERICAL/THEORETICAL)
I
T
RATIO(I)F(IIB,JJB)/RTIOtI)
RATIO(I+1)=F(It91,JJl/RTtO(I+1)
RATIO(I+2 )F(1192,JJ82) /RTIO(I+2)
RATIO(I+3)F(1193,JJ831 fRTIO(I+3)
I
T
RATI0(I+'.)F(II9'.,JJB.)/RATIO(I+4.)
WRITE(l2,11.9) RAIIO(E),II5.JJB,RATIO(I#1),IIBt,JJB1,
1 RATIO(I+2),tI32,JJ3Z,RATIO(I+3),II83,J.J93,RATtO(I+k),
2 II3,JJBk
RATIO(I)=.1E-19
RATIO(I+t),IE-19
RATIO(I+2),IE-19
RATIO(I+3).1E-19
RATIO(I+k),IE-t9
I
T
T
T
T
I
T
T
I
T
T
1
91 CONTINUE
WRITE 129
READ j02,RESPZ
IF(RESP2.E.1HYJ GO TO 9]
001025
G0T025
T
1
95 WRITE 12]
I
1
T
96 WRITE 121
IFLGO
001025
C
I
I
I
T
T
CTTITTTTTITTTTTTTTTTITTITTTTTTITTTTITTTTTTITTTTTTTITTTTTTTTTTTTTTTTTTTTTTTTT
C
C
C
C
C
C
C
C
aaQaQGQQQQCCQQQQQQQQQQOQQQQOQQQQQQQQQQOQQQOQQQQQQQQQQQQQQQQQQQQQQQQQQGQQQ
a
s
Q
QUA)RATIO = Q
COMMAND
QUAORATIO IS THE RATIO OF TNE 2UADRIATIC TERN NEGLECTED IN THE
0
To THE LINEAR TERM
SEE THEISI5 REFERENCE
LINEARIZATION
Q
OF THE LINEARIZED FREE SURFACE APPROXIMATION.
LINEAR TERM
I
-(CONO'GRAV/PHI)
(OPIOZ
,
Z0
QUADRIATIC TERM * -(0NO(RHOPMt))C(DP/OZ)2. +(3P/OR)2.3 ,Z0
C
C
POINT
151 = STARTING
(tST,1)
POINT
ISP = STOPPING
'(ISP,t)
1510
IINT= INT = INTERVAL SPAING OF CALCULATED POINTS
1ST
C
C
C
c
C
)
(
9
SELECT WINDOW LENGTH
WRITE 125
READ jlT,IST,ISTP,IINT
CALL QUAOP(F,Mt,N1 ,8,D,N,W,IST,ISTP, IINT,CCNO,PHI, RHO,
1 DRVR,DRVZ,RATIO)
WRITE OUT DERIVATIVE AND QUADRATTO VALUES
QUADRATIO VALUES ARE STORED IN ARRAY RATIO
INP( (ISTP-IST)/IINT) +1
00 99 11,INP,5
ISTNIST+ C (1-1) IINT)
ISTIISTT+IIMT
1ST 2=1ST 1 + TINT
IST3 IST2+IINT
IST'IST3+IIMT
11=1+1
12=1+2
13=1+3
WRITE(12,130) ORVR(I) ,ISTT,DRVR(I1),IST1,DRVR(12),ISTZ,
1 DRVR(I3),IST3,ORVR(I1.),IST
WRITE(12,i31) ORVZ(1) ,ISTT,DRVZ(Ii),IST1,ORVZ(12),IST2,
1 ORVZ(I3),ISI3,ORVZ(I'.),t3Te
WRITE(1Z,132) RATIO(I),ISTT,RATIO(I1),ISTI,RATIO(12),15T2,
1 RATIO(t3),1ST3,RATIO(IL.),IST
DRVR(I)0.O
ORVR(I1O.0
DRVR( 12) = 3. I)
OPVR(I3) =0.13
DRVR(I.)0.3
DRVZ(IFO.3
DRy! (11) =0.3
ORVZ (I2)0.3
DPVZ(t3)0.0
DRVZ (I)O.l3
RATIO(I)0.1E-19
0
Q
a
a
0
a
a
a
a
a
a
a
Q
RATIOCI1FO.iEt
RATIO(I2) 0.LE-t
RATIO(13)0.LE-t9
0
0
0
RATIOtI'.) =O.IE-19
99 CONTINUE
WRITE 129
READ 102,RESP2
IFCRESP2.EQ.1HY) GD TO 98
0
GOTO26
0000000000 QQQQQQQQQQQQQQQQQQQQQQQQQQ000000000Q00000000000000000000000QQQQQQ
QDODOOOODDDOODOOOOOOOODDODDD0DQOOOOOOOD0DO0O0OODODOD0DOO000DDDODODDDDDD000D
0
S
DATA FILE
0
COMMAND *
C
C
0
C
C
C
C
WRITE OUT PARAMETER LIST IF NECESSARY
(IFLG1 LS USED TO DETECT IF F(I,J) HAS BEEN RENEWED OR NOT)
(IFLG2 IS USED TO DETET IF FILE ii HAS BEEN CREATED)
60 REWIND Ii
IF(IFLG2.NE.L) GO TO 85
WRITE(14,t2'.) A,3,M,MBDCMD,CONQ,IFLG1,IFLG2
WRITEf1'.,1031 C,01N,NBDCMO2PHI,IFLG3,IFLGZ.
WRITE (1'., 109) EL1IuDA,RHO, TNC,TtN
IF($1.GE.N1) ITP4I
IF(NL.GE.Mt) ITZNL
0
0
a
0
0
D
0
0
0
00 55 I1,IT
WRITE(1'..i09) 9oA(I),BDB(I),BOCCI),BOOI)
55 CONTINUE
00 58 131,HRR
IC1I+MRR
IC2I+MRR2
1C31+MRR3
0
0
0
0
0
0
0
ICt.I+HRR'.
00 59 .J1,Nt
REAO(1i,126) FFI,FF2,FF3,FF(.,FF5
WRITE(14,122) FFI,I,J,FFZ,IC1,J,FF3,1C2,J,FF'.,1C3,J,
1 FF5, IC'.,J
59 CONTINUE
58 CONTINUE
GOTO2S
0
0
0
0
0
0
0
0
DODDODO0OODDD0DDO0DDOOOODOODD0D0O000ODD0OODOD000ODOOOOD0ODO0OODO0OD000OOOOD
C
CRFRFRFRF RFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRF RFRFRFRFR
C
C
C
C
C
COMMAND
RESET
F
I
RESET
VALUES FROM
F
SAVE F
RF
R
$
FILE FOR MORE RUNS
75 IF(IFLGI.EQ.i) GO TO 26
IF(IFLG2.NE.i.) GO TO 85
REWIND 11
00 76 11,MRR
1C11+MRR
IC2t+HRR2
IC3t+MRR3
ICt.1+P4RRI.
00 77 J1,Ni
REAO(1i,126). F(t,J),F(ICI,J),F(IC2,J),F(IC3,J),F(IC'.,J)
77 CONTINUE
76 CONTINUE
IFLGI1
GOTO2S
601026
85 WRITE 10'.
C
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
RFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFRFR
101 FORMAT(" QUEST ?")
102 FORMAT(A1)
103 FORMAT(2X,2F15.'.,2110.Et5.8,IX,21i)
10'. FORMAT(M FILE OF F VALUES NOT CREATED")
105 FORMAT(E15.8)
106 FORMAT("PROFtLE",/2X,"II3,JJB,INC,JNC,NNP)
107 FORMAT (A?)
S.XXXXXXXXXESYY")
109 FORMAT(2X,'.E15.8)
U.0 FORMAT( HOW MANY (I) VALUES DO YOU NEED TO CHANGE",f
1" Ill")
111 FORMAT(13)
112 FORMAT( WHAT F(I,J) VALUE WOULD YOU LIKE TO CHANGE",/
108 FORMAT("
70
III .JJJt
113 F0RMATt3,1X,t3)
1i.. FOR?4A1(
1'
111)
WHAT (1) VALUE WOULO
115 FOPM4T(" PARAME1R
,A7,
HOT
OU LIKE TO CHANGE,/
OUNU")
E19=,E12.6,/
I "SOLUTIONs IH F(1,1) THROUGH F(,I3,tL.,),f
116 FORMAT(" IERROR,13,
TINC
2
TIME= ,F15.Le)
,F15.1.,
117
FOPMAT(I3,jX,I3,X,I3,1X1I3,tXI3)
118 FORMAT(2X,E15.5, IC ,I3,i., ) ,E15.81 IC .13,1k,
,ciS.8, T( .13,11., ) ,E15.8,TC
i E15.8,TC .13,11.,
2
t3,Ik,)
)_ ,El5.81 PC ,13,IL.,
1I FORMAT(2X,Et5.81
,E15e8, R( ,13,ik, )
1 Et5.8, R(,13,i4, ) .13,!'.,
2 E15.8, RC ,13,I'., )
HOT ASSIGNED")
)
C
120 FORMAT(
1
FCIVALF,.JVALF)
121 FORMAIC" I
PHI .EO. ZERO "1
122 FORMAT(2X ,E15.8,FC,I3,1'.,) ,E15.8,"F,13,I'.,) -,
1
,t3,I'., I ,E15.8, ( ,13,Ik, I ,E15.8,
15.8.
F( .13,!'.,
2
123 FORMAT ( j,1X,2F15.k,2110,15.8,tX,2It,15X,0OSLST I
121. FOPP4T( I ,1x,Ft5.L.,2I10,t5.8,tX,2It,t5X, PARLST
125 FORMT(UAOR4TIO,/2X,IST,ISP,!4T")
t2 FORMAT (2X,E15. 8,LIX,E15.8,IIX,E15.8,iiX,E 15.8,1 1X ,E15 .8)
ERROR
J
C
1
1
127 FORMAT (WHAT FC1,J) VALUES,/
III TO III,JJJ TO JJJ)
1
128 FORMAT (I3,4X,13,tX,I3,kX,1)
12q FORMAT (MORE ? Y OR H")
,E15.8,DR('!3,",1)
,1I
I
E15.8,"t34(",13,,tI
1Ei5.8,
OR(,13,,t)
131 FORMAT C2X1E15.8, OZ( ,13, .1) ,E15.8, DZ( ,13, ,I) ,
.13, ,1)")
1. E15.8, D( .13, ,1) ,E15.8, ZC .13, .1) ,clS.8, OZ(
132 FOPMAT(2X1t15.8, OR(I3, .1) ,Ei5.8, ORC,13, ,t) ,
1 E15.8, OP'C .13, ,1) ,E15.8."OR( .13, ,1) ,15.8, QR( .13, .1)
130 FOPMAT(2X Ei5.8,OP(I3,,1)
1
1
C
CEEEEEEEEEEEEEEEEEEEEEEEE!EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
C
C
C
C
COMMAND
80 STOP
END
E
E
E
E
E
CEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE!EEEEEEEEEEEEEEEEEEEEEEEE
C
END
71
SUeROUTINE VELOCCF,Mt ,O,4,COND,TINC,TIM,PHI,RHO)
ccccccccccccccccccccccccccccc: cccccccccccc ccccccccccccccccccccccccccccccccCC
C
C
C
C
C
C
)IFTBL. AND FUNCTION OERVZO
VELOC UTILIZES SUBROUTINE AHO
DISPLACEPENT OF THE FREE SURFACE AT
To DETERMINE THE VELOCITY
FOR
A
RESERVOIR.
Z=O
POSGO TO
1P4 PROGRAM
VELOC OPERATES ON THE MATRIX F(I,JI
ON
THE
SOLUTION.
RESET VALUES AND ITERATE
C
C
THE ZO SURFACE PRESSURE
C
C
F(I,t)
IS
CALCULATED WITH
P = (RHO)(GR*V) OELT*(H)
THE
WHERE DELTA (HP IS DETERMINED 8'I' THE TIME STEP CHOOSEN AND
CALCULATED VELOCITY.
CONO
COP4OUCTIVITY
RHO
DENSITY
KM. /(SECSEC) P
C
q.8QGG5E-03
GRAV
GRAVITY
PHI
POROSITY
= TIM
TIME
TINC
TINE INCRMT
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
c
C
C
C
C
C
C
C
C
C
C
CALCULATED ARE OUTPUT WITH SOLUTIONS OF
C
SURFACE
PRESSURE
VALUES
C
VALUES.
C
NEXT ITERATION AS THE F(I,1.) 11,Mt
C
CCCCCCCCCCCCC
C
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C
DIMENSION X(6),VEI.FL(t55),TABLE(6,6),F(155,155)
T INS TI NC 'TIN
SET UP
X
SPACING OF F(I,J) J1,5
POINTS
X N N
SPAC=O/XN
X(11 0.Q
00 21 12,5
X (I)
X CI-t)+SPAC
21 CONTINUE
START POLYNOMIAL FITTING
00 31. ICI,Mt
POLYNOMIAL IN Z
DETERMINE DIFFERENCE TABLE FOR
MP.GE.NP
- OK - IS A PARAMETER TO HECK FOR
CALL OIFTBL(X,F,IC,TABL!,4,5,6,OK)
C
C
C
DIRECTION
NOT USEO HERE
CJJ.J-JJJ.JJJJJJJJJJJJJJJJJJJJJJJJJJ.,JJJJJJJJ-IJJJJJ,JJJJJJJJJJJ
C
C
C
WRITE OUT TABLE FOR FIRST TIME THROUGH TO CHECK
IF(IC.NE.1) GO TO 70
00 41 J11.5
WRITE(12,1.Q1) (TABLECIT,JT).1T1,5)
41 CONTINUE
CJJJJJJ.JJJ JJJJJJJJJJJJJJJJJJJJJ JJJ JJJJJJJJJ_$JJJJJJJJJJJJJJJ
C
C
C
C
C
C
C
C
C
C
CALCULATE THE DERIVATIVE
EQUATION OF VELFL(IC
RHO IS OMITTED FROM THE E2UAT ION FOR PRESSURE
SINCE IT CANCELS IN THE
F(IC,t)
70 OELTPOERVZO(X,TABLE,6)
VELFL (IC) (CONO'OELTP)/PlI
DELIHTINC*VELFL (IC)
THE FREE SURFACE IS BELOW THE
FOR A POINT SINK AT TIMEO
Z0. THUS THE
LEVEL AND Pso AT SOME POINT ABOVE
ZSO
PRESSURE AT ZS0 MUST BE IEGATIVE.
RHO IS OMITTED FROM THE NEXT EXPRESSION
F(IC,1)CGRAV'UELTH)+F(I,t)
31 CONT INUE
101 FORMATC2X,6F15.4)
72
,1
L03 FCRMATCX,F5.L4, F(,13
F( ,13, ,i) ,F1.k,
I Ft5..,
2
F(,13,,1)")
RETURN
END
Fj5.k,
(
F(,132,1)
,13,,i) ,Ft.'.,
,
73
SUBROUTINE DIFTBL(X,F,I, TABLE,MP,MP,K,OiC)
C
C
C
C
C
C
C
C
C
CC CCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCC
DIFTBL CALCULATES THE DIFFERENCE TA3LZ FOR NEWTONS
DIVIDED OIFFERENCE METHOD. (ADAPTED FROM "APPLIED NUMERICAL METHOOS
BOCK 8Y CARMAP4AN ET.AL. 19B)
NP
POLYNOMIAL ORDER
NP = NUMBER OF POIHS USED
OK
DETECTS ERRORS IN POLYHOMIAL ORDER
C
C
C
c
c
C
C
C
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
C
C
C
DIMENSION X(K),F(155,155),TABLE(K,K)
0K1
IF(MP.GE.NP) GO TO 10
CAI..CULATE FIRST ORDER DIFFERENCES
N1NP-1
00 21 J=t,Nl
TA8LE(1,J)(F(IC,J+1)-F(1,Jfl/(X(J+1)-XLU)
21 CONTINUE
IF(MP.LE.1J GO TO 11
NOW CALCULATE HIGHER ORDER DIFFERENCES
00 31 12,MP
00 .1. J1,N1
15J+ i-I
TABLE(I,J)(TABLE(I-t,J)-TABLE(I-t,J-1))/(X(J+i)-%(IS))
'.1 CONTINUE
31. CONTINUE
11 0K0
10 RETURN
END
74
C
FUNCTION DERVZO(X,TABLE,<)
cccccccccccccccccccccccccc:cc00000ccc0000ccccccccCCCcCCCcCC ccccccccccccccccc
C
C
C
C
C
c
C
C
C
C
C
C
C
C
C
CALCULATIONS ARE COMPLETED USING A SHORT FORM OF THE DERIVATIVE
FOR THE FOURTH ORDER POLYNOMIAL. THE DERIVATIVE IS COMSEUENTLY
ONLY FOR THE SPECIAL CASE
10
Q( P(Z))/DZ
WITH THE FIRST PO(.YNOMIAL FIT POINT
X(i)
AT
Z0
ALSO THE DIVIDED DIFFERENCE TABLE MUST BE SET UP FOR TABLE(i,i)
AS THE FIRST DIFFERENCE USED AND ALL NEECED DIFFERENCES ON THE
MATRIX DIAGONAL.
PZ)
ACt)
A(2)Z + M3)Z
C
C
OP/DZ = A(2)
C
C
A(2) REDUCES TO
2
+ ..
ZO
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
XCt)0
C
WITH
C
A(2)(X(2)XU))PtX()-(X(2)XC3JX($.))P(X(5)3+PCX(2))-(X(2))PCX(3fl
C
C
C
C
C
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
DIMENSION X(),TABLE(K,K)
X23X (2) X(3)
X23ksX23X(i.)
DER VZOX2 31* BLE (3 ,.3)
t X(2)'TABLE(2,2)
RETURN
E NO
X23 'IA BLEC .,k) +1* 8LE (1. 1) -
75
'.
SUBROUTINE RAOF(CONO,°HI,FSAVE,RINC,ZINC,R,Z,NNP,RATIO,DQ,TIM,
8,D,XM,XN,IFLG)
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC0000CCCCCCCCCCCCCCCCCCCCC
C
C
C
C
RADF COMPUTES THE THEORETIAL HALF SPACE POINT SOURCE SOLUTION
FOR THE PRESSURE FIELD ASS3IATED WITH A LINEARIZED FREE SURFACE.
t(IIRPQ) + (1./RPQP) - (2/RPQTP))
FMOT
RATIO
C RR + (Z-OQ)2 )Q.5
C
RPQ
C
C
RPQP = ( RR + (Z-QQ)
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C RR + (Z+(A'TIM)+OQ)'Q.S
THIS IS THE SOLUTION TO POISSONS EQUATION WITH A SOURCE TERM
- (FZ / COND)DELT(P-Q)I(T)
RPQTP
C
C
C
C
C
c
C
C
C
C
C
C
C
C
= CAUSAL UNIT STEP FUNCTION
C
P
C
WITH
DELT(P-Q) = DIRAC DELTA FUNCTION AT POINT
C
SOURCE POINT Q
C
C
CONCENTRATE) SINK
FZ
C
(REF.s DR. C. 900VARSSOM' , UNCONFINED AQUIFER FLOW WITH A LINEARIZEOC
C
FREE SURFACE CONDITION 1 1.977)
C
C
IT TURNS OUT THAT THIS CAN BE RELATED TO F IN
LAPLCP) : F
C=FSAVE)
1(1)
C
WHERE
C
C
50,
C
)'O.5
C
STANOS FOR THE LAPLACIAN OPERATOR.
LAPLC
C
C
C
C
-F3AIE(((RR2/2.)'2)'ZZ3/'..
FNOT
C
C
C
C
C
C------------------------------------ CC
c
C
c
C
C
C
C
C
C
C
C
C
C = 9.80665E-03
C
KM.
,'
SEC'SEC )
= CHARATERISTIC FLU!) VELOCITY
CHEC( TO PREVENT DIVISION BY ZERO
IFLG
DEPTH TO POINT SOURCE
DQ
(
NOTE )
C
C
C
C
C
C
C
C
C
RATtO MUST 9! DIMENSIONED THE SAME AS IN CALLING PROGRAM C
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
C
DIMENSION RATIOC155)
IF(PHI.EQ.0.'J) GO TO .0
RR8/ CXM4Z.)
Z Z 0/ X N
RRZZ(RRRRZZ)
G9. 8C665E-03
I'..
A(CONOG) /PHI
FNOT-(FSAVERRZZ)
IFLGO
00 21 I1,NNP
XI=I
RSQ*RR
ZAOZ+(ATIM) +00
ZP4OZ-OQ
ZPLD =Z+OQ
RP@ (RSQ+(ZMOZMOI)0.5
RPQP C RSQ+CZPLOZPLD) )0. 5
RPQTPCRSQ+CZAOZAO))
IF(RPQ.Et2.0.0 .OR. RPQP.E0.3.0 .OR. RPQTP.EQ.O.13) GO TO 30
RATIO(t,FNOT((1./RPQ),(t./RPQP)(2./RP!TP))
IF(RATIOCI).EO.l3.0) RATIO(I),IE-19
31. CONTINUE
R=R+RINC
Z=Z+ZINC
21 CONTINUE
RETURN
30 WRITE 105
t!21.
RATIO(I)
GO TO 31
76
'.0 IFLG1
RETURN
105 FORMAT(
A SINGULARITY 0ETCTEO, OEN0MINATC
I - EQUAL TO
ENO
.1121
.")
OF RATI3 SET,/
77
SUBROUTINE QUAOR(F,Mt,Nt,3,O,M,N,IST,ISTP,IINT,COND,PHI,RHO,
I DRVR,ORVZ,RATIO)
cccccccccccccccccccccccccccoc::ccccccccccccccccccccccccccccccccccccccccccccC
RZDIFT AND FUNCTIONS DERVR,OERVZO
QUADR UTILIZES SUBROUTINE
TO DETERMINE THE RATIO OF THE NEGLECTED QUADRATIC TERM TO THE
LINEAR TERM IN THE LINEARIZED FREE SURFACE APPROXIMATION.,
TERN
QUADRATIC
C
C
C
LINEAR TERM
C
C
_(CONQ/(RHOPHI))((Dp/OZ)*42. + (DP/OR)42.] , 20C
QUAORIATIC TERN
-(OND'GRAV/PHI) ' (OP/'OZ)
= CONO
2 RHO
= GRAV
2 PHI
,
Z0
C
C
C
C
C
9.80665E-3
C
C
C
POINT
STARTING F(IST,1)
POINT
ISTP = STOPPING F(ISTP,1)
tINT = INTERVAL SPACING OF CALCULATED POINTS
157
X
V
C
C
QUAORATIO- ------------ ----
LINEAR TERM
CONDUCTIVITY
DENSITY
GRAVITY
POROSITY
C
CORRESPONDS TO
CORRESPONOS TO
C
C
C
1 COORDINATE
C
R COORDINATE
C
C
CCCCCCCCCCCCCCC000CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCCCCCCCCCCCCC
DIMENSION X(6),Y(6),ORVR(L55),ORVZ(155),TA8LEX(6,6),TABLEY(6,6),
1 F(155,1551,RATIO(155),FRAY(155)
GRAVxq.50665EO3
CRPCONO/ (RHOPHt)
CGP (CONOGRA VI /PHI
SET UP ARRAY FOR RADIAL DERIVATIVE
FRAY( 1)F (3,1)
FRAY(2)F (2,11
DC 31 1=3,155
11=1-2
FRAY(t)F (11,1)
31 CONTINUE
SET UP X , V SPACING FOR
F(I,J)
POINTS
XN=P4
YP!M
SPACXO/XN
5 P AC Y 8/ V N
X (1)20.0
V (112-2. 'SPACY
00 21 122,5
X (I)=X (I-i)+SPACX
V (I)Y (1-1) +SPACY
21 CONTINUE
START POLYNOMIAL FITTING
I P= 0
DO 41. ICIST,ISTP,IINT
DETERMINE DIFFERENCE TABLES
CALL RZOIFT(X,Y,F,FRAY,IC,TABLEX,TABLEY,k,5,6,OX)
CALCULATE DERIVATIVE VALUES AND QUADRATIC TERMS
IP=IP+1
DRVZ(IP) 2OER VZO(X,TABLEX1 5)
ORVR(IP)OERVR (V,TA8LEY,)
RCUAD-CRP" (ORVZ (IP) DRVZ ( IP) + DRVR(IP)'ORVRCIP))
RLIN-CGPDRVZ ( IF)
IF(RLIN.EO.0.0) GO TO 45
RAIl CC IP) RQUAO/RLIN
46 CONTINUE
41 CONTINUE
105 FORMAT( SINGULARITY OET!TEC
RETURN
45 WRITE 105
RATIO(IPIO.1E21
GO TO 46
END
UUAORATIO SET EQUAL TO
1.0E20)
SU8ROUTINE RZOIFT (X,Y,F,FRAY,I: ,TABLEx,TAaLEY,p4p,Hp,K,OK)
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC
RZOIFT CALCULATES THE DIFERENCE TABLES FOR NEWTONS DIVIDED
DIFFERENCE METHOD.
(ADAPTEC FRON APPLIED NUMERICAL METNODS BOOK
NAHAN ET.At.. t%)
BY C
NP
POL"NOMIAL ORDER
NUMBER OF POINTS USED
NP
K = 0IMESION F OIT ARRAYS USED
R
IC
PCSITION , F(I,l) , OF SPECIFIED POINT
OK
X
Y
DETECTS ERRORS IN POLYNOMIAL ORDER
CORRESPONDS T THE Z COORDINATE
CORRE$PCP4OS TO THE R COORDINATE
DINENSION XfK),Y(',F(j55,j55),TABLEX(K,K),TABLEY(K,K),FRAY(t55)
0K1
IF(M°.GE.NP) GC TO 10
CALCULATE FIRST ORDER DIFFERENCES
TABLEyc1,n=(FRAYcIC+J)-cR4Y(tc+.J-1,,/(Y(J.1)-Y(J))
21 CONTINUE
C
C
C
C
c
C
C
C
C
C
C
C
C
C
C
ccccccccccccccccccCccCcCcccCcc:ccccCccccccccCcCcCCcCCcccCcCcCCccCcccCccCccc
NINP-1
00 21 J1,M1
TLEX(t,,J)(F(IC,J+1)-F(tC,JH/(X(J+1)X(J))
C
C
C
IFUP.LE.l) GO 10 11
NOW CALCULATE HIGHER ORDER DIFFERENCES
00 31 12,MP
00 1.1 J1,N1
15.J+t-t
TABLEX (I,J)(TABLEX t-t,J) -TABLEX (I-1,J-t) ) / tX (J+t)-X (IS))
TA!LY(I,.J)(TACLEY(I-1,J)-T*BLEY(I-1,J-t))/(Y(J+t)-Y(IS))
t CONTNUE
31 CONTINUE
11 OKO
10 RETURN
END
79
FUNCTION DERVR(Y,TA8LEY,<)
cccccccccccccccccccccccccccccc:cc cccccccc cccccccccccccccccccccccccccccccccc
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
CALCULATIONS ARE COMPLETED JSIN6 A SHORT FORM OF THE DERIVATIVE FOR A
FOURTH ORDER POLYNOMIAL AT R=O
THE DERIVATIVE IS CONSEQUENTLY ONLY
FOR THE SPECIAL CASE
OtP(R)]/R
AT
R = Q
C
C
C
c
C
WHERE THE CENTRAL POINT OF THE FIVE POINTS FITTED IS ASSIGNED
R
C
C
THE DIVIDED DIFFERENCE TABLE MUST BE SET UP FOR
TABLEY(1,1) AS THE C
FIRST DIFFERENCE USED AND ALL NEEDED DIFFERENCES ON THE MATRIX DIAGONALC
P(R) = ACt) + A(2)R
OP/DR = A(2)
R 2
,
WITH Y(3)=Q
A(2)
REDUCES TO
A(3)R
2
+
..
3
C A(2)P(Y(2)3(Yfl)+YC2))'tY(3) ]+(Y(1)Y(2))P(Y(e)(YC1)Y(2)Y(k))PCY(5)
C
C
C
c
C
C
C
C
C
C
CCCCCCCCCCCCCCCCCCCCCCC CCCCCCCCC CCCCCCCCC CCCCCCCC CCCCCCCC CCCCCCCCCCCCCCCCCC
C
DIMENSION Y(K),TA8LEY(K,()
Y12=Y(1)+Y(2)
Y2=Y(iJY(2)
Y23Y2'Y U'.)
DERv=TA9LEYc1,t)Yt2'TA8LEY(2,2)+Y2TA8LEYc3,3)Y23TAaLzY('.,.)
RETURN
END
1
C
,BOCND BOA BC8,C,C N NBDCND,30C,
SUBROUTINE PWSCYL (INTL,4,B
PYL3OOO5
PYLQO1
PYL]0015
PYLJOOZO
PYL3 3O25
PYL0003O
PYL0003S
PYL000kO
PYL300k5
PYL0005Q
PYL60055
PYL0006O
3OO,ELMB,F,IOIM,PETRB,IERO,W)
C
c
C
C
VERSION
OCTOBER 1.9Th
2
C
C
C
C
C
C
C
C
C
C
C
C
C
PAUL SWARZTRAUBER
CC'
CC'
CC,
CC+
CC,
004
CC'
CC'
CC,
00+
CC'
CC'
CC'
CC'
CC'
CC'
CC'
CC'
CC'
CC'
CC'
CC'
CC'
CC'
CC'
004
CC'
Cc,
CC,
CC'
CC'
C
C
C
C
C
C
C
PYL.30080
PYL00085
PYL0009O
PYL000B5
O3O7PYLOO1OO
PYLOO1C5
PYLOOhj
PYLOOi15
3'4'"PYLJ 0
--------- -------------- ----------CLASS TWO ROUTINE
QEVISIONI
DATE LAST CHANGED)
'CC
'CC
----------+cc
+CC
+cc
'CC
'CC
PWSCYL
0
77-03-1.5
CLASS TWO
RCUTIMES ARE MADE AVAILABLE BY NMG AS A SERVICE TO THE LLL
COMPUTING CCMMUNITY. SUCH ROUTINES FAIL TO MEET ONE OR MORE OF THE
CRITERIA FOR INCLUSIOM IN THE CLASS ONE (CERTIFIED) LI3RARY.
IT rs
RECCMMENOED THAT A
CLASS TWO ROUTINE BE USED ONLY WHEN THERE IS NO
4POPRIATE CLASS ONE ROUTINE.
NG WILL ATTEMPT TO
OVtDE CONSULTATION SERVICES FOR CLASS TWO
ROUTINES, BUT SUPORT IS NOT GJARANTEEQ.
4 -------------------------------------- ---------------------
NOTICE
+
12
'CC
'CC
LAWRENCE LIVERMORE LABORATORY
NUMERICAL MATHEMATICS GROUP -- MATHEMATICAL SOFTWARE LIBRARY
CC,-
c
C
C
C
C
PYL300T5
--------- -
CC'
CC,
CC'
Cc,
C
PYLE3007Q
AND
NATICNAL CENTER FOP ATMOSPHERIC RESEARCH
BOULDE.R,COLCRAOO
WHICH IS SPONSORED BY THE NATIONAL SCIENCE FOUNDATION
CC'
C
C
C
C
C
PYLOO65
ROLAND SWEET
TECHNICAL NOTE TNFIA-tO9
JULY 1.975
C
C
OCTOBER 1.976
BY
C
C
INCLUDING ERRATA
DOCUMENTATION FOR THIS PROSRAM IS GIVEN pi
EFFICIENT FORTRAN SUBPROGRAMS FOR THE SOLUTICN OF
ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
C
4
+
THIS REPORT WAS PREPARED AS AM ACCOUNT OF WORK SPONSORED BY THE
UNITED STATES COVERNWENT. NEITHER THE UNITED STATES NOR THE
+
UNITED STATES ENERGY RESEARCH AND DEVELOPMENT ADMINISTRATION,
+
NOR ANY CF THEIR EMPLOYEES, NOR ANY OF THEIR CONTRACTORS, SUB- '
+
CONTRACTORS, OP THEIR EMPLOYEES, MAKES ANY WARRENTY, EXPRESS OR +
+
IMPLIED, OR ASSUMES ANY LEGAL LIABILITY OR RESPONSIBILITY FOP
#
4
THE ACCURACY, COMPLETENESS OR USEFULSS OF ANY INFORMATION,
+
+
APPARATUS, PRODUCT OR PROCESS OISCLOSEC, OR REPRESENTS THAT ITS '
USE WOULO NOT INFRINGE RIVATELT-OWNED RIGHTS.
#
+
4
+
'---------- ------ - ------------ - --------- - ------------ -------+
+
'
+
I
'CC
'CC
+cc
fCC
'CC
+cC
'CC
+CC
'CC
4C
'CC
'CC
#CC
'CC
'CC
+CC
'CC
'CC
+CC
+
'CC
+CC
'CC
PLEASE REPORT ANY SUSPECTED ERRORS IN THIS ROUTINE IMMEDIATELY TO NMG. CC
'CC
------------- - --------- - -------- ---------- ----------- ---'CC
PYLJO1Z5
PYLOOI3D
PYLOO135
SUBROUTINE PWSCYL SOLVES A FINITE DIFFERENCE APPROXIMATION TO THE PYLOO1kO
MODIFIED 4ELMHOLTZ EQUATION IN CYLINORICAL COCROINATES
PYLOO145
PYLJO15O
(1/R)(D/DR)(R(OfOR)U) + (D/OIZ)(O/OZ)U
PYL30155
PYLOQ16O
4 (LAMBOA/R21U
F(R,2)
PYLDOI55
RCM
THIS TWO DIMENSIONAL MODIFIED 4ELMHOLTZ EQUATION RESULTS
THE FOURIER TRANSFORM OF THE THREE DIMENSIONAL POISSON EQUATION
INTL
'
' $
ON INPUT
PYL30175
PYLODI8O
PYL0185
PYL0O 190
THE ARGUMENTS ARE DEFINED ASO
'
PVLOO1.7O
'
PYLQOI95
PYLOO200
PYLGO2O5
PYL3O21O
PYLO215
PYL30220
EI
CM INITIAL ENTRY TO PWSCYL OR IF P4 AND NBOCNO ARE CHANGED PYLOCZ25
PYLOQ23O
FROM PREVIOUS CALL.
IF P4 AND NBOCND APE UNCHANGED FROM PREVIOUS CALL TO PWSCYL.PYL00235
PYLOO24O
PYL002.5
NOTED A CALL WITH INTL = 1 IS ABOUT 1 PERCENT FASTER THAN A
PYL30253
CALL WITH INTL = 0
= 0
= 1
A,8
THE RANGE OF P. I.E.
PYLU 0255
A
.LE. R .LE. S.
AND A MUST BE NON-NEATIVE.
M
THE NUMBER OF PANELS INTO
SUBDIVIDED. HENCE, THERE
R-OIRECTION GIVEN BY RU)
WHERE DR = (9-4)/N IS THE
A MUST BE LESS THAN B
PYLO 0275
WHICH THE INTERVAL (A,B) IS
WILL BE 1+1 GRID POINTS IN THE
= A+(I-1)OR, FOR I = 1,2,...,M+1,
PANEL WIDTH.
MBDCND
INDICATES THE TYPE OF BOUNDARY CONDITIONS AT R
= t
= 2
= 3
2 4
= B
2 6
PYLUO26O
PYLGO2S5
YLO327O
A AND R 2 B.
PYLOO28O
PYL30285
PYLOO29O
PYL00295
PYL00300
PYLO 0305
PYL003I.3
PYL00315
PYL130320
IF THE SOLUTION IS SPECIFIED AT P = A AND P. = B.
PYL003!5
IF THE SOLUTION IS SPECIFIED AT P = A AND THE DERIVATIVE OFPYL0033O
THE SOLUTION WITH RESPECT TO R IS SPECIFIED AT P = B.
PYL00335
PYL0034O
IF THE DERIVATIVE CF THE SOLUTION WITH RESPECT TO P IS
= A
SPECIFIED AT
(SEE NOTE BELOW) AND R = 3.
PYL003'.5
IF THE DERIVATIVE CF THE SOLUTION WITH RESPECT TO R IS
PYL00350
4
PYL00355
SPECIFIED AT P
(SEE NOTE BELOW) AND THE SOLUTION I
PYL0036O
SPECIFIED AT P
3.
IF THE SOLUTION 13 UNSPECIFIEC AT P = A 2 0 AND THE
PYL30365
SOLUTION IS SPECIFIED AT P = B.
PYL003TO
PYL30375
IF THE SOLUTION IS UNSPECIFIED AT P = A = 0 AND THE
DERIVATIVE OF THE SOLUTION WITH RESPECT TO P IS SPECIFIED PYL0038O
AT R = 8.
PYL0O35
IF A = 0, 00 NOT USE NBQCNO = 3 OR 4, BUT INSTEAO USE
MBOCND = 1,2,5, OR S
PYLO 0390
PYL00395
PYLOO400
PYLO 0405
PYLOO4LO
30A
A ONE-DIMENSIONAL ARRAY OF LENGTH P4+1 THAT SPECIFIES THE VALUES PYLOO415
OF THE DERIVATIVE OF TIE SOLUTION WITH RESPECT TO P AT R
A.
PYLOD42O
PYLOO425
WHEN MBOCND ' 3 OR ,
NOTED
BDA(J) ' (D/OR)U(A,Z(J)), J
WHEN
1,2,...,N+1
BDC4O HAS AMY OTHER VALUE, BOA IS A DUMMY VARIABLE.
PYL 30430
PYLOOL.35
PYL 30440
PYL0O.5
PYLOOL.50
PYL03455
A ONE-DIMENSIONAL ARRAY OF LENGTH P4+1. THAT SPECIFIES THE VALUES PYLOO46O
OF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO P AT R = 6.
PYLOO4S5
PYLOO47O
WHEN KBDCNO = 2,3, OR ,
PYLO 0475
PYLOO48O
908(J) = (O/QR)U(3,Z(J)), J = 1,2,...,N+1.
PYL0045
PYLOO49O
WHEN MBDCP4D HAS ANY OTHER VALUE, BOB IS A DUMMY VARIABLE.
908
PYL 00495
C,O
THE RANGE OF Z. I.E., C .LE. 1 .LE. 0.
N
C MUST BE LESS THAN C.
THE NUMBER OF PANELS INTO WHICH THE INTERVAL (C,D) IS
SUBDIVIDED.
HENCE, THERE WILL BE P4+1 GRID POINTS IN THE
C+(J-tJDZ, FOR 3 = i,2,...,N+1,
Z-OIRECIIOP4 GIVEN BY 1(3)
N MUST BE CF THE FORM
WHERE OZ 2 (0-0)/N IS THE PANEL WIDTH.
(2'P)(3'O)(5) WHERE P, 0, AND R ARE ANY MON-NEGATIVE
N
MUST
3E
GREATER
THAN
2
INTEGERS.
3
1.
2 3
= 4
PYLO 051.0
PYLOO5I5
PYLQO52O
PYLOO525
PYLOQ53O
PYLOOS3S
PYLDO54Q
PYL005I#5
PYLO 0550
PYLOO5B5
PYL0O5SO
PYLO 0565
IF THE SOLUTION IS PERIODIC IN 1, I.E., U(I,L) = U(I,N+L). PYLOO57O
IF THE SOLUTION IS SPECIFIED AT 1 = C AND Z = 0.
PYLQQ575
IF THE SOLUTION IS SPECIFIED AT Z = C AND THE DERIVATIVE CFP'fLOO5dO
THE SOLUTION WITH RESPECT TO 1 IS SPECIFIED AT Z
0.
PYLOOSB5
IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO Z IS
PYLOOS9O
0.
SPECIFIED AT 1 = C AND Z
PYL00595
IF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO Z IS
PYL3O600
SPECIFIED AT Z =
AND THE SOLUTION IS SPECIFIED AT Z
0. PYLOO6O5
NBDCNO
INDICATES THE TYPE OF BOUNDARY CONDITIONS AT 1
= 2
PYLOO500
PYLQQ5O5
C AND Z = 0.
PYLO 0610
PYL30615
A ONE-DIMENSIONAL ARRAY OF LENGTH M+1 THAT SPECIFIES THE VALUES PYLOO62C
OF THE DERIVATIVE OF TN! SOLUTION WITH RESPECT TO Z AT Z
C.
PYL30625
WHEN NBDD4O = 3 OR 4,
PYLOOÔ3O
PYLO 0635
= 1,2,...,M+t
PYLGO64O
BOC(I) = (0/DZ)U(P(I) ,C),
BDC
WHEN NBOCND HAS ANY OTHER VALUE, 8CC IS A DUMMY VARIABLE.
PYR0
P YL 3 0 6 55
900
YL00660
A ONEDIMENSIONAL ARRAY OF LENGTH P4+1 THAT SPECIFIES THE VALUES PYL00665
OF THE DERIVATIVE OF THE SOLUTION WITH RESPECT TO Z AT Z
PYLOO67O
0.
WHEN N8OCNO
2 OR 3,
PYLOO6?5
900(I) = (J/DZ)U(R(I),0) , I =
1,2,...,M+1
WHEN NBDCNO HAS ANY OTHER VALUE, BOO IS A DUMMY VARIABLE.
ELMBOA
THE CONSTANT LAMBDA IN THE MELMWOLTZ EQUATION.
IF
LAMBDA .01. 0, A SOLUTION MAY NOT EXIST. HOWEVER, PWSCYL WILL
ATTEMPT TO FIND A SOLUTION. LAMBDA MUST 8E ZERO WHEN
PYL30685
PYLO 0690
PY1130695
PYLOO 700
PYLOO7O5
PYLQO71O
PYL30715
PYLOO72I3
PYL00725
PYLOD 730
PY100735
A TWODIMENSIONAL ARRAY THAT SPECIFIES THE VALUES OF THE RIGHT PY1130713
SIDE OF THE HELNHCLTZ EQUATION ANO BOUNDARY DATA (IF ANY). FOR PYL0O75
S OR 6
MBOCNO
F
PYLOO 680
I
2,3,...,14
F(I,J) = F(RUE),Z(J)).
2,3,...,M AND U
PYLOQ75O
PYLt3O 755
ON THE BOUNDARIES F IS DEFINED BY
MBDCNO
1.
2
3
5
6
NBDCNO
0
1
2
3
PYL30761)
PYLOOTS5
---
F(1,J)
F(M+j,)
U(A,Z(J))
U(A,Z(J))
F(A,Z(J))
F(A,Z(J))
F(0,Z(J))
F(0,Z(.J))
U(8,Z(J))
F(B,Z(J))
F(8,Z(J))
U(B,Z(J))
U(B,Z(J))
F(8,Z(J))
F(t,N+j)
F(P(t),C)
U(P(I),C)
U(R(t),C)
F(P(t),C)
F(R(I),C)
F(P(I),C)
U(R(I),0)
F(R(I),D)
F(R(I),0)
F(I,t)
PYLOO77O
PYLDO 775
PYLOO7AO
U(R(I),O)
PYL3Q75
PYLO 0790
U = 1,2,...,N+t
PYLJO7B5
PYLOO800
PYt30805
PYLBO8IO
PYLOQ81S
PYL00820
PYLOOA2S
PYLOQ83Q
PYLI3 3835
P TI.. 3 0 840
I
1,2,...,M+1
PYL00845
PYLOO85O
PYLOO8S5
PYLOO86O
F NUST BE DIMENSIONED AT LEAST (M+j)(N+1).
PY130865
PYL3O 873
PYL00875
NOTE
PYL00865
PYLOD 880
PYLO 0890
IF THE TABLE CALLS FOR 90TH THE SOLUTION U AND THE RIGHT SIDE F PYLOO89S
AT
PYLOO900
A CORNER THEN THE SOLUTION MUST BE SPECIFIED.
PYLDQ9O5
PYLOORIO
IOIMF
THE ROW (OR FIRST) DIMENSION OF THE ARRAY F AS IT APPEARS IN THEPYL3O9L5
THIS
PROGRAM CALLING PWSCYL.
APAMETER IS USED TO SPEIFY THE PYL3OB2D
VARIABLE DIMENSICN OF F. IOIMF MUST BE AT LEAST M+j
pYLOoq2S
N
PYLOO 933
A ONEDIMENSIONAL ARRAY THAT MUST BE PROVIOEQ BY THE USER FOR
WOP
3
F
SPACE.
THE LENGTH OF W MUST BE AT LEAST 6(N#t)+8(M*1).
3
ON OUTPUT
'''
'
CONTAINS THE SOLUTION U1I,J) OF THE FINITE DIFFERENCE
APPROXIMATION FOR THE GRID POINT (R(I),2(J)), I = 1,2,...,M+1,
U = 1,2,...,N+t
PERTRB
PYL00935
PYL30940
PYp33945
P TI 3 0 95 0
PYLDO 955
PYLJOB6O
PYLQO%5
PYLGUB?0
PYL00975
PYLQO9AO
PYL00985
PYLO 0990
PYLDO99S
PYLOIOI3O
IF ONE SPECIFIES A COMBINATION OF PERIODIC, DERIVATIVE, AND
UNSPECIFIED BOUNDARY CONDITIONS FOR A POISSON EQUATION
PYLQ1005
0), A SOLUTION MAY NOT EXIST. PERTR8 IS A CONSTANT, PYLO1OIO
(LAMBDA
CALCULATED AND SUBTRACTED FROM F, WHICH ENSURES THAT A SOLUTION PYLUIOI5
PYL31O2O
EXISTS. PWSCYL THEN COMPUTES THIS SOLUTION, WHICH IS A LEAST
SOUARES SCLUTION TO THE ORISINAL APPROXIMATION.
THIS SOLUTION PYLOIO2S
IS NOT UNICUE AND IS UNNORMALIZED. THE VALUE OF PERTRB SHOULD
PYLQI.030
PYLO1O35
BE SMALL CCMPARED TO THE RIGHT SIDE F. OTHERWISE , A SOLUTION
IS OBTAINED TO AN ESSENTIALLY DIFFERENT PROBLEM. THIS COMPARISONPYLDIO4O
SHOULD ALWAYS BE MADE TO INSURE THAT A MEANINGFUL SOLUTION HAS PYLO1O'.5
BEEN CBTAINED
IERPOR
PYLDIO5O
PYLQ 1055
PYLO1O6O
AN EPPO
FLAG WHICH !NOtCATE !NVALID INUT
ARAMETER.
F
NUMBERS 0 AND 11., A SOLUTION IS NOT ATTEMPTED.
=
0
=
2
3
=
=
=
=
1
(.
S
6
7
8
= 10
= 11
NO ERROR.
A LT. 0
EXCEPT PYL31O65
PYL31OTQ
PY101375
PYLO1O8O
A .GE. B.
1IBOCNO .11. 1 OR WSDCND .GT. 6
C .GE. 0.
N NE.
OR N .LE. 2
MBDCMQ .LT. 0 OR NBDCMO .GT. .
A = 0, MBUCNO
3 OR
A
.GT. 0. M'CH .GE. 5
A
0, LAMBDA .ME. 0, M8DCNO .GE. 5
IDtPF .LT. M+1
LAMECA .GT. 0
PYL.01085
PYLO1O95
PYLO1100
PYLO11O5
PYLOI1LO
PYLGII15
PYLIJ112O
PY131125
PYLO113O
PYL31135
PYl11Li1
SINCE THIS IS THE ONLY MEANS OF INDICATING 4 POSSIBLY INCORRECT PYLO11L.5
CALL TO P%4SCYL, THE USER SHOULD TEST TERROR AFTER THE CALL.
PY101150
W
CONTAINS INTERMEDIATE VALUES THAT MUST NOT BE DESTROYED IF
PWCYL WILL BE CALLED AGAIN WITH INTL =
1.
PYL3 1155
PYL.31160
PYL]1165
PYLO117O
PYLO 1175
PYL3 1180
PYLO 1185