Agricultural Experiment Station )lytechnic Institute and State University

·'.STEP:

Bulletin 85-3

ISSN 0096-6088

1linear Parameter Estimation Program ··aluating Soil Hydra

1

ulic Properties Jne-Step· Outflow Experiments ~ . J. C. Parker, and M. Th. van Genuchten

James R. Nichols, Dean and Director College of Agriculture and Life Sciences Virginia Agricultural Experiment Station Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061

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ONESTEP: A Nonlinear Parameter Estimation

Program for Evaluating Soil Hydraulic

Properties from One-Step Outflow Experiments

J. B. Kool and J. C. Parker

Department of Agronomy

Virginia Polytechnic Institute and State University

Blacksburg, Virginia 24061

and

M . Th. van Genuchten

U. S . Salinity Laboratory

Riverside, California 92501

Virginia Agricultural Experiment Station

Bulletin 85-3

V!RGINIA POLYTECHNIC INSTITUTE AND STATE. U..NlV~•TY ueRARES

March 1985

( . ..

TABLE OF CONTENTS

Abstract . . . . .

Acknowledgements

Introduction .

Di re ct sol utiori of flow prob I em

Solution of inverse problem

Program description

Example problem . . .

Cone I us ions and future extensions

References .

Appendix A.

Appendix B.

Appendix C.

Appendix D.

Appendix E.

Input instructions for ON ESTEP.

Glossary of variables in ONESTEP.

Input file for example problem .

Output file for example problem.

Program listing ........ .

iii

.V

vi

. 1

.3

9

11

13

17

19

21

23

28

29 31

LIST OF FIGURES

Figure 1. Nodal arrangement of the flow system.

Figure 2. Cross-sectional view of Tempe

pressure cell. Pneumatic pressure

is applied through A. Outflow is

collected and measured in burette B.

Air is removed from beneath the porous

plate through the small diameter tube

tube D via peristaltic action on

tube C .

Figure 3. Observed ( O ) and predicted moisture

characteristic for silt loam soil

Figure 4 . Hydraulic diffusivity calculated with

method of Passioura ( O ) and predicted

curve

iv

. 5

14

15

16

ABSTRACT

J. B. Kool, J. C. Parker, and M. Th. van Genuchten. 1985.

ONESTEP: A nonlinear parameter estimation program for evaluating

soil hydraulic properties from one-step outflow experiments. Bulletin

85-3. Virginia Agricultural Experiment Station, Blacksburg.

This report contains a description of the parameter estimation

program ONESTEP. The program, written in FORTRAN IV, will

estimate up to five unknown parameters in the van Gen uchten soi I

hydraulic property model from measurements of cumulative outflow

with time during one-step outflow experiments. The outflow data can

be optionally supplemented with measurements of equilibrium moisture

contents and pressure heads. The[ program combines a nonlinear

optimization routine with a Galerkin finite element model for the one-

dimensional flow equation. Results obtained for a silt loam soil are

shown. Input instructions for the program, example input and

output files, and a program listing are given in appendices.

v

ACKNOWLEDGME ns

The authors would like to express their appreciation to the

Virginia Department of Health and the Virginia Soil and Water

Conservation Commission for funding under which this research was

carried out.

vi

INTRODUCTION

In recent years, interest has 6 risen in the feasibility of us ing

parameter estimation methods to determine soil hydraulic properties

from transient flow measurements. This approach can be summarized

as follows:

1. Hydraulic properties are assumed to be described by a

given model that contai h s a small number of unknown

parameters.

2 . Some flow-controlled attribute is measured at a number of

times during transient flow.

3. The same flow process is simulated numerically using initial

guesses for the unknown parameters.

4. Step 3 is repeated with adjusted parameter values at every

step until an optimum match between observed and

simulated response is obtained .

5. Back-substitution of fi na I parameter va I ues into the

assumed model yields the soil hydraulic properties .

Different authors (Zachman et al, 1981, 1982; Hornung, 1983;

Dane and Hruska, 1983) have considered a variety of experimental

procedures for obtaining input data for the estimation problem. Kool

et al. (1985) have presented a parameter estimation procedure which

uses measurements of cumulative outflow against time from

undisturbed soil cores, subjected to an instantaneous increase in

pneumatic pressure. Experimentally, the procedure is identical to

the familiar one-step method for qetermining soil water diffusivity.

The procedure is experimentally simple, quickly executed, and

applicable to soils of widely varying hydraulic properties.

Theoretical results dealing with solution uniqueness and sensitivity to

2

error, as

presented

This

FORTRAN

well as actual results for a number of natural soils, are

elsewhere (Kool et al., 1985; Parker et al., 1985).

report describes the program ON ESTEP, written in

IV, that solves the parameter estimation problem. Input

instructions, example input and output files, and a program listing

are given in appendices.

DIRECT SOLUTION OF FLOW PROBLEM

The hydraulic system being considered consists of two layers --

a soil core and a porous plate. For vertical flow in the assumed

nondeformable medium a solution tc Richard's equation is required

which .may be written in the head formulation as:

C(h) ~~ = ;x [K(h)(ah/ax-1] (1)

and subject to initial and boundary conditions:

ah/ax =

h = h - ha L

t = 0, 0 ~ x ~ L

t > 0, x 0

t > 0, x = L

(2a)

(2b)

(2c)

where x is distance taken positive downward with x=O located at the

top of the soil core and x=L at the bottom of the porous plate, t is

time, K is hydraulic conductivity, C = d8/dh is the water capacity

with 8 the volumetric water content and h the pressure head, h0

( x)

is the initial pressure head distribution, hL is the regulated pressure

head at the bottom of the porous plate and ha = tip/ pg is the

pneumatic head where tip is the gauge gas pressure applied to the

core, g is gravitational acceleration, and p is the density of water.

The notation employed regards the pressure potential h as the

component of total potential attributable to matric and hydrostatic

components but excluding pneumatic contributions. On the

assumption that gas pressure increments cause pneumatic potentials,

ha, to instantaneously propagate through the porous medium, ha is

effectively translated to the lower boundary condition.

3

4

Properties of the porous plate are independent of h with C=O

and K=K . For the soil, e(h) is assumed to be descr ibed by van p Genuchten ' s (1978a) expression

J (1 + !ah i nrm h

5

boundary problem is solved by means of simple forward difference

scheme that uses the same spatial discretization of the flow domain as

applied to the finite element solutio h (Figure 1). From the specified

initial condition (2a), it is first determined if part or all of the soil

column is unsaturated. If the entire column is initially unsaturated,

initial condition (2a) is simply passed on to the finite element routine

to begin flow simulation . The criterion for determining whether the

soil at node x . is unsaturated is: I

h (x.) $ h 0 I e (6)

where he is some value of the pressure head close to zero, which was

a rbitra ri ly fixed at:

(7)

On the other hand, if all or p art of the soil is saturated, the

solution proceeds in the following manner. Let x. be the nodal I

coordinate(x) node 0 •

2 • • • •

i+1 • soil • • • •

L' • • } pcrous olat~ • L NN •

Figure 1. Nodal arrangement of the flow system.

6

coordinate at which the boundary between unsaturated and saturated

zones is located, i.e. the soil is unsaturated at O

7

By employing this scheme for the first stage of outflow from

saturated cores, marked reductions in computational time were

obtained for the problems that we have investigated, with negligible

concomitant loss in accuracy.

After al I elements become unsaturated, program control is passed

to the finite element routine. Equations ( 1), (2b), and (2c) are then

solved with the modified initial condition:

h(x,t0

) =he

M-1 to = r tit .

j=i J

( 13)

Cumulative outflow at time t during the finite element simulation is

obtained as:

Q(t;l =A f~ [B(x,t) - B(x,0)] dx ( 14)

Note that the equilibrium pressure head distribution hf(x)=h(x,t=00 )

and therefore 8(hf) may be obtained directly from the lower boundary

condition (2c). The corresponding equilibrium outflow can thus be

obtained immediately without the ne~d to solve ( 1) for large times.

The finite element routine uses a variable weighted finite

difference model to approximate ~ime derivatives (van Genuchten,

1978b, eq. 14), where the temporal weighting coefficient w may vary

between 0 and 1. Stability requires that 0.5 5 w 51. Choosing w =

0. 5 corresponds to a second-order correct time-centered model.

When w=l is used, a first-order correct, backward difference scheme

res u Its. Coefficients K and C in ( 6) a re a I ways evaluated at the

half-time level, independent of the time weighting scheme used. Due

to the strong nonlinearity of (1), an iterative procedure is used to

obtain the new pressure head distribution ht+tit' at the end of tit.

The general iteration scheme is:

h k + (1 )hk-1 £ t+tit -e: t+tit ( 15)

where e: is a weighting coefficient (0 5 e:5 1) and k denotes iteration

number. Iteration continues u nti I a specified degree of

correspondence is obtained between pressure head distributions

8

before and after a certain iteration step. As in the original code,

convergence is determined by the program variables TOL 1 and TOL2.

In most cases choosing E:=l will lead to fastest convergence. In some

cases, however, notably when s imu la ting flow th rough coarse

materials, the above scheme with E:=l fails to converge rapidly due to

oscillation in successive iterations. These oscillations are most

pronounced when a time-centered (w=O. 5) scheme is used and often

may be effectively removed by switching to a backward difference

scheme (w=l). In severe cases, oscillations can be further damped

by using E:=0.5 in (15). Time step size is controlled by the number

of iterations required for convergence. When the iteration fails to

converge in a given number of steps (controlled by variable

NITMAX), the time step is halved and the iteration started anew.

Slow convergence may greatly increase computer costs, due to both

the greater computational effort required per time step and to the

increased number of (smaller) time steps. To avoid exces.sive

computer costs, the number of times the iterative procedure fails to

converge within NI TMAX timesteps is counted (variable NOCON) and

execution is interrupted when NOCON exceeds a specified limit

(controlled by variable NOMAX).

SOLUTION OF INVERSE PROBLEM

Values of the unknown

properties in

function:

(3)-(5) are

parameters defining the soil hydraulic

desired which min 1m1ze the objective

N A 2 E(b) E [w.{Q(tt)-Q(b,t.)}]

i =l l l

M " 2 +E[v.{e(h.)-e(b,h.)}] j= l J J J

( 16)

where Q(ti) is the measured cumulative outflow at times ti (i=l, .. . N),

Q(b , ti) represents the numerica jly calculated cumulative outflow

computed for the tria I pa ramete 1 vector b, 8 ( h .) is the water

content corresponding to pressure head hj for equili~rium desorption, 8( b , hj) is the predicted water content for trial parameter vector bas

calculated from (3) , and w. and v . are weight ing factors. The I J

second term in the objective function allows available data from

equilibr ium desorption experiments to be included in the optimization

process. As shown by Parker et al. (1985), this inclusion will

reduce the likelihood of nonuniqueness problems dur ing the inversion

process The differences 8( hj)-8(b , hj) are automatically pre-weighted

by the factor :

( 17)

which gives 8( hj)-values approximately the same weight as the

Q ( ti)-observat ions. Unless otherwise specified in the program input,

weighting coefficients wi are fixed at unity, and coeffic ients vj are

set equal to a .

Parameters are adjusted until the optimum set of va lues , b 0 , for

which E( b) has a minimum, is obtained . Assuming that the problem

has a unique solution, that is that E( b) has only one (global)

9

10

minimum, then backsubstitution of the parameter values b 0 into

(3)-(5) yields the soil hydraulic properties .

The optimization algorithm used to evaluate b0 is based on

Marquardt ' s maximum neighborhood method (~larquardt, 1963) which

is an opt imum combination of the method of steepest descent and the

Gauss-Newton Taylor series method. The computer code for

imp lementing this method was adapted from Meeter (1964). For

highly nonlinear problems this method is efficient and quite robust.

The value of any of the five parameters 8s' Sr ' Ks , ex, and n

may be either optimized or kept constant during optimization. Since

8 and K can be measured relatively easily , it is recommended that s s measured values be used for these two parameters . Limiting the

number of parameters to be optimized wi 11 reduce problems of non-

u n iqueness and wi 11 al so reduce computational effort. The

optimization algorithm evaluates the partial derivatives aE/aj for each fitted parameter j at every iteration of the optimization routine, thus

requiring for each iteration at least J + 1 solutions of the flow prob I em,

where J is the number of fitted parameters.

Both upper and lower limits may be specified for the parameters

to be optimized. When no limits are specified, unconstrained

optimization is used. Note that both upper and lower limits must be

specified fo r constrained optimization. For the three parameter model

(a, n, 8 r), it was found in practice to be unnecessary to specify

limits for ex. The van Genuchten model ( eqs. 3-5) requires n~l.

Using a lower limit of 1. 1 for n will avoid numerical problems that

may occur when n is very close to 1. Finally, it is recommended to

specify an upper limit for Sr to insure 8r

PROGRAM DESCRIPTION

The FORTRAN IV program ONESTEP couples nonlinear

regression analysis and a numerical solution of the direct flow

problem. ONESTEP consists of a main program (MAIN) and seven

subprograms (MATINV, FLOW, STAGEl, TOTALM, BANSOL, MATEO,

SPR). The main program first reads data from an input file

(unit=5) which is echoed back to an output file (unit=6). Input

instructions and a glossary of impo 1rtant program variables are given

in Appendices A and B, respectively.

Input data consist of a rlumber of program parameters,

dimensions of soil core and porous plate, and initial values for the

five soil parameters (ex, n, 8 , 8 and K ) . Each of the parameters r s s can be either held constant or optimized (controlled by the value of

INDEX for each parameter). Input data also specify the initial

pressure head distribution and nodal spacings. Material properties

are node-centered so the boundary between plate and soil must be

located between two nodes with different material indices. Observed

Qi and ti data (FO(I) and HO(I), respectively) are read in along

with the weighting factors, WT(I), if nonuniform weighting is

desired. If equilibrium h-8 data are included in the optimization,

they a re stored in the same arrays FO (I) and HO (I), respectively,

following the Q (t) data. The variable I TYPE disti ngu is hes whether

data are Q(t) data (ITYPE=O) or 8(h) data (ITYPE=l) . If WT(I)

values are omitted, weighting factors default to WT(l)=l for all

observations. With respect to eqs. (16) and (17), WT corresponds to

w. for Q(t) data and to v ./a for 8(h) data. I J

Most of the parameter estimation routine is carried out in MA IN

which also outputs results when the optimization is completed, either

because the parameters have converged (controlled by variable

STOPCR) or because the maximym number of iterations of the

optimization routine is reached (controlled by variable MIT).

Subroutine MATI NV performs the matrix inversion for the optimization

11

12

routine, whereas subroutine FLOW controls the solution of the flow-

equation. Subroutine ST AG El checks whether the soi I is in itia I ly

saturated or unsaturated, and in the former case solves for the first

stage of the outflow process. Subroutine MATEO performs the finite

element simulation, and performs the necessary calculations for

assembly of the global matrix equation, which is subsequently solved

in subroutine BANSOL to obtain the updated values of the pressure

heads and their gradients. Subroutine TOTALM calculates the

amount of water present in the sample at desired times by spatia I ly

i nteg rating the nodal moi st.u re contents. The function SP R generates

the different soil hydraulic functions [8(h), K(h), C(h), and h(8)]

according to (3)-(5).

Versions of the program are currently operating on an IBM 3084

mainframe, Prime 750 minicomputer and I BM PC microcomputer.

Minimum hardware requirements to run the program on an IBM PC are

70 K of random access memory, disk drive and an 8087 math

coprocessor. Machine readable copies of source and compiled code

are available from the first author on request.

EXAMPLE PROBLEM

As an example of the parameter estimat ion program, res u Its a re

given for a silt loam soil. An undisturbed core sample (3 . 95 cm

long, 5. 4 cm diameter) was ta ken from the field and eq u i Ii brated at

zero tension in a Tempe pressure cell (Figure 2). For comparison

with the transient met ho~, the moisture retention characteristic 8 ( h)

was first determined by stepwise increases of pneumatic pressure on

the sample up to a pressure of 98 kPa (ha = 10 m) with water loss

measured at every step. Moisture contents for h=-30 and -150 m

were measured on disturbed samples of the same soil. After

resaturating, the pneumatic pressure was increased instantaneously to

ha=lO m and cumulative outflow was recorded with time . To maintain

a constant head lower boundary condition during the one-step outflow

test, the posit ion of the measuring burette was adjusted every time a

reading was made. Entrapped air beneath the porous plate was

f lushed through tube D by applying a peristaltic action on the lower

tube C (see Figure 2). Upon cor pletion of the one-step test the

porous plate was removed and the soil resaturated. Saturated

hydraulic conductivities of the soil and porous plate were measured

by falling head tests using leaching solutions of 0.01 M CaCl 2 .

The three un known parameters a., n and e r were est imated from

the observed cumulative outflow versus time data and also the

measured 8 at h=-150 m. Input and output files for this problem are

given 1n Appendices C and D, respectively . Note that in our

example all lengths are given in centimeters and times in hours ; in

general any consistent set of units may be used . Figure 3 compares

the 8 ( h )-curve obtained by parameter est imat ion with the

experimental 8(h)-data. In Figure 4, the predicted diffusivity curve

is plotted with D(S) values that were determined independently from

one-step outflow measurements by t Ke method of Pass iou ra ( 1976) .

Running this example problem required 45 seconds of CPU time

on an I BM 3084 mainframe. The same problem on an I BM PC ( with

8087) took about 60 minutes .

13

14

a-ring

Figure 2 .

A

~---soil core

~-------,...,.....-----..........- ~~1-- o- ring porous plate

B

Cross - sectional view of Tempe pressure cell . Pneumatic

pressure is applied through A. Outflow is collected and

measured in burette B. Air is removed from beneath

porous plate through the small diameter D via peristaltic

action on tu be C.

15

103

102

-..§ 0 a: w 101 I

w a:: ::J tf)

lJ) 10° [!] w a: Q..

10- 1

10-2 +-~--~~------~------~~--~~

Figure 3.

0.1 0.2 0.3 0.4 e &n3 m-3)

Observed (0) and predicted moisture characteristic for

silt loam soil.

16

Figure 4 .

10-10...-~~~~--,r--~~~~-.-~~~~----i

0.1 0.3 0.4

Hydraulic diffusivity cal cu lated with method of Passiou ra

(D) and predicted curve .

CONCLUSIONS AND FUTURE EXTENSIO NS

Inversion of the unsaturated flow equation a !lows the water

retention characteristic and hydraulic conductivity to be determined

simultaneously from transient flow measurements . The one-step

outflow method is a convenient procedure to obtain the necessary

data from laboratory soil cores. Outflow versus time measurements,

complemented with the measured 15 bar moisture content of the soi I,

in practice yield sufficient information to obtain a unique solution for

the inverse prob I em. A requirement is that initial parameter

estimates are sufficiently close to their final values. Reasonable

initial estimates can be made on the basis of the texture of the

sample. The FORTRAN IV program described in th is bulletin

specifically estimates the parameters in the van Genuchten model for

hydraul ic properties from outflow and / or equi librium moisture content

vs. pressure head data. Future work will involve the use of other

input data for the inversion procedure. Measurements during

infiltration into the soil can be used to obtain the wetting hydraulic

properties of th~ soil. Used in conjunction with the outflow test,

this may provide a convenient procedure for evaluat ing hysteresis .

Al so, extension of the method to use field measured i nfi It ration and

drainage data wi 11 be studied. I Si nee most studies of soi I-water

cha racteri sties a re ultimately directed towards f ield-sea le flow

processes, in-situ measurements are more relevant than data obtained

from generally small soil cores, the collect ion of which will

unavoidably introduce some disturbance of the soil .

17

18

1.

REFERENCES

Dane, J. H. and S. Hruska. 1983. In-situ determination of

soil hydraulic properties during drainage.

J. 47:619-624.

Soil Sci. Soc. Am.

2. Hornung, U. 1983. Identification of nonlinear soil physical

parameters from an in ut-output experiment. In: P. Deufl ha rd

and E. Hairer, eds. Progress in scientific computing.

Birkhauser, Boston.

3. Kool, J. B., J. C. Parker and M. Th. van Genuchten. 1985.

Determining soil hydraulic properties from one-step outflow

experiments by parameter estimation. I. Theory and numerical

studies. Soil Sci. Soc. Am. J. (submitted).

4. Marquardt, D. W. 1963. An algorithm for least-squares

estimation of nonlinear parameters. J. Soc. Ind. Appl. Math.

11 : 43 1 - 44 1 .

5. Meeter, D. A. 1964. Norn-linear least-squares (Gaushaus).

Univ. Wisc. Computing Center. Program revised 1966.

6. Mualem, Y. 1976. A new model for predicting the hydraulic

conductivity of unsaturated porous media. Water Resour. Res.

12 : 513- 522 .

7. Parker, J.C., J . B. Kool and M. Th. van Genuchten. 1985.

Determining soil hydraulic properties from one-step outflow

experiments by parameter estimation. 11. Experimental studies

Soil Sci. Soc '. Am. J. (submitted).

8. Passioura, J. B. 1976. Determining soil water diffusivities

from one-step outflow experiments. Au st. J. Soi I Res. 15: 1 -8.

9. van Genuchten, M. Th. 1978b. Calculating the unsaturated

hydraulic conductivity with a new closed-form analytical model.

Res. Rep . 78-WR-08. Dept. of Civil Engine~ring, Princeton

University, Princeton, New Jersey.

19

20

10. van Genuchten, M. Th. 1978b. Numerical solutions of the

one-dimensional saturated-unsaturated flow equation. Res.

Rep. 78-WR-09, Dept. of Civil Engineering, Princeton, New

Jersey.

11. van Genuchten, M. Th. 1980. A closed-form equation for

predicting the hydraulic conductivity of unsaturated soils . Soil

Sci. Soc . Am. J. 44:892-898.

12. Zachmann, D. W., P. C. Duchateau , and A. Klute. 1981.

The calibration of the Richards flow equation for a draining

column by parameter identification. Soil Sci. Soc. Am . J .

45: 1012-1015.

13. Zachmann, D . W., P . C. Duchateau , and A. Klute. 1982.

Si mu ltaneou s approximation of water capacity and soi I hyd rau I ic

conductivity by parameter identification. Soil Sci . 134: 157-163.

Card

2

3

4

5

6

7

8

APPENDIX A: Input instructions for ON ESTEP

Columns Format

1-5

1-80

1-80

1-5 6-10

11-20 21-30 31-40 41-50 51-60

1-10 11-20 21-30 31-40 41-45 46-50 51-55

56-60

1-10

11-20 21-30 31-40

15

20A4

20A4

15 15 ElO .O FlO.O F10.0 FlO.O F10 .0

FlO .O FlO. 0 FlO .O FlO .O 15

15 15 15

FlO .O FlO .O FlO. 0 FlO .O

Variable

NCASES

TITLE

TITLE

NN

LNS

DNUL

PNl

AIRP

EPSl

EPS2

SLL PLL DIAM

CPLT

NOBB

MDATA

MODE

MIT

B ( 1)

B (2)

8 (3)

8 (4)

21

Comments

Number of cases considered. The remaining cards are read in for each case.

Title

Title

Blank card

Number of nodes

Number of nodes in soi I

In itia I time step

Pressure head lower boundary

Pneumatic head

Temporal weighting coefficient

Iteration weighting coefficient

Length of soil core

Thickness of por-ous plate

Diameter of soil core

Conductiv ity of porous plate

Number of observations

Mode for i nput data

Mode for output

Max imum iterations of optimi-

zat ion rout ine

Blank card

Init ial value of a

Initial value of n

I nit i a I v a I u e of 8 r

Initial v alue of 8 s

22

9

10

11

41-50

1-10

11 - 20

21-30

31-40

41 - 50

1-10

11-20

21-30

31-40

41-50

1-10

11-20

21-30

31-40

41-50

12 1 1-5

6-10

11-20

21-30

13+NN 2 1-10

11-20

21-30

31-40

FlO. 0

110

110

110

110

110

FlO.O

FlO.O

FlO.O

FlO.O

FlO.O

FlO.O

FlO.O

FlO.O

FlO.O

FlO.O

1-5

15

FlO.O

FlO.O

FlO.O

FlO.O

FlO.O

15

B (5) Initial value of Ks

INDEX(l) Index for each

INDEX(2) coefficient

INDEX(3)

INDEX(4)

INDEX (5)

BMIN(l)

BMIN(2)

BMIN(3)

BMIN(4)

BMIN(5)

BMAX(l)

BMAX(2)

BMAX(3)

BMAX(4)

BMAX(S)

K

MAT

X(K)

P(2K-l)

HO( I)

FO( I)

Minimum value for

each coefficient

Maximum value for

each coefficient

Nodal number

Material index, l=soil;2=plate

Nodal coordinate

Initial pressure head

Time (ITYPE=O) or pressure-

head (ITYPE=l)

Outflow ( ITYPE=O) or mois-

ture content (I TY PE=l)

WT (I) Weighting factor; omit for

default weighting

ITYPE(I) Data type: ITYPE=O for Q(t),

ITYPE=l for 8(h)

1 This ca rd specifies the geometry and the initial conditions of the flow system, and is repeated for each of the NN nodes.

2 This card contains the observed Q(t) and/or 8(h)-data and is repeated NOBB times.

APPENDIX B: List of most significant variables in ON ESTEP

Units are given in brackets (L=length T=time) , and name of the

subprogram in which the variable is used is given in parentheses.

Variable

ALPHA

AEP

AIRP

AREA

B(5)

BI ( 10)

BMAX(5)

BMIN(5)

Definition

Parameter ex [L- 1] (SPR)

Air-entry value of h [L] (STAGEl).

Pneumatic head [ L].

Cross-sectional area of soil core [L].

Array containing initial values of parameters

(MAIN); B(l)=cx, 8(2)=n, B(3)=8r, B(4)=8s,

B(5)=K5

.

Array containing names of parameters (MAIN).

Array containing maximum permissible parameter

values. See note on BMIN .

Array containing minimum permissible parameter

values. If BMIN and BMAX are omitted for any

parameter, their values will not be subject to any

constraints.

BN(l5), TH(15), TB(l5) Arrays containing present parameter values.

CONDS Hydraulic conductivity [LT- 1] (SPR).

CPLT Hydraulic conductivity of porous plate [LT- 1].

23

24

DELT

DELCH

DELMAX

DELMIN

DIAM

DNUL

EPSl

EPS2

FC(25)

F0(25)

H0(25)

I NDEX(5)

!STEP

!TYPE

LNS

Timestep [T] (FLOW).

Relative increase of DEL T between two consecutive

time steps (FLOW).

Maximum value for DEL T [T] (FLOW).

Minimum value for DEL T [T] (FLOW).

Diameter of soil core [L] (MAIN).

Initial time step [T] (FLOW) .

Temporal weighting coefficient, Q::;EPSl::;l (FLOW)

EPSl=O, 0.5 and 1 corresponds to forward, central

and backward difference model respectively .

Iteration weighting coefficient.

Array containing simulated results (FLOW).

Array containing observed results (FLOW).

Array containing observation times [T] or pressure

heads [ L] .

Index for each parameter . If INDEX(l)=O, the par-

ameter 8(1) is known and kept constant; if

INDEX(l)=l , B(I) represents only a initial guess and

the best-fit value for the parameter is determined by

optimization.

Time step counter (FLOW).

Flag for input data. ITYPE=O: Q(t) data, ITYPE=l:

9(h) data.

Number of nodes located in soil .

MAT

MAX TRY

MOAT A

MIT

MODE

NN

NCASES

NIT

NITMAX

NITT

NOCON

25

Material index; MAT=l: soil, MAT=2: porous plate.

Maximum number of function evaluations within an

iteration of the optimization routine to find new par-

ameter values that decrease SSQ. Currently set to

20 in MAIN.

Mode for outflow data. MOAT A=l: transient outflow

data only; MDA T A=2: last Q ( t) entry represents

equilibrium outflow (t is dummy value).

Maximum number of iterations for optimization rout-

ine. (MAIN)

Mode for optimization routine. MODE=O: flow equation

is solved only for initial parameter values. MODE=l:

optimization process continues until parameter values

converge or the number of iterations reaches MIT;

all intermediate parameter values are printed.

MODE=2: as MODE=l, except parameter values are

only printed at end of every iteration. MODE=3: as

MODE=2 but 8(h) and K(h) according to final param-

eter values are also printed. MODE=9: error condi-

tion, program continues with next case.

Total number of nodes used in finite element solu-

tion. (FLOW)

Number of cases considered (MAIN).

Iteration number within time step (FLOW).

Maximum for NIT (FLOW).

Iteration number for optimization routine (MA IN) .

Counter for number of times NIT exceeds NI TMAX

(FLOW).

26

OB

NOMAX

NP

NSTEPS

P(NN2)

PE(NN2)

PNl

PLL

QOUT

RELSAT

SLL

Number of observations.

Maximum for NOCON (FLOW)

Number of parameters to be optimized

Maximum for ISTEP (FLOW).

Array containing values of h and ah/ax at previous

time step (FLOW).

Array containing latest values of h and ah/ax

(FLOW).

Lower boundary condition during outflow process [L]

(FLOW).

Thickness of porous plate (STAG El).

Cumulative outflow during first stage (STAGEl).

Relative saturation ( 8/8 s) at which the soi I is

assumed to be unsaturated (STAG El).

Length of soil core [L] (STAGEl).

SPR(MAT,BN,N,PR) Function to calculate the soil hydraulic

properties . MAT: material index, MAT=l: soil,

MAT=2: porous plate. NB: array containing param-

eter values. N: Index. N=l: 8(h) is calculated,

N=2: K(h) is calculated. N=3: C(h) is calculated,

STOPCR

N=4: h(8) is calculated.

when N=4).

PR: value of h (or 8

Stop criterion for parameter optimization. Optimiza-

tion stops when relative change in each parameter

becomes less than STOPCR (MAIN).

SUMT

T(60)

TE(60)

TOL 1, TOL2

TMINIT

TMIN

WCR

wcs

WT(25)

X(30)

27

Cumulative time [T] (FLOW).

Array containing intermediate values of h and ah/ax

(FLOW)

Array containing values of h and ah/ax at previous

iteration (FLOW).

Absolute and relative convergence criteria for itera-

tive solution process (FLOW).

Initial amount of water in soil [L 3 ] (FLOW).

Current amount o~ water in soil [L 3 ] (FLOW).

Parameter er (SPR).

Parameter 8 (SPR) . s

Array containing weights assigned to every observa-

tion.

Array containing nodal coordinates [L].

APPENDIX C. INPUT FILE FOR EXAMPLE PROBLEM

1 SIL l LAM ll/011/1311 MrlllOD II INPUI DATA (PARKEH FT AL, 19811)

13 10 3.95

0.0250 1

1 1 2 I 3 1 11 1 5 I (> I I 1 H 1 9 1

10 1 11 ?. 12 2 13 ?. o. () 17 (). 03 :1 O.O'j() 0. 1(j7 0. 500 1 . 033 2. 150 'j . 111 7

99'). 9 -1 5000. ()

1 . I -05 0. 57

1. 50 1

1.1 10.0

o.n (). 5 1.0 1. 5 2 . () ? . '.J :\.()

:l.'..> O 3. 7 '..> 3.'.J() 11. no 11. 2) 11 . '.>2

1. 80 3 . /() 11.110 7.80

lU. 20 11 .110 12 .135 13. 59 1'.J. 62 0. 157

?.5? 5.110

0.200

0.300 -?. . () -1. 5 -1. (J -0. 5

(),()

(). 5 1 .Cl 1. 5 1. 1') 1 . 90 2.00 2.25 2.52

28

1000.0 0.003

1.ll 10

0.3138 5 .ll()()

1. 0 2 ?

0 ()

15

APPENDIX D. OUTPUT FILE FOR EXAMPLE PROBLEM

****************************K*K****************H*******K***K******K********KH** * *

K

* * SI LT LO/l.M 8/011/811 * : METHOD 11 INPUT DAT/I. (P/l.Rl

30

8 5.417 13.590 0 1.0 9 999.900 15.620 0 1. 0

10-15000.000 0. 157 1 1. 0 ITERATION NO SSQ ALPHI\ N WCR

0 14.3551178 0.0250 1.5000 0.2000 1 8.6202688 0.0395 1. 5927 0. 1829 2 5. 13811296 0.0586 1.5318 o. 1871 3 IL 3154287 0.0495 1.5358 0.1828 4 4.0541239 0.0530 1.5177 o. 1823 5 3.81~510112 0.0511 1. 5077 0. 1806 6 3.6868916 0.0504 1.4988 o. 1792 7 3. 5398378 0.0494 1.4891 o. 1780 8 3.4026499 0.0484 1.4813 o. 1767 9 3.2833033 o. 01179 1 .11737 0. 1756

10 3.1732512 0.01112 1.4667 o. 111111 11 3.0758791 0 :)471 1.lJ610 0.1732

RSQUl\RE FOR REGRESSION OF PREDICTED VS OBSERVEU =0 . 99930

CORRELATION Ml\TRIX

1 1 1.0000 2 0.6918 3 0.574Q

2

1. 0000 0.9510

3

1.0000

NON-LINEAR LEAST-SQUl\R[S l\Nl\LYSIS: rlNAL RESULTS

VARIAl3LE ALPllA N WCR

VALUE 0. 01~705 1.116097 0.17321

S.E.COEFF. 0.0125 o. 1035 0 . 0166

95% CONFIDENCE LIMITS LOWER UPPER 0.02 0.0767 1. 22 1. 7058 0.13 0 .21 211

--------013SfRVED & FITTED OUTFLOH --------Hr.SI-

NO TIME (HR) 013S r I TTED DUAL 1 0.017 1.800 2.077 -0. 2Tl 2 0.033 3.700 3. 8117 -0. 147 3 0.050 ll. 800 11. 8118 -0.048 4 0. 167 7.800 7. 59ll 0.206 5 0.500 10.200 9 . 97'.J 0.225 6 1. 033 11. llOO' 11. l11Q -0.010 7 2.750 12.850 13. 078 -0.228 8 5.lt 17 13.590 111. 01t1 -0.1124 9 999.900 15.620 16. 1111 -0.521

10-15000.000 0. 157 0.164 -0.027

FINAL CONDITIONS ================

PRESSURE 11 [ /\0 MOISTURE CON1FN1 NOOE OE Prn FUNCTION GRAD I ENI rUNCI ION GRAD I EN r

1 0.00 -1002.000 1 .000 0.210 0.000 2 0.50 -1001.500 1. 000 0.210 0.000 3 1. 00 -1001. 000 1. 000 0.210 0.000 ,, 1. 50 -1000.500 1. 000 0.210 o.uuo 5 2.00 -1000. 000 1. 000 0.210 0.000 6 2.50 -999.500 1. 000 0.210 0.000 7 3.00 -999.000 1. 000 0.210 0.000 8 3.50 -998.500 1 .000 0.210 0.000 9 3.75 -998.250 1. 000 0.210 0.000

10 3.90 -998.100 1. 000 0.210 0.000

INITIAL AMOUNT OF MOISTURE IN SAMPLE 35.099 CM3 FINAL 18. 910

c c c c c c c c c c c

c c

c c

c c

c c

APPENDIX E. PROGRAM LISTING

****************************************************************M* * * * [V/\I U/\ TI ON or so IL llYOR/\UL I c PRO PE.HT I [S rrwM * ONESTEP OUTFLOW DAT/\ BY PARAMETER ESTIMATION *

* * *

* J.B. l

32

c

NU1 -= NV/\R+1 NlJ?- NVAR*? Nl' -= 0 lJO fl l =- NlJl,NU? 11 -= l-NVAH H3( I ) = B( I 1) IF(INDEX(ll).EQ.0) GO TO 8 NP- NP+l 1

c C ----- BEGIN OF ITERATION -----

c

311 NITf= NITT+l NET=O GA= O. l*GA 00 38 J = 1, NP l[MP= Tll( ,J) TH(J) = l.Ul*Tll(J) O(J) = O. CALL FLOW(Tll,OELZ(1,J),NITT,NOCON,MDATA,MODE) I f(MOO[. EQ.1)WRITE(6,1038) (TH( I), 1= 1,NP) lf(MODE.EQ.9) GO TO 144 00 36 1= 1,NUBB OEl.Z( l,.J) =Wf( I )*(OELZ( l,J)-fC( I)) l F( I .GI. NOl3) DELZ( I ,J ) = DELZ( I , ,J )*DEFWf

36 Q(J) = Q(J)+DELZ( l,J)*R( I) Q(J) = 100.*Q(J)/TH(J)

C ----- Sl[[PEST DESCENT -----

c

38 Tll( .J ) -= TLMI' DO t111 1= 1,NP 00 112 ,J =- 1, I SUM=- 0. 00 11() K-=- 1, NOBB

40 SUM= SUM+DELZ(K, I )*DELZ(K,J) D( I ,.J) -= IOUOO.*SUM/(HI( I )*Tll(,J))

4?. D(J, I ) = D( 1,J)

C ----- 0 = MOM[NT MATRIX 411 E( I ) = SQRT( D( I, l)) 50 DO 52 I = 1 , NP

DO 52 J = 1,NP '}2 A( I ,J ) c:: O( 1,J )/( [(I )*E(J))

c C -----/\IS lllE SCALE.D MOMENf MATRIX-----

rm ')11 I :-: 1. NP C( I) ' Q( I)/[( I) Cll I (I ) :-o C( I)

511 /\(1,1) - /\(1,l)~GA C/\I L MAflNV(/\,NP,C)

c c ----- c;r Is fllE CORIU:CT I ON VEClOR -----

SI ff' -: 1. 0

c

56 N[l · NI T+l DO '.> 8 1- 1, NP l B( I ) ·· C( I) l+ Slf P/[( I )+Tit( I) ll(BMIN(l).[Q.flMAX(I)) GO 10 58 I r ( 113 ( I ) . L r . OM I N ( I ) ) 113 ( I ) :- (3M I N ( I ) I F ( 1 R ( I ) . G I . 11M/\X ( I ) ) l B ( I ) = 13MAX ( I ) C( I) = ( rlJ( I )-111{ I) )*E( I )/STEP

58 CON I I NU[ 60 DO 62 1=1,NP

IF ( fll ( I ) * 113 ( I ) ) 66, 66, 62 62 CON I I NU E

SUM!1o-:: O.U C/\I. I. I LOH( I B, r C, NI TT, NOCON, MDI\ l /\,MOOE) I r ( MO()[ . [CL 1 ) WR I T [ ( 6 ' 1 0 3 8 ) ( rn ( I ) , I = 1 , N p ) I F ( MODF. [Q . 9) GO TO 11111 DO 611 ' " 1, Norm R( I ) =WI{ I)"' ( 10( I )-re( I)) II (I . GI .NOU)R( I ) ~ R( I )l+OEFWf

611 SUMU- SUMU-+R( I )*R( I) 66 SllMl = 0.0

SllM2=0.fl SUM3 "-0.0 DO 68 1- l, NP SllMl = SllMl+C( I )*CHI( I) SUM2 = SlJM?+C( I ) *C( I)

68 SUM3 =- SlJM1+Cll I ( I )*Cll I ( I ) ARG .- SUM1/SQHT(St1M2*SUM3) ANGLE~ 57. 2 9578*ATAN2(SQRT(1.-ARG*/\RG),ARG)

33

134 135 136 137 138 139 1 llO 1111 1'12 1113 141~

145 1116 1111 1118 1'19 150 151 152 153 1511 155 156 157 158 159 160 161 162 163 1611 165 166 16 7 168 169 170 1 7 1 172 173 1 ,,~ 1 f '.) 176 177 118 179 180 181 rn 2 183 1811 18'} 186 187 188 189 190 191 192 193 1911 1')5 196 197 1913 199 ?00 2 01 2 0 2 203 2 011 2 05

34

c ---------- 206 00 72 I == 1 , NP 20 7 Ir( Tll( I )*TB( I) )74, 71-t, 72 ?OB

72 CONllNUE 209 IF(N[T.GE.MAXTRY) GO TO 79 2 10 lf(SlJMB/SSQ-1.0)80,80,7/t 211

74 IF(ANGLr- :rn.0)76, 76, 78 2 12 16 s H r- s 1 E r /? . o 2 1 3

GO I 0 '..>6 21'1 78 GA~ lO.~GA 2 15

GO TO 50 216 79 WRl1E(6,1086) 217

GO TO 96 218 c 2 19 C ----- PRINT COEFflCIENTS AFTER [ACH ITERATION ----- ?20

80 CONJ INUl 221 DO 82 l = l,NP 222

82 111( I) = fB( I) ?23 WRITE(6, 10112) NITT,SUMB, (HI( I), 1= 1,NP) 2211

c 225 90 DO 92 1= 1,NP 226

I F(ABS(C( I )*SfEP/E( I))/( 1.0E-20+ABS(TH( I)) )-STOPCl1) 92,92,911 227 92 CONTINUE 228

GO 10 96 229 94 SSQ~ SUMU 230

I r ( N I 1 T • l T . M I r ) GO T 0 3 ,, 2 3 1 c 232 c ----- fNO or lfERATION LOOP 233

96 CONllNlJE 2311 CALL MATINV(D,NP,C) 235

c 236 C ----- WRITE RSQUARE, CORRELATION MAfRIX ----- 237

SUMOc=- 0.0 238 StJMC=O. O 239 SllM02= 0. 0 2110 SUMC2=0. 0 2 11 1 SUMOC= O.O 242 00 98 1= 1,NOl3B 2113 SUMO= SUMO+FO( I) 21111 SUMC= SUMCHC( I ) 2115 SUM02= StJM02+rO( I )*FO( I) 2"6 SUMC2= SllMC2+rC( I )*FC( I) 2117

98 SUMOC=SUMOC+FO( I )*FC( I) 248 RSq= ( SlJMOC-SUMO*SUMC/NOBB )**2/ ( ( SUM02-SUMO*SllMO/NOBB) *( SUMC2-SUMC* 2119

1 SUMC/NOBl3) ) 250 WRITE(6, 1050) RSQ 251 DO 100 I = 1, NI' 252

100 E( I ) = SQRT( D( I, I)) 253 WRlfE(6,lOl16) (1,1 = 1,NP) 2511 00 1011 1= 1,NP 255 DO 102 J := l, I 256

102 A( .J, I ) -= D(,J, I)/( E( I )*E(J)) 257 104WRITE(6,1048) 1,(A(J, I ),J = l, I) 258

c 259 C ----- CALCULATE 95% CONFIDENCE INTERVAL ----- 260

Z= l./FLOAT(NOBB-NP) 261 SDEV= SQRT(Z*SUMB) 262 WRITE(6, 1052) 263 TVAR= 1. %+Z*( 2. 3 779+Z*( 2. 7135+Z*( 3. 187936+2. l166666*Z**?.))) 2611 DO 108 1= 1, NP 265 SECOEF= E( I )*SDEV 266 TSEC= fVAR*SFCOlF 267 TMCOl= lH( I )-TSEC 268 TPCOE= Tll( I )+TSEC 269 K=2*1 270 J = K-1 271

108 WRITE(6,1058) Bl(,J),Bl(K),TH(l),SECOEF,TMCOE,TPCOE 272 c 273 C ----- PREPARE FINAL OUTPUT ----- 2 flt

110 WRITE(6,1066) TYPE(1),TYPE(2) 275 DO 118 I = 1, NOUB 276 R(l) = FO(l)-fC(I) 277

118 WRITE(6, 1068) ~,HO( I), FO( I), FC( I ),R( I) 278

35

c ?..79 C ------FINAL PRESSURE HEAD AND MOISTURE CONTENT DISTRIBUTION----- 280

122 lMIN= 0. 28 1 WR 11 [ ( 6, I 080) 252 NL~ I NS-1 283 DO 130 L= 1,NF. ?8l1 l ~ ?*L-1 285 11 = 1+1 286 12 7 1+~ 287 137. I ~3 288 El =X(L+l)-X(L) 289 P13 -= . 7llfl/1107*P( I )+.2592593*P( 12)+.071107111*E.L*( 2.*P ( 11)-P(13)) 290 P23 -= .2592593*P( I)+. 7Ll07ll07*P( 12)+.07110711l*[L-M( P( 11)-2 .*1'(1 3)) 29 1 Zl = Sf'R(l,113,1,P(I)) 29?. Z2= SPR( 1, 113, 3, P( I) )*P( 11) 293 Z3=SPR(1,TB,1,P13) 2911 Zl~ = Sf'R( 1, ll3, 1, P23) 295 Z5 -= SPR(1,TB,1,P(l2)) 296 WRITE(6,1060) L,X(L),P(l),P(l1),Z1,Z2 297

130 1MIN :-.: TMIN+Ll.*(0.5*(Zl+Z5)+Z3+Zl1)/3. 298 1 MI N= l MI N*AREA 299 Z2 -= SPR( 1, ru,3,P( 12))*P( 13) 300 WRI lE( 6, 1fl60) LNS,X( LNS), P( 12), P( 13) ,Z5,Z2 301 lMINI T= TMINll-Tll( ll1)*AREA*PLL 30?. WR I H ( 6 , 1 0 8 2 ) 1 M I N I T, 1 M I N 3 0 3 lf(MODF...LE.2) GO TO 1l1l1 3011

c 305 C ----- WRllE SOIL HYDRAULIC PROPERTIES ----- 306

WRITE(6, 1069) 307 PRESS-: 1. 18850 308 SNl =c ll.O 309 RKLN~ l.O 310 SALPllA= tn( 6) 311 SN= ll3(7) 312 SWCR= lB(8) 313 SWCS =TB(9) 3111 SCONDS= Tl3(10) 315 WRITE{6, 1072) SNl,SWCS,RKLN,SCONDS 316 DO 1l10 I = 1, 75 3 11 lf(RKLN.Ll.(-16.)) GO TO 1l12 318 PRESS= 1. 18850*PRESS 319 SM= l.-1./SN 320 SNl = SM*SN 321 RWC=- 1. / ( 1. +(SAL Pit/\* PRESS) **SN) **SM 322 WC= SWCR+( SWCS-SWCR)*RWC 323 TE.RM= 1.-RWC*(SALPHA*PRESS)**SN1 32'-l lf((TEHM.LT.5.E-05).0R.(RWC.LT.0.06)) TERM = SM"RWC**(l./SM) 325 RK= SQRl(RWC)*TE.RM*rERM 326 TERM= SALPllA*SN 1 *( SWCS-SWCR) *RWC*RWC**( 1. /SM)* ( SALPltA*PRESS) ** 327

l(SN-1.) 328 AK= SCONOS*RK 329 DIFfUS= AK/TERM 330 l'RLN= ALOGlO(PRESS) 331 AKLN=AIOG10(AK) 332 RKLN =-= Al.OC10(RK) 333 IJ I f I. N =fl I. OG 1 0 ( DI Fr US ) 3 31~

1110 WRI TE(6, 1071l) PRESS,PRLN,WC,RK,RKLN,AK,AKLN,DIFFUS,DlfLN 335 142 CONTINUE 336 1111~ CONTINUE 337

c 338 C ----- FND OF PROBLEM ----- 339

1000 fORMA I ( I 5) 3110 1002 roRMfll ( 1111, lOX, 82( 111* )/ 11X, 111*, BOX, 111*/ 11X, lH*, 811X, 111*) 3111 1004 r ORM/\ I ( 20flll) 3112 1006 FORMAT( 11X, 1H*,20M, 1H*) 3'-13 1008 FORMAT( 1IX,1H*,80X, Hl*/1lX,82(111*)) 34Lt 10111 FORM/\T(/215,E10.1,4F10.0) 3115 1016 FORM/\ T( 11FHJ.0, 1115/) 3116 1018 fORMAT(//11X, 'PROGRAM PARAMETERS'/11X, 18( 11t= )/11X, 3117

+'NUMB R OF NODES •••.••..•••.•.••.•••. (NN) . ••... •....•.... 1 ,13/llX, 3118 +'NOOE AT SOIL-PLATE BOUNDARY •.•...•.. (LNS) .•........•• . •. 1 ,13/llX, 3119 +'INITIAL TIME STEP ...•.....••.....••. (ONUL) •...•.....•. 1 ,E9.2/11X, 350 +'PRESSURE ltEAD LOWER BOUNDARY ••.•••.. ( PNl) ...••.....•.. ', F8. 3/1 lX, 351

36

+'PNEUMATIC PRESSURE ....••............ (AIRP) ..........•. ',F8.3/11X, 352 +'fE.MPORAL WEIGHTING COEFF ...••....... (EPS1) ............ ',F7.2/11X, 353 +'ITERATION WEIGHTING COEFr •..•......• (EPS2) ....•....•.. ',F7.2/11X, 3511 +'MAX. ITCRATIONS ••.••........•....... (MIT) ............... ',13/llX, 355 +'DATA MODE •....••••••..••....••....•• (MDAfA) .......••.... I, 13/11X, 356 +'No. or OBSERVATIONS ....•...••..•.... (NOIJB) .......•....•. ', 13) 357

1020 roRMAf(5t10.0) 358 1021 FORM/\T(2f10.0, 15, f10.0) 359 1022 FORMAT(5110) 360 1026 tORMAT(//11X,'SOIL AND PLATE PROPERTIES'/11X,25(11t= )/11X, 361

+'SOIL COLUMN LENGTH ..•......•...•.... (SLL) ............. ',F8.3/11X, 362 +'COLUMN DIAMETER •••...•••••....••...• (DIAM) ........•... ' ,f8.3/11X, 363 +'llllCKNESS or PL/\1E. .•........••..•.. (PLL) ....•........ ',F8.3/11X, 3611 +'PL/\lE CONDUCTIVITY •..•••.••...•..... (CONDS(2)) .....• ',E9.4/11X, 365 +'SAlUR/\TEO MOISTURE CONTENT ...•...... (WCS) ............. ',r8.3/11X, 366 +'RESIDUAL MOISTURE CONTENf ........... (WCR) ...........•. ',FB.3/11X, 367 +'FIRST cour1c1ENT ................•.. (ALPHA) ..•.•...... ' ,rB.3/11X, 36B +'SECOND c0Err1c1ENT ..............••.. (NJ ......•........ ' ,FB.3/11x, 369 +'SAfUR/\TED CONDUCTIVITY SOIL ...•••.•. (CONDS(l)) .•.•.. ',E9.4) 370

1021~ FORM/\T(//11X,' INITIAL CONDITIONS'/11X, 18( 11t= )/11X, 'NODE' ,2X, 'DEPTH 371 l 1 ,2X,'PRESSURE ltEAD',5X,'MOISTllRE CONfENf') 372

1028 FORMAT(215,2F10.0) 373 1030FORMAf(//5X,8(111~),'ERROR ENCOUNTERED WHILE READING INITIAL CONDIT 374

110NS, CHECK NODE',IL1,1X 'EXECUTION lERM!NAfE0',9(111*)) 375 1032 roRMAr u 111 x, •OBSERVED 1 , 2A4/ 11 x, 16( lit= l / 14x, 'OBS•, 5X, '1ms ··, 5x, 2A11 376

1,11x, 'TYPE' ,11x, 'WEIGHT') 377 1036 FORMl\f(11X,15,2F10.3,5X,13,F10.1) 378 1038 FORMAT(ll?X,5(F8.4,3X)) 379 101!0FORMAT(1H1, 10X,' ITERATION NO' ,6X, 'SSQ' ,5X,5(6X,A4,A2)) 3BO 10112 fORMAT(15X,12,5X,F12.7,8X,5(f8.11 3X)) 381 10116 FORMAT(// I lX, I CORRE LAT ION MATR 1x 1/11X,1B( HI= )/111X, 10( l1X, 12, 5X)) 3B2 1048 roRM/\T(11X,13,10(2X,r7.4,2X)) 3B3 1050 FORMAT(//11X, 'RSQUARE FOR REGRESSION OF PRrOICTEO VS OBSERVED = ', 3B4

1F7.5) 385 1052 ronMAl(//11X,'NON-LINE/\R LEAST-SQUARES ANALYSIS: rlN/\L RESULlS'/ 386

lllX,118( 1H :-: )/53X, '9'.J% CONFIDENCE LIMI rs• /11X, 'VARIABLE' ,BX, 'VALUE'. 387 ?lX, 'S.E.COErr. ',11x, 'LOWER' ,BX, 'UPPER') 31rn

1058 FORM/\ f( UX, All, /\2, llX, FlO. 5, 5X, F9.1t, 5X, r6. 2, llX, F9.11, 5X, F9. II) 389 1 060 roHMA T( 1 ox • 1 11 • 1 x. F7 . 2. 2 r 1 1 • 3 , 3 x, 2 r 9 . 3 ) 3 9 o 1062 roRMAl(lOX,ll1,1X,F7.2,f11.3,8X,F9.11) 391 1066 IOHMATl//HlX,8(111-),'0BSEHVEO & FITTED ',2All,13(111-)/45X,'RESl-'/1 39?.

lOX, 'NO ,3X, 'TIM[ (HR)' ,11x, '011s' ,11x, 'FITTED' ,11x, 1 1Jll/\L 1 ) 393 1068 roRMAT( lOX, 12, FlO. 3, lX, 3F9. 3) 3911 1069 roRM/\l(IH1,10X,'PRESSURE',11X, 1 LOG P',6X,'WC',7X,'REL K',5X,'LOG RI< 395

1',6X,'AllS K',ltX,'LOG KA',5X,'DIFFUS',5X,'LOG O') 39(> 1070 roRM/\ r( HJX. E 10. 3, r8. 3, r 10. 11, 3 ( E 1 3. 3, rn . 3) ) 391 1072 roRM/\ l ( 1ox.r10. 3, BX, F 10. 11, E 13. 3, BX, E 1 3. 3) 398 1080 roRM/\T ( // 1 lX, Ir I NAL CONDI r IONS' I I lX, 16( IH'" )/ 25X, I PRESSURE HEAD'. 12 399

1x, 'Mo1s11mr coNJ[Nf' 111x, 'NODE' ,2x, 'D · r1H' ,3x, 'ruN

37

lr(/\Ml\X) 30 , 30, 111 11 2 11 111 INDX(IC,l) = IR /125

Ir( IR. EQ. IC) GO TO 18 1126 DO 1(1 L-= 1,NP 11?.7 f' -:-- 1\( I H, I.) /128 /\( IR,L) =I\( IC:,I) 11?9

16 /\( IC,L) :: p 1130 r ~- n( 1 R) 1131 B(IR) = ll(IC) '13? B ( IC) -= P 113 3 I ::: I+ 1 11311 INUX(l,2) = 1C 113'..i

18 P= l .//\(IC, IC) 1136 /\ ( IC, IC) :· 1 . ll t13 1 DO 20 L= 1, N r 113n

20 A(IC,L) = /\(IC,L)*P 1139 B( IC) = B( IC)*P 11110 DO ?II K= l,NP 11111 If ( K. [.Q. IC) GO TO 211 11112 P=A(K,IC) 11113 /\( K, IC) :..: Q. O 111111 DO 22 1_ -- 1,Nr 11115

22 A(K,L) -::1\(K,L)-A( IC,L)*P 4116 13( K) = B( K)-!3( IC)*P 1147

211 CON I I NUL l111n GO TO 11 1~119

26 IC= INOX ( I, ?) 1150 IR= INDX ( IC, 1) 1151 D028K= l,NP 11'..)2 P=I\( K, IR) 1153 I\( K, IR ) -o- 1\( K, IC) 11511

28 l\(K,IC) = P 1155 1-= 1-1 1156

30 IHI) 26,32,26 1157 32 RE 1 lJRN !158

[N() 1159 SUl3HOlJ r I NE FLOW( E3N, re, NI TT, NOCON, MDATA, MOOE) '160 DI Mrns I ON r( 60), FC( 25). p IN( 60). PE( 60), l E( 60), BN( 15) "61 COMMON/MA/X( 30), P( 60), I SPH( 30), NN , AREA, LNS, PLL, SLL, PNl, 110( 25), DNlJ 1162

1 L, NOB, NOB2, l MIN IT, EPSl, EPS2, INDEX( 5), NVAR 1163 COMMON/ST 1 I I OBS. SMT 1, I STP 1, Ir LAG, oLDT 11611 DAT/\ Nl1Ml\X/10/,TOL1/1.00/,TOL2/0.005/,NSTEPS/300/,0ELMl\X/0.5/ 1165 OATA NOMAX/45/ 466

c 1167 C ----- UPD/\TE PARAMETER ARRAY ---- 1168

K-= O L169 NUl = NVAR+l 1170 NU2::-: NV/\R*2 1171 DO? l = NU1,NU2 472 IF( INDEX( 1-NVAR).EQ.O) GO TO 2 473 K= K+ 1 1111~ BN( I ) =BN( K) L175

2 CONTINUE 476 C ----- O[FINE INITIAL CONDITIONS & ~l\LCULAlE OUTFLOW /177 C OU~ I NG SATURATED STAGE-------- l17H

EPSM= l.-EPS2 '179 ISTP1 =1 '180 1013S= 1 1181 QOUT=O. 482 I FL/\G=O 483 SMT 1 =O 1184 NN2= 2*NN 485 IF(NITT.GT.O) GO TO 5 486 DO I~ 1= 1,NN2 1187 PIN(l) = P(I) 1188

4 CONl INUE 489 NOBl = NOB-1 1190 IF(MDATA.LE.1) NOBl = NOB 491

5 DO 6 1= 1,NN2 1192 P( I )=PI N( I) 1193

6 CONT I NU [ 11911 P(NN?-l) = PNl 495

38

c C ----- SOIVE FOR FIRST STAGE OF OUTrLOW -----

CALL SfAGt:l(BN,FC,QOUT) 00 8 1-= 1, NN2 PE( I ) =~ I'( I )

8 CONllNU~ c c ------ D[T[RMIN[ AMOUNT or WATER IN SAMPLE -----

CALL IOlALM(P,BN,TMIN)

c c

9 c

10

12

1 3 c c

11~

15 c

16

17

18

19

20

22

24

c c c

28

30

c

TMINll =lMIN+QOUT DrL T:: ONUL 0(1 .M I N== O. 005*DNUL OIN - O[LT SUM I =SM 1 1 +OELT ISTEP= ISlPl

----- DYNAMIC PART OF MODEL -----Nl =NN?-1 Nf T- O Ir( MOOE. EQ. 0 )WR I TE( 6, 1002) NIT I DEL TI I ST[P I SUMT. (P E( I) I I= 1. NN 2 ) DO 12 1== 1,NN2 T{ I )= PF.( I) CO N I I NUF NIT =Nlltl Cl\Ll Ml\ I [Q ( BN I [PS 1, P[, OF.LT) I r ( N I T. GT. 1 ) GO TO 1 4 DO 1 3 I= 1, NN2 1 E( I )=P[( I)

----- CHECK ITERATIVE PROCESS DO 15 l - 1,NN2,2 10L-= 10Ll+TOL2*A13S(T( I)) IF(l\IJS(Pf{l)-T(l)).GT . TOL) GO TO 16 CONTI NU[ IF{MOOE.EQ.O)WRITE(G, 1002) NIT,DELT, ISTEP,SIJMT, (PE( I), 1=1,NN2) IF(DELT.LT.DELMIN) GO TO 19 GO 10 28 IF(NI r.GE . NI fMAX) GO TO 18 !JO 17 1=1,NN2 ff.MJ>:-EPS2'"1'E.( I )+EPSM*lE( I) H( I )=P[( I) PE( I )-= lLMP CONTINUE GO 10 10 NOCON=NOCON+l DELT=0.5*DELT IF(DELT.GE.DELMIN.AND.NOCON.LE.NOMAX) GO TO 22 tr(NOCON.GT.NOMAX) WRITE(6,1007) IF(OELl.Ll.DELMIN) WRITE(6,1008) DELT,DELMIN,SUMl,NITT

WR I TE( 6, 1009) DO 20 1=- 1,NN .1 =2*1-1 WR I TE ( 6 I 1010) I Ix ( I ) • p ( J ) , p ( J + 1 ) I PE ( J ) I PE ( J + 1 ) CONTINUE MODE=9 REl URN SUMT=SUMT-OELT DO 211 I= 1 , NN2 Pf( I )=O. 5*( P( I )+PE( I)) GO TO 9

---- CALCULATE CUM. OUTFLOW IF IFLAG=l ------

IF( IFLl\G.EQ.O) GO TO 34 DELF=( SUMl-110( IOBS) )/DELT DO 30 1= 1,NN2 T( I )=Pf ( I )-DELF*( PE( I )-P( I)) CONTINUE CALL 10TALM(T,13N,TMIN) FC( 1013S) =TMINIT-TMIN 1013S= IOl3S+1 IFl.AG=O

11 ~>6 1191 1198 1199 50lJ 501 502 503 5011 505 506 507 508 509 51() 511 512 513 51'1 515 516 ) 1 7 518 519 '.120 521 '.)?.2 5?3 521~

525 526 527 528 529 530 531 532 533 5311 535 536 537 538 539 540 541 5112 5113 51111 51~5 5116 5111 5ll8 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 5611 565 566 567 568

C ----- PREPARE FOR NEXT TIME STEP -----

c c

c c

c c

c

34 IF( IOOS.GT.NOOl.OR. ISTEP.GE.NSTEPS) GO TO 40 DELCll=-1. 0 If (NIT. u : . 2) DELCll= 1 • 25 lf(Nlf.GE.6) DCLCll=0.80 DELCll= AMINl(OELCH,DELMAX/D[LT) DELT=UfLT*D[LCH DO 36 J=1,NN2 PEl = PE(J)-P(J) P(J ) = P[(J) PE(J) = P(J)+DELCH*PE1

36 CONf INllE SUMf =- StJMf+Dr:Lr IF(SUMT.GE.110( 1013S)) IFLAG= 1 ISTEP=IST[P+1 GO TO 9

--------------------40 IF( ISIEP.GE.NSTEPS) WR I TE( 6, 10111 )NSTEPS, SUMf, NI TT

IF( MDATA. LE. 1) GO TO 42

---- COMPlJH [QUI LIBRIUM OU Tr LOW If MDAfA= 2 ----XZfRO= PLL+SLL-PNl 1)0 41 1= 1, NN ,J =2*1-1 J 1=- .J+I PE(J) -= X( I )-XZERO PE(,J 1) = 1. 0

41 CONllNU[ CALL IOTALM(PE,BN,TMIN) rC( IOOS) = lMINI r-TMIN

112 DO 11t1 I = 1, NN2 P( I) = PF ( I)

I~ II CONTINUE

------ CALCULATE lllElA(ll) IF N0132 > 0 -----1r1Non2.r:q . o) GO ro 60 Nl =NOB+l N2 -: NOFJ+NOO? DO 116 1- Nl, N? fC( I ) -= SPR( 1,13N, 1,110( I))

116 CONTINllF 60 REl URN

c ----------

c

1001 roRMAI( 11X,lt[10.3) 1002 roRMAl(/11X,'PEll) DURING ITEHf\llON (Nll = ',13, 1 DLLl -=- 1 ,[10.?,' ISf

1[ P= I ' 111. I SlJM I = , [ 10. 3' I ) I I ( lOX, 10 I 11 . 3) ) 1uo3 r ORMf\I ( 1 1 x. 2 r 10. 5 ) 1007 fORMAl(//11X, 1 TROUl3LE CONV[RGING, START AGAIN Willi DlrfER[NT INlll

11\L VAl.UF s I) 1008 roRMAl(//llX,8(111*),'DELT = ',r11.11,', IS LESS TllAN D[LMIN ( = ',Ell.

111, 1 ), EX[ClJllON TERMINATED AT TIME = ',[11.11,' (Nll =- )',15) 1009 FORMAT(//1X,' LAST CALClJLATEO VAl.lJES'/11X,22\11P•)/11X,'NODE',5X,'D

IEPIH' ,9X, 'P( I ) 1 ,6X, 'GR/\DIENf' .9X, 'Pf( I ) 1 ,6X, GR/\UIE.NI"') 10 10 fOHMA I ( 11 X, 111, F 10. 2, 2 ( 3X, 2 F 12 . 4) ) 10111 IOHM/\1(/11X,'NO. or SfEPS F.XCE[DS',11~.· AT llM~. - ',Fl0.3,' Dl!HING

1 1 r LHA 1 1 ON' • 111 ) 1016 FORMAT"( llX, 15,F10.3,2(2X,FI0.3))

[N()

SUl3ROUTIN[ STAGEl(BN,FC,QOUT)

c PURPOSE: SIMlJL/\TE flRST (S/\llJRATED) STAG[ or rtow c

DIMr.NSION BN( 15), rC(25) COMMON/ A/\A/X( 30), P( 60), I SPR( 30), NN, AH£.A, LNS, PLL, SLL , PN 1, 110( 25 ) COMMON/S l l /I 0135, SIJMT, IS TEP, I FLAG, OLD r OAT/\ RELSAl/0.99/ LPLUS= LNS+l UNSAf ~ RI LS/\l*BN(9) /\fP- SPR( 1,mi,11,UNS/\T)

10 DEIWC~ FJN(9)-UNSAT 00 ?O 1-=- 1, LNS

39

569 570 571 572 573 5111 575 576 577 578 579 580 581 582 583 58ll 585 586 587 588 589 590 591 592 593 5911 595 596 597 59fl 599 600 601 602 603 6011 605 606 60 7 608 609 610 611 612 613 6111 61'.i 616 617 618 619 620 621 622 623 6 211 625 626 627 628 629 6 311 631 63 2 633 6311 635 636 637 638 639 6llfJ 6111

c

c

40

J = ?* 1-1 lf(P( .J) . Gf ./\[P) GO 10 21

20 CON I I NIJf. 1)[111 - l'(LNS)-PNl GO TO 3 7

? 1 I r( I . GI . 1 ) GO TO 22 orut= P( 1 )-PNl-X(NN) EL=X(2 )- X( 1) GR/\D- BN( 1'.:> ) *DF.LH/(BN( 10)*PLL+l3 N( 15 )*SI L) l =- 2 GO 10 2 11

22 IMIN= l-1 Df.Lll =- /\EP+X( IMIN)-PN1-X(NN) E.L= X( I )-X( IM IN) TS -= SLL-X ( I ) +0.5*EL GR/\D-:: 13N( 15 )!10FLH/(ON( 10)*PLL+RN( 15)*TS)

24 Ol.DT = SUM r QOLD= QOll r QOUf = QOUf+ OELWC*ARE.A*EL oru ·- ou WC*CL/(ON( 10)*Gn/\D) SlJMl -= SIJMf+DEL I lf"( SUMf.GC.110( 1005)) IFl./\G= l Ir (I I l/\G . fQ.O) CO 10 26 re( IOB S) -= ( SllM f-110( IOOS) )/O[Ll*QOLO+(llO( IOll S) -01 01 )/DlLl*QOUT IOf\S::: IOBS+ l I FL/\G -= O

26 1-= 1+1 ISIE.P= l.::;1[1'+ 1 Ir ( I • c; r. l NS) GO TO 311 GO I 0 ??.

34 DO 36 1-:: 1,LNS J =?* l-1 P( .J )= Al P

36 P(J+l) = l.U 37 DO 38 l = l.PLUS,NN

J =- ?.* 1-1 P(J) -:- -D [l. ll*( X ( I )-SLL)/PLL + P(LNS)

38 P(J+l ) - -D ELl l/PLL l12 CO NTINU E

R[ ru1rn [ND

c -------------------c

c

SUOROUTINE lOTALM(P,BN,TMIN) DIM[NSION P(60),BN(15) COMMON//\A/\/X(30),EMTY(60), ISPR(30),NN,/\RE/\

C PURPOSE: TO CALCULATE THE /\MOUNT OF WATER IN SAMPLE c

TMIN= O. NE= NN-1 00 10 L-= 1 , NE MA I = IS PR ( l. ) MATl -:: I SPR( L+l) 1= 2*L-1 11 = 1+1 12= 1+2 13= 1+3 [L-X(L+l)-X(L) P13 = . 7ll07ll07*P( I )+.2592593*P( 12)+.0740711l*EL!1(2 . *P( I 1)-P(13)) P23 = . 711()711Q7*P( I 2 )+. 2592593*P( I)+. 071107!11 *EL*( P( I 1 )-2. *P( 13)) Z 1 = S PR (MAT, ON, 1 , P ( I ) ) Z3 -: 0. 7ll071107*SPR(M/\f,ON, 1, P13)+0.2592593*SPR(M/\Tl,BN, 1, P13) Zll =O. 2592593*SPR( MAT, BN, 1, P23) +O. 71~07407*SPR( MA Tl, BN, 1, P23) Z 5 = SP R ( Ml\ I 1 , l3 N , 1 , P ( I 2 ) ) TMIN= TMIN+/\REA*EL*(0.5*(Zl+Z5)+Z3+Z4)/3.

10 CONflNUE RETURN END SUBROUTIN E B/\NSOL(NEQ)

6112 6113 (,1111 6115 6"6 6117 6118 6119 650 651 652 653 6511 655 656 651 658 659 660 66 1 662 66 .1 66 11 6(15 666 66 7 668 669 670 611 672 6 73 614 6 7'j 676 677 67fl 679 680 681 682 6fl3 6811 685 686 687 688 689 690 691 692 693 6911 695 696 697 698 699 700 101 702 703 1011 705 706 701 708 709 710 711 712 713

41

c 7111 C PURPOSE: TO SOLVE TllE GLOl3/\L M/\TRIX EQU/\TION 715 c 716

COMMON /CCC/ S( 60, 4), r( 60) 711 Nl = NCQ-1 718 DO 11 I == 1 , N 1 11 9 J = l-1 720 M-: M I Nil( 11, Nf.Q -J) 7? 1 P=- S(l,1) 722 no 11 L, ? ,M 123 C=S(l,L)/P 7211 f f - J+I 725

, Jj :- (J 726 DO 2 K- I , M l? I J J - .J J + 1 7 2 8

2 S( 11,.JJ ) : S( I I ,JJ )-C*S( I, K) 729 11 S ( I , L ) - C / 3 0

DO 6 I =- 1 , N 1 7 3 1 J-1-1 732 M=MINO(l1,NEQ-J) 733 C=- F ( I ) 7 311 f(l) =C/S(l,1) 735 DO 6 l =-? ,M 736 11 =.J+I 737

6 r( 11 ) = r( 11 )-S( I, L)*C "/38 r(NEln = r(NEQ)/S(NFQ, 1) 1]9 DO 8 1= 1,N1 7110 11 -= NEQ-I 7111

.J = I I - 1 7112 M-= MINO(ll,NEQ-J) 7113 IJO 8 l

42

11 r( L 1 ) = ( EPSM*P( L 1 )+1. )*CONDl c C ----- ELEMlNT LOOP; CONSTRUCT GLOOAL MATRIX -----

DO 12 L::- 1, NE LL= 2·*L-2 Ll = l.L+l L2= LL+2 L3 = LL+3 L'1 = LL+l1 U =X( L+l )-X( L)

c C ----- CALCULATE llEHMITIAN BASIS FUNCTIONS-----

fE(2, 1) = .1181895*EL

c

fE(2,2) -= .1?.5*CL f[(2,3) = .0246676*EL OX(2, 1) = .1993777*EL DX(2,2) = -. 125*EL DX(2,3) = -.1279491*EL DO 6 K=- 1, 3 r c ( 11, K ) = - r E ( 2, 11- K)

6 DX(4,K) =DX(2,4-K)

C ----- CALCULATE MATERIAt PROPERTIES AT LOOAlTO POINTS -----W 1 = FE ( 1 , 1 ) * T ( L 1 ) +FE ( 2, 1 ) * T ( L2 ) +F E ( 3 • 1 ) * r( L3 ) +r E ( 11 , 1 ) * l ( Lii ) W2= rE.( 1, 2 )*1 ( L 1)+FE(2, 2 )*1 ( L2 )+FE( 3, 2 )*T( I 3) +rC( 11, 2 )*T( Lii) W 3 = F [( I , 3 ) * I ( L1 ) +r [ ( 2, 3 ) * T ( L2 ) +r E ( 3 , 3 ) * T ( L3 ) +F E ( 11 , 3 ) * T ( Lii ) MATl = ISPR(L)

c

MAT2= I SPR( L+ I) G2= .08G1869 Gl - 1.0027021 CONDI :: (Gl*SPR(MATl,UN,2,Wl )+G2*Sl'R(MAT2,BN,2,W1) )/EL COND2= . 7111111*(SPR(MAT1,0N,2,W2)+SPR(MAT2,BN,2,W2))/EL COND3 = (G2*Sl'R(MAT1,BN,2,W3)+Gl*SPR(MAT2,0N,2,W3))/E.L EL1 = .25*CL/DELT CAPl = (Gl*SPR(MAfl,BN,3,Wl )+G?*SPR(MAT2,0N,3,W1) )*[L 1 CAP2 = .7111111*(SPR(MAT1,BN,3,W2)+SPR(MA12,BN,3,W2))*EL1 CAP3 = (G?.*SPR(MAT1,BN,3,W3)+Gl*SPR(MA12,AN,3,W3))*EL1

C ----- ADD f.l.CMENT CONTR113UTIONS TO GLOOAL MAlRIX -----K=O

c

DO 10 I =-- 1 .11 I l = l L+ I DO 10 ,J -= 1,11 Wl :-: DX( J, 1)*OX(I,1)*CONOl+DX(J,2 )*DX( I, 2 )*CON02+DX( .J, 3 )*OX( I, 3 )*

1COND3 W?= f[( J, 1 ) *FE( I, 1) *CAP 1+fE(J,2) *FE( I, 2) *CAP2+FE( J. 3 )*FE ( I, 3) *CAP3 JJ =J+l-1 K-= K+l S( 11,J.J) = S( I 1,JJ)+Wl*EPSl+W2

10 SRtK) -=Wl*f PSM+W2

C ----- C:ONSTRUCl RIIS VECTOR ----

c

fL1 =. ?14?85 l*EL*( GON01+1. 75*CON02+CON03) f(Ll ) - r(l 1)+SR(1 )*P(Ll )+SR(2)*P(l2)+SR(3)*P(U)+SR(ll)*P(lll)-EL1 f( L? ) :-:. I ( L2 )+SR(?. )*P( L 1)+SR(5 )*P( l2 )+SR( 6 )* P( L3 )+SR( 7 )*P( LI~)+

1 (. 0996889"CON01-. 0625*COND2-. 06397116""COND3) *LL*El I ( L3 ) -=- r ( 1.3) +SR ( 3) *P( L 1 ) +SR ( 6) *P( L2) +SR ( 8) * P( L3) +SR ( 9) * P (Lit) +[Ll

1?. F (Lit) -= f ( 1_11) +SR( It) *P( L1 ) +SR( 7) * P( L2) +SR( 9) * P( L3) +SR( 10) * P( Lit)+ 1( .0996889*COND3-.0625*COND2-.0639746*CON01)*fL*[l

C ----- INCLUDE OOUNDARY CONDITIONS S( 2, 1) = 1. Wl =AM IN 1( 0. 5, 100 . / ( X( 2 )-X( 1 ) ) )

C W2= 1 . F(2) = ( 1.-Wl )*PE(2)+W1 F( 1) = F(1)-S(1,2)*F(2) F(3) = F(3)-S(2,2)*F(2) F ( 11 ) = r (It ) - S ( 2, 3 ) * F ( ?. ) S( 1, 2) -= 0. 5(2,2) =0. S(2,3) =0.

78() 787 788 789 790 791 792 793 7911 795 796 797 798 799 800 1301 802 803 8011 805 806 807 808 809 810 811 812 813 81'1 815 816 81 l 8 Hl 1319 820 821 822 823 8211 825 826 827 8213 829 830 831 832 833 8311 835 836 837 838 839 8110 8111 8112 8113 81111 8115 1346 8117 8'113 8119 850 851 852 853 8511 855 856

c

211 CON f I NUf r(Nl ) = I (NN?)-S(N1,2)*P(N1) F(N2) = r(N2)-S(N2,2)*P(N1) F(N3)-F(N3)-S(N3,3)*P(N1) S( Nl, 1 ) = S( NN2, 1) S(N2,2) =-= S(N2,3) S(N3,3) := S(N3,ll)

30 CONTINllf

c ----- SOLVE roR UNKNOWNS CALL BANSOL(NEQ)

c c ----------

c c

DO 3'1 I :-: 1 , N2 311 Pr ( I ) - r ( I)

PE(NN 2 ) = F(N1) RL rtJHN END FUNG 1 I ON SPR( MAT, ON, N, PR)

C PURPOSE: TO CALCULATE lllE SO I L-HYIDRAULI C f'ROPERl I ES c

DIMENSION RN(15) DATA CONOM/1.E-08/,SS/1.E-07/ K=MATl

44

Virginia's Agricultural Experiment Stations

1 -- Blacksburg Virginia Tech Main Station

2 -- Steeles Tavern Shenandoah Valley Research Station Beef, Sheep, Fruit, Forages, Insects

3 -- Orange Piedmont Research Station Small Grains, Corn, Alfalfa, Crops

4 -- Winchester Winchester Fruit Research Laboratory Fruit, Insect Control

5 -- Middleburg Virginia Forage Research Station Forages, Beef

6 -- Warsaw Eastern Virginia Research Station Field Crops

7 -- Suffolk Tidewater Research and Continuing Education Center Peanuts, Swine, Soybeans, Corn, Small Grains

8 -- Blackstone Southern Piedmont Research and Continuing Education Center Tobacco, Horticulture Crops, Turfgrass, Small Grains, Forages

9 -- Critz Reynolds Homestead Research Center Forestry, Wildlife

10 -- Glade Spring Southwest Virginia Research· Station Burley Tobacco, Beef, Sheep

11 -- Hampton Seafood Processing Research

and Extension Unit Seafood

I• ,·"---

image36865image36866image36867image36868image36869image36870image36871image36872image36873image36874image36875image36876image36877image36878image36879image36880image36881image36882image36883image36884image36885image36886image36887image36888image36889image36890image36891image36892image36893image36894image36895image36896image36897image36898image36899image36900image36901image36902image36903image36904image36905image36906image36907image36908image36909image36910image36911image36912image36913image36914image36915image36916image36917image36918image36919image36920