·,

C0.. \":\.J · .... ~

CALCULATING THE UNSATURATED HYDRAULIC CONDUCTIVITY WITH A Ni'.:W CLOSED-FORM

ANALYTICAL MODEL

by

Rien van Genuchten * Water Resources Program

Department of Civil Engineering Princeton University

Princeton, New Jersey 08540

PROPERTY WMT LIBRARY

*Present address: U. S. Salinity Laboratory, USDA, 4500 Glenwood Drive, . Riverside, California 92501.

78-WR-08

September 1978

mation Only

. ' .

..

ABSTRACT

A new and relatively simple equation for the soil moisture content-

pressure head curve, e (h) , is described in this report. The particular -

form of the equation enables one to derive closed-fonD analytical expres-

sions for the relative hydraulic conductivity, X , when substituted in r

the predictive conductivity models of Burdine (1953) or MUalem (1976a).

The resulting expressions for K (h) contain three independent parameters r

which may be obtained by fitting the proposed soil moisture retention

model to experimental data. Two different methods of curve-fitting are

discussed in the report, a simple but effective graphical method, and a

least-squares method requiring computer assistance. An existing non-

linear least-squares curve-fitting program was modified for this purpose

and is included in an appendix, together with detailed instructions

regarding its use.

Results obtained with the closed form analytical expressions based

on the Mualem theory were compared with observed relative hydraulic

conductivity data for five soils with a wide range in hydraulic prop-

erties. The relative hydraulic conductivity was predicted well in

four out of five cases. It was found that a reasonable description

of the soil moisture retention curve at low moisture contents is

necessary if an accurate prediction of the hydraulic conductivity is to

be made.

ii

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ACKNOWLEDGEMENT

This work ~as supported, in part, by funds obtained from the

·. Solid and Hazardous Waste Research. Division, u. s. Environmental Protec-

tion Aqency, MUnicipal Environmental Research Laboratory, Cincin~ti,

Ohio (EPA Grant No. R803827-0l) •

·: ....... .. ·. ·.-\s ,

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TABLE OF CONTENTS

-. ABSTRACT ••••• . . . . . . . . . . . . . . . u.

ACKNOWLEDGEMENT . . . . . . . . . . . . . . . •. . . . . . . iii

LIST OF FIGURES . . . . . . . . . . . . . . . . v

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . vii

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . 1

MATHEMATICAL DEVELOPMENT • . . . . ' . . 4

PARAMETER ESTIMATION •• . . . . . . . • 16 ·

INFLUENCE OF THE RESIDUAL MOIS'l'URE CONTENT • . . . . . • 26

RESULTS . . . . . . 31

REFERENCES . . . . . . . . . . . . . . • 42

APPENDIX A . . . . . . . . . • 44

•.

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...

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. : :··.:

LIST OF FIGURES

Fig. l. Typical plot of the soil moisture retention curve based on Eq.

(3)-

Fig. 2. Plot of the relative hydraulic conductivity versus press~e

head as predicted from knowledge of the soil moisture reten-

tion curve shown in Fig. l.

Fig. 3. Plot of the soil moisture diffusivity versus moisture content

as predicted from knowledge of the soil moisture retention

curve shown in Fig. l, and the hydraulic conductivity at satu-

ration.

Fig. 4. Comparison of the proposed soil hydraulic functions (solid

lines) with curves obtained by applying either the Mualem

theory (M: dashed lines) or the Burdine theory (B: dashed­

dotted lines) to the Brooks and Corey model of the soil mois-­

ture retention curve.

Fig. S. Plot showing the location of the points P, Q, and Ron the

soil moisture retention curve. The point P is situated half­

way between e (•0.10) and e (•0.50), the point Q represents r s

Fig. 6.

Fig. 7.

the inflection point of the curve (semilogarithmic plot), while

R represents the inflection point if the curve were plotted on

a normal (9 versus h) scale.

Plots of the dimensionless slopes SP and s2 as functions of

the parameters nand m (•1-l/n~-

Plot illustrating the graphical determination of the para­

meters a and n for three different values of the residual

moistur~ content: ea •0.05 (curve a), e~ •0.10 (curve b), c 3 3 and 9r cO.lS em /em (curve c). The open circles represent

the observed soil moisture retention curve of Silt Loam

G.E.3 (Reisenauer, 1963) •

v

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..

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: :-· ...

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Fig. 8.

Fiq. 9.

Comparison of observed (open circles) and calculated curves

(solid lines) of the relative hydraulic conductivity of Silt

Loam G.E.3. The predicted curves were obtained for three

different values of the residual moisture content, 9 : 0.05 3 3 r

(curve a), 0.10 (curve b), and O.lS em /em (curve c).

Observed (open circles) and calculated curves (solid lines)

of the soil hydraulic properties of Hyqiene Sandstone. The

relative hydraulic conductivity was predicted from knowledge

of the curve-fitted soil moisture retention curve.

Fiq. 10. Observed (open circles) and calculated curves (solid lines) of

the soil hydraulic pro:Perties of Touchet Silt Loam G.E.3. The

relative .-hydraulic conductivity was predicted from knowledge of

the curve-fitted soil moisture retention curve.

Fig. ll. Observed (open circles) and calculated curves (solid lines) of

the soil hydraulic properties of Silt Loam G.E.3. The relative

hydraulic conductivity was predicted from knowledge of the

curve-fitted soil moisture retention curve.

Fig. 12. Observed (open circles) and calculated curves (solid lines) of

the soil hydraulic properti~s of Beit Netofa Clay. The relative

· hydraulic conductivity was predicted from knowledge of the curve­

fitted soil moisture retention eurve.

Fig. 13. Observed (open circles) and calculated curves (solid lines) of

the soil hydraulic properties of Beit Netofa Clay. The relative

hydraulic conductivity was predicted from knowledge of the curve•

fitted soil moisture retention curve. The last four data points

of the observed soil moisture retention curve were not considered

in the fitting process •.

Fig. 14. Observed (circles) and calculated curves (solid lines) of the

soil hydraulic properties of Guelph Loam. The drying and wetting

branches of the relative hydraulic conductivity curve were pre­

dicted from knowledge of the curve-fitted branches of the soil

moisture retention curve.

vi

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LIST OF TABLES

Table 1. Calculation of the parameters a and n from the observed soil

moisture retention curve of Silt Loam G.E.3, using three dif­

ferent. values fore ce •0.396). r s

Table 2. Soil-physical properties of the five example soils.

Table Al. List of the most significant variables in SOHYP.

Table A2. Input data instructions for SOHYP.

Table A3. Data input for example 3 (Silt Loam G.E.3).

Table A4. Output for example 3 (Silt Loam G.E.3).

Table AS. Fortran listing of SOHYP.

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INTRODUCTION

The use of numerical models for simulating fluid flew and mass

·. transport in the unsaturated zone has became increasingly popular the last

few years. Recent literature indeed demonstrates that much effort is put

into the development of such models using both finite difference (Bresler,

l975J Amerman, 1976) and finite element techniques (Reeves and Duguid, 1975:

C'· ~t~}:

Segel, 1976). Unfortunately, it appears that the ability to fully charac- ·

terize the simulated system has not kept pace with the numerical and model-

ing expertise. Probably the single most important factor limiting the

successful application of unsaturated flow theory to actual field problems

is the lack of information regarding the parameters entering the governing

transport equations.· Reliable estimates of the unsaturated hydraulic

conductivity are especially difficult to obtain, partly because of its

extensive variability in the. field, and partly because measuring this

parameter is time-consuming and expensive. Several investigators have,

for these reasons, used models for calculating the unsaturated conductivity

from the more easily measured soil moisture retention curve. Very popular

among these models has been the Millington-Quirk method (Millington and

Quirk, 1961), various forms of which have been applied with some success

in a number of studies (cf. Jackson et al., 1965; Jackson, 1972: Green and

Corey, 19717 Bruce, 1972). Unfortunately, this method also has the dis-

advantage of producing tabular results which, for example when applied to

nonhomogeneous soils in multi-dimensional unsaturated flow models, are

quite tedious to use • . . Closed-form analytical expressions for predicting the unsaturated

• hydraulic conductivity have also been developed. For example, Brooks and

l

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Cor~y (1964) and Jeppsun (1974) each used an analytical expression for

. the conductivity based on the Burdine theory (B~ine, 1953). Brooks

and Corey (1964, 19~6) obtained fairly accurate predictions with their

equations, even though a discontinuity is present in the slope of both

the moisture retention curve and the unsaturated hydraulic conductivity

curve at some negative value of the pressure head (this point is often

referred to as the bubbling pressure}. Such a discontinuity sometimes

prevents rapid convergence in numerical saturated-unsaturated flow pro~

lems. It also appears that predictions based on the Brooks and Corey

equations are somewhat less accurate than those obtained with various

forms of the (modified) Millington-Quirk method.

Recently Mualem (l976a} derived a new model for predicting the hydrau-

lie conductivity from knowledge of the soil moisture retention curve and the

conductivity at saturation. Mualem's derivation leads to a simple inte-

gral formula for the unsaturated hydraulic conductivity which enables

one to derive closed-form analytical expressions, provided suitable

equations for the soil moisture retention curves are available. It is

the purpose of this report to derive such closed-form analytical expres-

sions. The theories of both Mualem and Burdine are used for this deriva-

tion. The resulting conductivity models generally contain three indepen-

dent parameters which may be obtained from the soil moisture retention

data by means of curve-fitting. Two different methods of curve-fitting

are discussed in this paper, a simple graphical method which enables one

to obtain the parameters without requiring computer assistance, and a

more elaborate non-linear least-squares curve-fitting method requiring

the assistance of a digital computer. An existing computer model was

modified for this purpose and is included in the appendix. Results

2

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''

obtained with the closed-form equations based on the Mualem theory are

compared with observed data for a few soils having widely varying hydrau-

lie properties.

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MATHEMATICAL DEVELOPMENT

The following equation was derived by Mualem (l976a) for predict-

ing the relative hydraulic conductivity (Kr) from knowledge of the soil

moisture retention curve

2

(l)

where bah(®) is the pressure head, given here as a function of the dimen-

sionless moisture content:

e • e ~

r e -e •

s r (2)

In this equation, s and r indicate saturated and residual values of the

soil moisture content (9), respectively. To solve (l), an expression

relating the dimensionless moisture content to the pressure head is needed.

An attractive class of ®(h)-functions, adopted in this study, is given by

the following general equation

(3)

where a, n and m are as yet undetermined parameters. To simplify notation

later, h in (3) is assumed to be positive. Equl'ltion (3) with m-l has

been successfully used in many studies to describe soil moisture re-

tention data (Ahuja and Schwartzendruber, 1972; Endelman et aZ., 1974;

4

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'•

·.

Haverkamp et aZ., 1977). A typical 8(h)-curve based on Eq. (2) and

(3} is shor.im in Fig. 1. Note that a nearly symmetrical "S"-shaped curve

is obtained, and that the slope (d9/dh) becomes zero when the moisture

content approaches both its saturated and residual values.

Simple, closed-form expressions for K (9) can be derived when cer­r

tain restrictions are imposed upon the values of m and n allowed in (3).

Solving this equation for h•h(9) and substituting the resulting expres-

sion into (1) gives

K <9> • e., r [

f(9}] f(l}

where f(9) is given by

2

f(9) • J9 [ xl/m J 1/n dx.

0 1-xl/m

m Substitution of x=y into (5} leads to

. 91/m

f(9) = m JO Ym-1+1/n (1-y)-1/n dy.

(4)

(5)

(6}

Equation (6) represents a particular form of the Incomplete Beta-function

(see for example Abramowitz and Stegun, 1970i p. 944} and, in its most

general case, no closed-form expression can be derived. However, it is

easily shown that for integer values of k~l+l/n the integration can

be carried out without difficulties. For the particular case when k=O

(i.e. m-1-1/n} integration of (6) yields

m f(9) = l-(l-9l/m)

5

(m=l-1/n)

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~. -..

G>

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. . . . . . . .

-E u -.I:: •

Q ct

"' X

&aJ a: :::;) en en

"" a: CL

• · -lOr-----~----~------~----~----~----~

-10

4 -10

! -10

2 -10

I -10

0

8,. 0.10

a•0.005 (1/cm)

ft. 2.0

Iii • 0.5

8,• o.5o

•

.1 .2 .3 .4 .5 -10~----~----_.------~----~----~----~ .6 0

MOISTURE CONTENT, 8 (cm7cm3)

Fig. 1. Typical plot of the soil moisture retention curve based on

Eq. {3) •

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'.

·-

and, because f(l) a 1, (4) becomes

2

Kr (9) • 9J:j [ 1- (1-Ell/m, m]

(m=l-1/n)

(O<m<l) (8)

The relative hydraulic conductivity may also be expressed in terms of

the pressure head by substituting (3) into (8), i.e.

2 -m {1-(ah)n-l [l+(ah}n] }

[l+(ah)n] m/2 (mcl-1/n) (9)

From the hydraulic conductivity and the soil moisture retention curve

one may also derive an expression for the soil moisture diffusivity,

which is defined as

0(8)= K(8) 1:1 . (10)

This leads to the following equation for D(S):

D(S) = (1-m)K -m m

-..,.-_,.;;.s_ eJ:I-1/m [ {1-Ell/m, + (1-Ell/m) •2] amce -e > s r

{11)

where K is the hydraulic conductivity at saturation. Equations (9) and s

{11) are shown graphically in Fig. 2 and 3, respectively, using the

same values of a, n and m(ml-1/n) as in Fig. 1. AS ean be seen from

Fig. 2, the relative hydraulic conductivity starts out with a slope of

zero at pressure head values near zero, but then falls off increasingly

rapid as h decreases. The soil moisture diffusivity, on the other hand,

attains (as does the soil moisture retention curve) a fairly symmetrical

"S"-shaped curve with infinite gradients, d(log D)/d9, when 8 approach-

7

Information Onl .. ·~ ; . .- ·... . . . . . . . . ··::-;

. '·

·.

.. >-1--> 1-(.) :J c z 0 (.)

-

IU > l­et ..J LLI a:

-z 10

OC = 0.005 (1/cm) n =2.0

·6 10 ~------~------------------------~ -ao 0 -10 1 -10 2 -1o 3 -10

4

PRESSURE HEAD , h (em)

Fig. 2. Plot of the relative hydraulic conductivity versus

pressure head as predicted from knowledge of the soil

~oisture retention curve shown in Fig. 1.

8

Information Only . - . . . . . .... •.... . ... ~-·· ' . . . ' ..... ' . ~-. . ,. ..... ,. ··.~·.·· ~ ... •· •.···· .. ·--··-···-··: ... • .···.-.-~--~

~~:-·.

(:;=/}::: '~2.._:::-

.. -: ~: : : .. t:::·:.::. -~

·.

106 ~------------------------------~----~

_..... ~ a 4 "a 10

' .. e u -0 10'3 .. Ks • 100 (em/day)

>- 0.005 ( I I em) t- a • -> 10

2 n • 2.0 en ::l I&. LL. -Q

101

Br..........._ Bs,. 10°~----~----~------~----~------~----~

0 .I .2 .3 .4 .5 .6

MOl STURE CONTENT I e ( cm3 I cm3)

Fig. 3. Plot of the soil moisture diffusivity versus moisture content

as predicted from knowledge of the soil moisture retention

curve shown in Fig. 1, and the hydraulic conductivity at satu-

ration.

9

Information Only •••• • • •• : • ' .... \ 0 --~-. .. • • •• -:-

.... _ '

·-

(10.

es either 9 or 9 • Note that the diffusivity becomes infinite when 9 ap­r s

·preaches e . Only at intermediate values of the moisture content (approxi­s

' mately between 9a0.25 .and 9•0.45 in Fig. 3) does the diffusivity acquire

the often assumed exponential dependency on the moisture content.

Similar features of the soil moisture diffusi vi ty were obtained and

discussed by Ahuja and Schwartzendruber (1972), using the following

special form of 0(9):

a eP 0(9) .. _.....;:;;...;;.... __

(9 -6) q s

where a, p and q are material characteristic parameters.

(12)

The soil hydraulic properties derived above were obtained by assuming

that k~l+l/n•O in (6). One may also derive closed-form expressions

for other integer values of k. For k•l, for example, the conductivity

becomes

2

K (9) = 9~ (1-m(l~l/D) m-l +(m-1) (1-E>l/D) m). r

(m=2-l/n) (13)

While this particular model is not only more complicated than model {8) ,

it also represents only a slight pertubation of the earlier function.

Hence, (13) does not present an attractive alternative for (8), and will

not be discussed further.

Similar results as above for the Mualem -theory may also be obtained

when the Burdine theory is taken as a point of departure. The equation

given by Burdine {1953) is:

1 --- dx. · h

2 (X)

(14)

10

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:.:he analysis proceeds in a similar way as before. Equation (3) is invert-

ed to give h=h (9) and substitution of the resulting expression into (14)

;(J:~:: yields

{15)

where

l [ l/ ]2/n f(9) a X l/: dx

0 1-x

(16)

Substituting x-ym into (16} gives -

1/m f(9) • m r: Ym-1+2/nll-y)-2/n dy. (17)

(8·:: Again it is asswned that the exponent of y in (17) vanishes. Bence

m•l-2/n, and (17) reduces to

·-

m f(9) • 1 - (l-9l/m) •

The relative hydraulic conductivity hence becomes

or in terms of the pressure head

K (h) = r

1-(ah)n-2 [1+(ah)n] 2m

[1+(ah)n]

-m

11

Information ......... . . ··· . ... : . . ·

(18)

(JD1111-2/n) (19)

(O<m<l; n>2)

(20)

nly

,· ' ' The soil moisture diffusivity for this case is given by

(1-m)Ks (3-1/m)/2 . l/m •(m+1)/2 l/m (~l)/2

oce> - __ _..;;.._ e [ (1-e > -(1-e J. 2am(9 -e )

s r (21)

Preliminary tests indicated that (8) generated results that we~e, in

most cases, in better agreement with experimental data than (19). Through

an extensive series of comparisons, also Mualem (l976a) concluded that pre-

dictions based on his theory (i.e., based directly on Eq. (1) by ·means of

numerical approximations) were generally more accurate than those based

on various forms of the Burdine theory (including the Millington-Quirk

method). It is not the intent of this pape2: to give accuracy comparisons

between various closed-form analytical conductivity expressions. Only a

brief discussion of the equations derived by Brooks and Corey (1964) will

be given here, since their model of the soil moisture retention curve

@: represents a limiting case of the moisture retention model discussed in

this study.

·.

Brooks and Corey {1964; .1966) concluded from comparisons with a large

number of experimental data that the soil moisture retention curve 9 {h)

could be described reasonably well with the following general equation

-A e- Ch~>

(22)

where ~ is the bubbling pressure (approximately equal to the air entry

value), L~d A a soil characteristic parameter. Comparing Eq. (22) and

(3), one sees that (3) reduces to (22) for large values of the pres-

sure head, i.e.

12

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-mn e ... (ah> {23)

For the Mua1em theory one has DPl-1/n, and hence ).o::n-1, while for the

Burdine theory {m-1-2/n) one finds that >.•n-2. The parameter a, further-

more, is inversely related to the bubbl.ing pressure, ~- Brooks and Corey

used the Burdine theory to predict the relative hydraulic conductiv.ity and

.the soil moisture diffusivity. They derived the following expressions

-2-3>. Kr (h) ., {ah)

Ks 2+1/A o ce > = -_.;;;..,-- e

a>. <e -e > s r

{24a)

(24b)

{25)

Through substitution of (22) into (1), similar equations can be obtained

@: when the Mualem theory is used:

. -

5/2+2/>. K ce> = e r

-2-5>./2 Kr(h) • {ah)

D(El) • K _ ...... s;,._._-e

a>.ce -e ) s r

(26a)

(26b)

3/2+1/A (27)

Figure 4 compares the different expressions given above with the earlier

obtained relations for the conductivity and the diffusivity [Eq. (3),

(9), and (ll)]. The parameters a and n were chosen to be the same as

before (i.e., a=O.OOS and n•2), while A was assumed to be equal to (n-1).

l3

Information Only .. ·.·.·.·.. . .. · .. · .. . ···.··-·· .-. -.· .. ·-... _ -·.-··;"'":·";'

~ = ~ ~

e = ~ •• = = 0 = ~

..... ,..

·. . . (8•:. . . c;y:, ·· .. ::.

-e U· -

.&.

5 •10 I I I I I I I

a

-·~ -·03 B.M

l ....... _ 2 '\1 -•o a • 0.005

n • 2.0 K1 •100 .k•l I J -•o' L

I f •100 I I I

0 I I

.I .2 .3 .4 .5

9 (emS/em!)

cs.:. ... :: . ~-: 0 •••

I &

101~--~--~~--,-~ b

101

~ 10° 0 "0

....... e u -· B -·---·-- 10 ~ M----

-tL 10 \\ -1

I0-1 • I -too . , - I !' I

h (em)

fTh' .. ' l: .... : ~·:: .'· \'• .... ....... ,

.•:'

to•. . I I I

c

101

~:}}.)

I I i

-; 104 0 "0 ....... N e u -0 I Ill M----

101

IO'· I II I I I I I

0 .I .2 .3 .4 .5

9 (em1/em1)

Fig. 4. Comparison of the proposed soil hydraulic functions (solid lines) with curves obtained by

applying either the Mualem theory (MJ dashed lines) or the Burdine theory (BI dashed-dotted

lines) to the Brooks and Corey model of the soil moisture retention curve.

·. '

(}:}~ ... __ ... ~

~e soil moisture retention curves for all three cases become then identi-

cal for sufficiently low values of the moisture content. Figure 4a shows

that the Brooks and Corey model of the 6(h)-curve approaches the curve

based on (3) asymptotically when 9 decreases. However, large deviations

between -the two models occur then e approaches its saturated, value. '!he

curves based on (22) reach e at a much lower value of h, i.e~ at -200 em s

(h•~•l/a). The most important deviations between the predicted conduc­

tivity curves are also present at or near the bubbling pressure (Fig. 4b).

As expected, the curves based on Eq. (9) and (26b) (the solid and dashed

lines, respectively) approach each other asymptotically when h becomes

increasingly negative, while the curve used by Brooks and Corey (the

dashed-dotted line) remains somewhat separated from the other two because

of the different exponent in the conductivity equation · (see Eq. (24b) and

(26b)]. The diffusivity curves (Fig. 4c) show their most important

differences at both the intermediate and higher values of the moisture

CL: content. Note that the diffusivity curves based on (22) remain finite

2 (Ds•SO,OOO em /day) when e approaches es, while the solid line (Eq. 10)

goes to infinity at saturation. It should be emphasized that Fig. 4 was

included only to demonstrate typical properties of the various conductiv-

ity and diffusivity models, and that the figure should not be viewed as

an accuracy evaluation of any one model.

·-

15

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. '·

·.

PARAMETER ESTIMATION

The soil moisture content (8) as a function of the pressure head

(h) is given by Eq. (2) and (3), i.e.,

(8 -e ) e s r

- 8r + ----~~---m-. (28)

[1 + (ah.)n]

where, as before, it is understood that h is positive, and where for the

Mualem model

m •1-l/n. (29)

Equation (28) contains four independent parameters (8 , e , a, and n) 1 r s

which have to be estimated from observed soil moisture retention datae

Of these four, the saturated moisture content {8 ) is probably always s

available as it is easily obtained experimentally. Also the residual

moisture content (8r) may be measured experimentally 1 for example by de­

termining the moisture content on very dry soil. Unfortunately, er­

measurements are not always made routinely, and hence have to be estimated

by extrapolating existing soil moisture retention data. Assuming for the

moment that sufficiently accurate estimates of both e and e are avail-r s

able 1

the following procedure can then be used to obtair. ~stimates of the

remaining parameters a and n.

Differentiation of (28) gives

-am(e -e ) m _de_ - __ ,_,.;;s;_...r.;;._ el/m(l-91/m) dh 1-m

(30)

16

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where the right-hand side is expressed in tenns of 91 rather than h. The

pressure head may also be expressed in terms of the_moisture content by

inverting (3) 1 i.e. 1 _

-l/m 1/n h - .!. (9 . -l)

a

Elimination of a from (29) and (31) results in

h d6 db - -m<e -e > s r

1-m

(31)

(32)

The right-hand side 'of this equation contains only the unknown parameter

m (both 6 and 6 are assumed to be known) • Hence it is possible to ob­s r

tain estimates of m by determining the product of the slope (dB/db) and

the pressure head (h) at some point on the e (h) -curve. Soil moisture re-

tention data are often plotted on a semi-logarithmic scale. One may take

advantage of this fact by noting that

d8 d(log h) = (ln 10) h dS

db (33)

LetS be the absolute value of the slope of 9 with respect to log h, i.e.,

(34a)

or, equivalently,

l (34b) s .. <e -e > s r

17

Information Only

Combininq (32), (33), and (34b) leads to the following expression for s

l/m m s ... 2.303 r:m· e c1-e >· (35)

The best location on the e {h) curve for evaluating the slope s is about

halfway between e and e • Let p be the point on the soil moisture re­r s

tention curve for which e~ (see Fiq. 5). From Eq. {2) and (31) it

follows then that the coordinates of P are qiven by

while Eq. (35) reduces to

-1/m {l-2 . ) •

{36a)

(36b)

(37a)

The subscript P in these equations is used to indicate evaluation at P.

Equation (37a} can also be expressed in terms of n

n/(1-n} SP(n) • 1.151 (n-1) (l-2 ). (37b)

Figure 6 gives a plot of SP as a function of both n and m. This figure

may be used to obtain an estimate of n once the slope SP is determined

qraphically from the experimental data. For relatively large values of

n, (37b) is closely approximated by

18

Information Only .. ··-· · ... ·. · .. . · .. ··:·-····.··. •: .... ·:-·

-·~-.. ~~//)

Gl>

@:,

•.

. (£;-..

•J06r-----~----,------r----~----~------~

!>J05

a • 0.005

.,..._ •10

4 E n• 2.0 u --.z: •

0 ct LrJ :z:

0 1&.1 CD

a:: ·::l -ao2 fl) Cl) LtJ 0.622 a: a....

·ao1

•10°~----~----~------~-----L------L-----~ 0 . I .2 .3 .4 .s .6

MOISTURE CONTENT, 8 (cm3jcm3)

Fig. 5. Plot showing the location of the points P, Q, and R on the

soil moisture retention curve. The point P is situated half­

way between er (•0.10) and es (=0.50), the point Q represents

the inflection point of the curve (semilogarithmic plot), while

R represents the inflection point if the curve were plotted on

a normal (8 versus h) scale.

Inforn 19

ion Only

vi:/· . ·

.u -~" 3

I -

~ s ~ = 2

~ ., a ~ "" ' ~

0

•• Q . I

= 0 = ..... ~

0 v I

' 0 Fig. 6.

@:?I

.60 .65 .-....... ·--

~Q

I I I I I

2

@);·. ..

.70 .75

-m__..

3 4

.~,. f" ..... ~I '(:.::·:·:~:::' .

.80

5 -n-+·

Plots of the dimensionless slopes SP and SQ as functions of 'the parameters n and

m (=1-1/n).

/.0.[0 ...... !• ••• •••.

\t·\:.::·.;

Sp(n) • 0.5769 n - 0.2ll (n>4) (37c)

from which one obtains ·

n • 1.733 sp + 0.37. (n>4) (38)

Alternatively, n can also be obtained from (37b) itself by rearranging the

equation into the following iterative scheme:

n/(1-n) n • l + 0.869 SP/(l-2 ) • (39)

The iterative solution converges rapidly. Even for a wild initial guess of

n generally only two or three iterations are necessary to obta;n answers

correct to within l\. Once n (or m) is determined, a can be evaluated

with (36b).

An alternative approach for estimating n and a from experimental data

follows by considering the inflection point on the 9 versus log h curve

(the point marked "Q" in Fig. 5). Here one has

2 d(log h) - 0 • (40)

(Jl)J calculation of the inflection point is greatly simplified by noting that

2 d(log h) - 2 (ln 10) 2 (h2 d ~ + h : ) •

dh

(41)

•• !t is easily verified that substitution of (3) into (40) and subsequent

expansion leads to . (]):

21

Information Only

·'·

·.

Hence, the coordinates of the inflection point are

m

eQ • 8r+(e s -er) [l:m]

h • !. m-l Q a m

(42)

(43a)

(43b)

From (43a) it follows that, at least theoretically, one could estimate

the value of m directly by locating the inflection point on the soil

moisture retention curve. However, from Fig. 5 it is clear that it is not

easy to determine this point accurately (even less so when the curve is

based on experimental data). It seems, therefore, better to again esti-

mate m from the slope of the curve. Substitution of {42) into (35) gives

2.303 l-m

or, in terms of n,

[l:m]

m+l

2-l/n

s2 <n> • 2.303 n [ 2~:i]

(44a)

(44b)

Fiqure 6 shows that SP(n) and SQ(n) define approximately the same curve,

especially for the larger n-values. This is not surprising since the

points p and Q are generally very close together on the soil moisture re-

tention curve. Fig. 5, furthermore, shows that both points define approxi-

mately the same gradient. Hence the n-values obtained from the sketched

22

Information Only

slope should be nearly identical.

Instead of usinq the qraphical procedure of Fig. 6, it is also possible

(ij? to obtain n as a function of s2

by iteratively solvinq Eq. (44b) itself.

. c{fjjj

· .

. ClL··

The followinq converging scheme was used for that purpose:

3 n•-+ 4 1 . 1

(n - 2) A + 4 (l-2A) I A• 2.303 n . (45)

As an illustrative example, the foregoing procedure was applied to

the curve shown in Fig. 5. Assuming the indicated slope to be the same

for both points, P and Q, one obtains for SP and SQ (E)q. 34b) :

s • s • -="='_o;:::...·::-:6;..;2:.:2~~ p Q (0.40) (1.8}

- 0.864 •

From Fiq. 6, or Eq. (39) and (45), it then follows that np • 2.00 and n2 c

1.96. Hence from (20) one finds ~ • 0.50 and m2 = 0.49. From Fig. 5

it follows that log(~) • 2.54 and loq(hQ) • 2.43. Finally, from Eq.

(36b) one obtains

1 1/m ap = ~ (2 -1)

ar.d from (43b)

1-m

23

Inform tion Only

. ·,

• lo-2 •43 co.49)-o. 51 • 0.0053.

The relative hydraUlic conductivities hence are (Eq. 8) ·:

2 ~ [ 2.00 o.so]

Kr{9) • 9 1-{1-9 ) (46a)

2 ~ [ 2. 04 . 0. 49 ]

Kr(9) • 9 1-(1-9 . ) (46b) .

Equation (46a) exactly reproduces the conductivity equation one would have

obtained if the original data shown in Fig. 5 were used in Eq. (8). Equa-

tions (46a) and (46b) generate nearly the same curve when plotted versus

or versus h. Minor differences between the curves occur only at the extreme

@ dry side of the curves, and are caused by the fact that the same slope was

used to calculate both SP and SQ (in reality, SQ should have been measured

somewhat larger than SP) •

The parameters a and n can also be estimated from soil moisture

retention data which are plotted on a normal e versus h scale. The pro-

cedure for finding the two parameters is similar to that used before.

Equation (37) still holds provided, however, that S is calculated with Eq.

([); (33) and (34). These two equations show that now estimates of both h

and the slope, d9/dh, are necessary for evaluating S. Equations (43)

and (44), on the other hand, have to be modified because the inflection

point of the 9(h)-curve does not coincide with the inflection point of the

·. 9(log h) curve. Contrary to (40), one has now

: .... :-\i> 24

Information Only .. • ... ••;·······

. .

•.

(47}

Expansion of (47) yields the followinq coordinates of the inflection point

on the 8 (h) -curve (this point is marked "R" on the 8 (log h) -curve in Fig •

(5) •

l l-m h • -m R a

-m

and (35) becomes

- (l+m)

SR(m) • 2.303 m(l+m}

25

lnformati n Only

(48a}

(48b)

(49)

• • • • • ~ • - 4 .-. ••• •

INFLUENCE OF THE RESIDUAL MOISTURE CONTENT

The foregoing discussion assumes that independent measurements

o~ the saturated and residual moisture contents are available. While

e is usually easY to obtain by direct measurelllent, er is often much s '

more difficult to quantify. In fact, in many cases e may become a'n r .

ill-defined parameter. The residual moisture content in this report

is defined as the moisture content for which the gradient (dB/dh) ·becomes

zero (excluding the region near e s which has also a zero gradient). Also

the hydraulic conductivity will approach zero when 8 approaches er. From

a practical point of view it seems sufficient to define e as the moisture r

-6 content at some large negative value of the pressure head, e.g., at -10

em. Even in that case, however 1 significant decreases in h are likely to

result in further desorption of moisture. It seems that such further

changes in e are fairly unimportant for most practical field problems.

In fact, they would be inconsistent with the general shape of the e (h)-

curve defined by (22) 1 and probably invalidate the concept of a residual

moisture content itself. A reasonable estimate of er is necessary for

an accurate prediction of the hydraulic conductivity, even though its in-

fluence on the predictions is generally less than that of a and n. The

following example problem demonstrates the effect of e on the conductivity r

(l) predictions.

..

\8·'

Figure 7a shows the soil moisture retention curve of Silt Loam

-3 G.E.J, for values of h between zero and 10 em. (~isenauer, 1963).

The open circles represent data po~ts of the curve, and were taken from

the catalogue of Mualem (1976b). Because only a limited portion of the

the curve is defined, an accurate estimate of e is not easy to obtain. r

26

Information Only

~ = ~ ~ e = ..... = = 0 = ~

6··:. ·::-: .

:_:_..:·

-.s::.

-C)

0 "' . .J -..J

!5

4

3

2

0

. . @'·\ . . . . ~ ~: _; C)J~.

,__---,r----.----,.--..,--- 5

Ba Bb Be r r r a I t t 4

\P ,...., \ i ·,·,. \ i t '·,~n._

"-a.. 'a., b'·,. 2

I l. 0.4~---·-·-·~·-:"'!.. . ·-·- ~ L ·-·- I ·- ~

3

1.8

~ l

. . (~::·· ,:- ··i· :_.:::·.=:::·

a b IC b

c

tt::.:::\ ~::: .. \::_:_:)

Bs•0.3\: 1 1 1 1 fJ o' I I I I I

0 .I .2 .3 .4 0 .I .2 o3 .4

-9---- -8~

Fig. 7. Plot illustrating the graphical determination of the parameters a and n for three different values a b c 3 3

of the residual moisture content: a =0.05 (curve a), a =0.10 (curve b), and a =0.15 em /em r r r (curve c). The open circles represent the observed soil moisture retention curve of.Silt Loam

G.E.3 (Reisenauer, 1963).

7

..

·.

':L1u:~a ~:1iZ:Z~.ren~ Vl\!U:!S for 9r were chosen rather arbitrarily (0.05, 0.10,

and 0.15 cm3 /cm

3, respectively), and subsequently used to calculate the

hydraulic conductivity. The calculations, based on Eq. (36) and (37),

are' summarized in 'l'able l. 'l'he llope of the e (log h) -curve at e-~

was assumed to be ~e same for all three cases (step 6 in Table l), a

sufficiently accurate assumption in this case. Figure 7b compares Che

calculated retention curves with the experimental curve. Each of the

Table l. Calculation of the parameters a and n from the observed soil

moisture retention curve of Silt Loam G.E.J, using three dif­

ferent values for e ce •0.396) r s

STEP ea r

eb r

ec r

l. Estimate 8 0.050 0.100 0.150 r

2. Obtain (9 -e > 0.346 0.296 0.246 s r

3. Calculate 8p•(8 +8 )/2. s r

0.223 0.248 0.273

4. Obtain log(~) from data (Fig. 7a) 2.76 2.65 2.55

s. Calculate ~ 575. 447. 355.

6. Estimate d8/d(log h) at eP 0.244 0.244 0.244

(Fig. 7a) (•0.44/1.8)

7. Calculate sp [•0.244/(9 -9 >] 0.706 0.826 ' 0.994

(Eq. 34b) 5 r

a. Obtain n from Fig. 6 or 1.77 1.95 2.21

Eq. (39)

9. Calculate m (•1-l/n) 0.435 0.487 0.548

10. Calculate a (Eq. 36b) 0.0038 0.0040 0.0043

three curves describes the experimental curve fairly accurately, although

c curve c (based on er) fits the data points somewhat better at the dry

end of the curve than the other two. On the other hand, this curve also

28

Information Only

slightly overpredicts i:Jla observed (:'I'!Ye at the higher moisture contents,

i.e. near h•-100 em. The predicted conductivity curves are presented in

( }:}. Fig. e. Again, all three curves give a reasonable description of the ' '-:.:..,:..· ~

· .

. ·. ··.· .

'<ilL·

experimen.tal points. The higher conductivity values are most accurately

described by curve b, while curve c is the most acc~ate one at the dry

side of the curve. However, it is clear that all three curves are·

acceptable, and hence that the influence of the residual moisture

content, at least for this particular example, is not that significant.

In the above example 8 · was selected beforehand in an arbitrary r

way, and still no clear procedure is available for obtaining a reasonabie

estimate of er from measured data, especially when only p~t of the 8(h}

curve is given. TO alleviate this problem, at least partially, a least-

squares curve-fitting technique was used to estimate the three parameters

er' a, and n directly from the observed data. An existing non-linear

least-squares curve-fitting program (Meeter, 1964) was modified and

adapted for this purpose. The program uses the maximum neighborhood

method of Marquardt (1964}, which is based on an optimum interpolation

between the Taylor series method and the method of steepest descent. A

detailed analysi~ of this technique is also given by Daniel and Wood

{1973). A listing of the computer program is given in Appendix A.

29

Information Only

--> ...

·U ~ 0 z 0 (,.)

10° ~-~~~;:---.-----,-, a

u 101

-..J ::::» < a: 0 >­X

LaJ > -... <t _. LaJ 0::

PRESSURE HEAD , h (em)

a b c

Fig. B. Co~arison of observed (open circles) and calculated curves

(solid lines) of the relative hydraulic conductivity of Silt

Loam G.E.3. The predicted curves were obtained for three

different values of the residual moisture content, e : 0.05 · . 3 3 r

(curve a), 0.10 (curve b), and 0.15 em /em (curve c).

30

Information Only ... - ., . .... . . . . - ' .. :""···· .. -.- .- .. : :··· -.. ~-"" .... -~· .......... ..,...

. (§}

CD

. -

RESULTS

In this section comparisons are given between observed and calcu-

lated conductivity curves for five soils. The examples were selected for

soils vi th widely different hydraulic properties. The observed data for

each example, with the exception of the last one, were taken from the

soils catalogue of Mualem (l976b). Table 2 summarizes same of the soil-

physical properties of the five soils. Estimates of the parameters e I r

a, and n are also inclu.ded in the table, and were obtained by fitting

Eq. (28) to the observed soil moisture retention data.

Results for Hygiene Sandstone (Brooks and Corey, 1964) are shown

in Fig. 9. This soil has a rather narrow pore-size distribution, causing

the soil moisture release curve to become very steep around h--125 em.

A relatively high value of 10.4 for n was Obtained for this soil, a direct

consequence of the steep curve. The value of a was found to be 0.079

(1/cm), approximately the inverse of the pressure head at which the soil

Tab~e 2. soil-physical properties of the five example soils.

e e K a n s r s

SOIL NAME 3 3 (em /em ) (cm3 /cm3) (em/day) (1/cm) (---)

Hygiene sandstone .250 .153 108.0 .0079 . 10.4

Touchet Silt Loam G.E.3 .469 .190 303.0 .00505 7.9

Silt Loam G.E.3 .396 .131 4.96 .00423 2.06

Guelph Loam (dxying) .520 .218 31.6 .0115 2.03

(wetting) ( .434) .218 .0200 2.76

Beit Netofa Clay .446 .286 • 082 .00202 1.59

31

Inti rmation Only . ,•. -·--· ·:-• .-... · ··;·• .~ .....

~ = ~ ., e = ~ ...... Q = 0 = ~

l.J t.J

0>-:· ' C·. .. ;_.\

'. · .. ·.· ,•

HYGIENE SANDSTONE

.251 0 0 0 Q

-., E ~ r ~~FITTED e ~ .20 CD

a • .0079 n • t0.4 B, • .153

(3·: .. ' . ·. ,• ::·.· .

-t01 ·101 ·10• h (em)

~'\_

(/});~ (\) ' .

to·~----~~~~J

K,

ao-•

PREDICTED-,

to-•

10-4

101 ·to• ·ao• h (em)

Fig. 9. Observed (open circles) and calculated curves (solid lines) of the soil hydraulic properties

of Hygiene Sandstone. The relative hydraulic conductivity was predicted from knowledge of

the curve-fitted soil moisture retention curve.

.. '

moisture retention curve becomes the steepest (Fig. 9). i~s, of course,

follows directly from Eq. (36b) and (43b) which, for values of m close to

C.·· ')i5) one (i.e., for n large), reduce to ~- hQ ., l/a. In that case ~ and hQ

@~·

·.

· . .. :

both become identical to the bubbling pressure, ~, used in the Brooks

and Corey equation~ (see Eq. 22 and 23). Fig. 9 shows a nearly exact

prediction of the relative hydraulic conductivity, with only some minor

deviations occurring at the higher conductivity values.

Results obtained for Touchet Silt Loam G.E. 3 (Brooks and Corey,

1964), shown in Fig. 10, are very similar to those for Hygiene Sandstone.

The curves in this case are also very steep (n•7.09), and again a good

description of the relative hydraulic conductivity is obtained.

Figure ll presents results obtained for Silt Loam G.E.3

(Reisenauer, 1963). This example was already discussed in the previous

section, where estimates of a and n were obtained graphically for three

different values of the residual moisture content. It was then found

that 9 -values of 0.10 and 0.15 gave the best answers, both for the r

description of the soil moisture retention curve and the relative hydraul-

ic conductivity. Interestingly, the three-parameter curve-fitting gave a

value of 0.131, approximately the average of these two 9 -values. r

However, it remains clear that the value of e for this particular r

example is poorly defined, and that a considerable change in e will have r

only minor effects on the calculated curves. Data for this soil were

also used as an illustrative example for the non-linear least-squares

curve-fitting program given in Appendix A. OUtput of the program (see

Appendix A) shows that the 95' confidence interval for 9 is given by r

0.131 (+ 16,). By comparison, these intervals are .00423 (! 5') and 2.06

(! 9\) for a and n, respectively. It may be noted here that the computer

33

Infor ation Only •.. •• ····1:'0 ·• -----· •• •••• ~ ••• •• ·• •·•

~ = ~ ~

e = w

~ ~

•• 0 = 0 = ~ ~

TOUCHET SILT LOAM

.5

.4

FITTED-.,.

- .3 ... e u ...... .. e u - .2

CD 0

a •.00505 n• 7.09

.I ,, •. 190 .

0 -to0 -101

h (em)

..

(GE 3) ao•

K,

ao-a

. ao-4

-ao• -10°

• • • • • • • • • • •

0

PREDICTED~

-ao• h (em)

-104

Fig. 10. Observed (open circles) and calculated curves (solid lines) of the soil hydraulic properties

of Touchet Silt Loam G.E.3. The relative hydraulic conductivity was predicted front knowledge

of the curve-fitted soil moisture retention curve.

~ = ~ ~

e = ~ •• 0 = 0 = ~

cr·:.::

w U1

@_.::·: (/!(})·.

SILT LOAM G.E. 3

.5

.4t= 00 On

::- .3 e u

' ... e u -Q) .2

I ••. 00423

.II- n • 2.06 Br• .131

o~----~----~~----~----~ -10° -t01 ·104

h (em)

~\ (:.::,.·.) (::.::":·:·:

to•, O<h.....

K,

to-a

PREDICTED \

I0-4

-too -t01

h (em)

. . . (}I[./)

-to•

Fi9. 11. Observed (open circles) and calculated curves (solid lines) of the soil hydraulic properti~s

of Silt Loam G.F.3. The relative hydraulic conductivity was predicted from knowled9e of the

curve-fitted soil moisture retention curve.

program also provides for a correlation matrix between the different

parameters. Results, for example, show that er is highly correlated

-·

with n but much less than with a, and that a and n are nearly inde-

pendent of each other. Some of these effects are also noticeable from

the calculations in Table 1.

'l'he first three examples each showed excellent agreement ~tween

observed and predicted conductivity curves. Predictions obtained for

Beit Netofa Clay (Rawitz, 1965), however, ·were found to be much less

@t: accurate (Fig. 12). 'l'he higher conductivity values are seriously under­

predicted, and also the general shape of the predicted curve is consider-

..

(1~::

ably different from the observed one. It seems that much of the poor

p~edictio~s can be traced back to the inability of equation (28) to match

the observed soil moisture retention data. For example, _the residual

moisture content was estimated to be zero, a rather surprising result

since clay soils have generally higher er-values than coarser soils

(the saturated hydraulic conductivity of this soil is only 0.082 em/day).

Limited data at the lower moisture contents further increases doubt about

the accuracy of the fitted e -value. A careful inspection of the observed r

curve shows that the gradient of the curve changes fairly suddenly at

approximately h-10,000 em (the slope suddenly becomes more negative).

The location of the last four data points, in particular, appears to be

inconsistent with the general shape of curves based on (28). With some

imagination one could also identify an inflection point on the observed

curve at a pressure head of about -2,000 em. The observed curve should

have become flatter from that point on if equation (28) were to describe

the data points. Because of the seemingly unreasonable law value of

e , the break in the slope of the curve at h=-10,000 em, and the presence r

36

In for ation Only

. ·.~

-... E u ;;. E u -Q)

. '

.5

.4

.3

.2

.I

p• .. \.:.:·<.:.·! . ,.•.

• BElT NETOFA CLAY

a • .00152 n• 1.17 B,• 0.0

0~----~------~----_.------~ -101 -101 -101

h (em)

..

ao•

K,

ao-a

ao-•

0 0

PREDICTED/

-to• h(cml

. . .

Fig. 12. Observed (open circles) and calculated curves (solid lines) of the soil hydraulic properties

of eeit Netofa Clay. The relative hydraulic conductivity was predicted from knowledge of the

curve-fitted soil moisture retention curve.

.. .

, ··

.of. .1m inflection point at h.;.-2, 000 c:m, an atteD"g?t was made to improve the

predictions by deleting rather arbitrarily the last four data points at

the dry side of the curve. Fig. 13 shows that the ·soil moisture retention

cuzve is now much better described (wi. th the obvious exception of the

last foUr data points). Also the description of the conductivity cuzve

is improved somewhat. At least the general shape of the curve .is ~escribed

more accurately, even though the predicted curve is still displaced to

the right of the observed one. The example shows that by deleting only

four points at the dry end of the curve a completing different value of

er is obtained (0.28~ versus .O.O cm3;cm3). - T.his case demonstrates again

the impor~ance of having some independent procedure for estimating the

residual moisture content.

Results for Guelph Loam (Elrick and Bowman, 1964) are given in

Fig. 14. This example represents a case in which hysteresis is present

in the soil moisture retention curve. The observed data of this example

QB·;·. were taken directly from the oriqinal study (Fiqs. 2 and 3 of Elrick and

Bowman, 1964). For the wettinq branch a maximum ("saturated") value of

·.

0.434 for the moisture content was used, being the highest measured value.

Also the wetting branch of the hydraulic conductivity curve was matched

to the highest value of Kr measured during wetting (Fig. 14). The value

of er' f\.:.rthermore, was assumed to be the same for drying and wetting,

and was o.:_. tained from the drying branch of the curve. Both the drying

and wetting branches of the soil moisture retention curve are adequately

described by (28). Also the conductivity curves are reasonably well

descr~d, even though the predicted curves are slightly below the observed

ones. Note that some hysteresis is predicted in the relative hydraulic

conductivity. Although this is generally to be expected when two different

38

Information Only

;) E 0

. 5

.4

' eo .3 e 0 -

.2

.I

0 -ao1

Fig. 13.

BElT NETOFA CLAY .

0 ELIMINATED~

DATA o 0

-102

h

a• .00202 n •1.53

9,· .286

-105 -104

(em) .,

..

Kr

PREDICTED/

-ao8 -ao0 -ao1 -ao2 -ao1 -104

h; (em)··~

Observed (open circles) and calculated curves (solid lines) of the soil hydraulic properties

of Belt Netofa Clay. The relative hydraulic conductivity was predicted from knowledge of

the curve-fitted soil moisture retention curve. The last four data points of the observed

soil moisture retention curve were not considered in the fitting process.

,·

~ = ~ ~ e = ,c:.

0 .......... •• = = 0 = ........ ~

((?·

-'E 0

' If) e 0 -

(})

..

.5

.4

.3

.2

.I

@·;';:

GUELPH LOAM

a• .0200

n• 2.76

a • .0115

/ n • 2.03

8,· .218

CJL·'·

K,

o• , , , . -10° -10 2 -104

h (em)

. . (0) (}/)

to0

102

lo4

.3 . -.4 .5

9 ( cm5 I cn:a1 )

Fig. 14. Observed (circles) and calculated curves (solid lines) of the soil hydraulic properties of

Guelph Inam. The drying and wetting branches of the relative hydraulic conductivity curve were

predicted from knowledge of the curve-fitted branches of the soil moisture retention curve.

t?::. ~~~:;_-:"

. Q~}:

· .

. . ,. ct:

retention CU1i7~S ."1:;::.! ~~r~~ent, ~. (S) also shows that different retention

curves may generate the same conductivity curve as long as er and m (and

hence n) remain the same (i.e. a may be different).

41

Information Only • • • ••••• 0 .·; :. ::·· ... _·;~·~·-

.:

~

.·· : · ..

REFERENCES

Abramowitz, M., and I.A. Stegun. 1970. tions. Dover Publ., New York.

Handbook of mathematical func-1046 pp.

Ahuja, L.R., and D. Swartzendruber. water diffusivity function.

1972. An improved for111 of the soil­

Soil Sei. Soc. Am. Proc. 36(1):9-14 • .

Amerman, C.R. 1976. Waterflow in soils: a generalized steady-state, two­

dimensional porous media flow model. u.s. Dept. Of Agr., ~-NC-30.

62 pp.

Bresler, E. 1975. Two-dimensional transport of solutes during nonsteady

infiltration from a trickle source. Soil Sci. Soc. Am. Proc.

39(4) :604-613.

Brooks, R.H., and A.T. Corey. 1964. Hydraulic properties of porous media.

Hydrology Paper No. 3, Civil Engineering Dept., Colorado State

University, , Fort Collins, Colorado. ·

Brooks, R.H., and A.T. Corey. 1966. Properties of porous media affecting

fluid flow. J. Irrig. Drain. Div., Am. Soc. Civil Eng. 92(IR2):

61-88.

Bruce, R.R. 1972. Hydraulic conductivity evaluation of the soil profile

from soil water retention relations. Soil Sci. Soc. Am. Proc.

36(4):555-561.

Burdine, N.T • . 1953. Relative permeability calculations from pore-size

distribution data. Petr. Trans., Am. Inst. Mining Metal!. Eng.

198:71-77.

Daniel, c., and F.S. Wood. science, New York.

Endelman, F.J., G.E.P. Box, and P.G. Saffigna. nitrogen phenomena. 66pp.

1973. Fitting equations to data. Wiley-Inter-

350 pp.

J.R. Boyle, R.R. Hughes, D.R. Keeney, M.L. Northup,

1974. The mathematical modeling of soil-water-

Oak Ridge National Laboratory. EDFB-IBP-74-8.

Elrick, D.E., and D.H. Bowman. 1964. Note on an imp:-: ~ved apparatus for

soil moisture flow measurements. Soil Sci. soc. Am. Proc. 2 8 ( 3) :

4SQ-453.

Green, R.E., and J.C. Corey. 1971. Calculation of hydraulic conductivity:

a further evaluation of some predictive methods. Soil Sci. Soc.

Am. Proc. 35(1):3-8.

Haverkamp, R., M. Vauclin, J. TOuma, P.J. Wierenga, and G. Vachaud. 1977.

A comparison of numerical simulation models for one-dimensional

infiltration. Soil Sci. Soc. Am. J. 41(2):285-294.

42

nformation Only

Jackson, R.D. 1972. On the calculation of hydraulic condut.:tivity. Soil Sci. Soc. Am. Proc. 36(2):380-382.

::-:;:.<·. Jackson, R.D., R.J. Reginato, and C.H.M. van Bavel.·· 1965. Comparison of ·:.=[j)": measured and calculated hydraulic conductivities of unsaturated

soils. Water· Resour. Res. 1:375-380.

.........

Qj}

\3.···:-:· ...

. .

Jeppson, R.W. 1974. Axisymnetric infiltration in soils, I. Numerical techniques for solution. J. Hydrol. 23:111-130.

Marquardt, o.w. 1963. An algorithm for least-squares estimation Of non-linear parameters. J. Soc. Ind. Appl. Math. 11:431-441.

Meeter, D.A. 1964. Non-linear least-squares (Gaushaus). Univ. Wise. Computing Center. Program revised 1966 •

Millington, R.J., and J.P. QUirk. 1961. Penneability of porous solids. Trans. Faraday soc. 57:1200-1206.

Mualem, Y. 1976a. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12(3) :513-522.

Mualem, Y. l976b. A catalogue of the hydraulic properties of unsatur­ated soils. Research Project No. 442, Technion, Israel Institute of Technology, Haifa, Israel. 100 pp.

Rawitz, E. 1965. The influence of a number of environmental factors on the availability of soil moisture to plants (in Hebrew). Ph.D. thesis. Hebrew Univ. , Rehovot, Israel.

Reeves, M., and J .c. DugUid. 1975. Water 100vement through saturated­undersaturated porous media: A finite-element galerkin model. Oak Ridge National Laboratory, ORNL-4927. 232 pp.

Reisenauer, A.E. 1963. Methods for solving problems of multidimensional partially saturated steady flow in soils. J. Geoph. Res. 68(20): 5725-5733.

Segol, G. 1976. A three-dimensional qalerkin finite element model for the analysis of contaminant transport in variably saturated porous media. User's guide. Dept. of Earth Sciences, Univ. of Waterloo, Canada. 172 pp.

43

Information Only

I'

APPENDIX A

SO!aP:

. A COMPUTER MODEL FOR CALCULATING

THE SOIL HYDRAULIC PROPERTIES

FROM SOIL MOISTURE RETENTION DA~.

·:.'· ... 1:: •.. .. \S· 44

Information Only . . . . . . . . . . . . . ·. . . ~-.. ' .. ~ •. _. . ~··.-. - . • ·. -- ....

·~

(-:.~:.: :::· ~:

This Appendix gives a brief description and listing of SOHYP, a

computer program for calculation of the soil hydraulic properties from

observed soil moisture retention data. ~e program does this by means

of a non-linear least-squares fit of the following equation to the ob-

served data [see also Eq. (28) in the text]

ce s -er) e - e + -;;._-=---. r m

(Al)

[l+ (ah)n]

where for the Mualem theory,

m=l-1/n, (A2)

a,nd for the Burdine theory

m-1-2/n. (AJ)

The most significant variables in the program are defined in Table Al.

Table A2 gives detailed instructions for set-up of the data cards, while

Table A3 shows a list of the input data of example problem 3 (Silt Loam

G.E. 3), described in the main body of this report. The computer output

fer this example is given in "Table A4, while the actual listing cf the

program is given in Table AS.

The computer program provides for three options, controlled by the

variable MODE. If MODE equals one, the program op.timizes the three para-

meters e , a, and n by means of a least-squares fit of equations (Al) and .r

(A2) to the observed data. The soil hydraulic properties are then calcu-

lated in accordance with the Mualem theory. If MODE equals two, the

45

Information Only

... ·.·

({;)

program only calculates best-fit values of a and n, and assumes that 9 r

is known beforehand. The value of 9 is now given as an input variable r ..

(see Table A2). Values of a and n are still calculated by means of

Eq. (Al) and (A2). (i.e. the Mualem theory still applies). If MODE

equals three, the computer model again calculates best-fit values of the

three parameters (9r' a, and n), but it is now assumed that the Burdine

theory applies. Hence Eq. (Al) and (A3) are now used in the program. In

each case the computer program provides for a table of the hydrauli~

properties of the soil (see Table A4), consistent with the value of MODE

selected •

46

Information Only ..... .. .

,'

TabZe Al.

VARIABLE

ALPHA

B (I)

BI (I)

DIFFOS

MIT

MODE

MODEL

NC

NDATA

NIT

NOB

RK

RM

RN

RWC

SATK •.

SSQ, SUMB

List of the most significant variables in SOHYP.

DEFINITION

Hydraulic conductivity (K).

Coefficient a in Eq. (Al).

Array containing initial estimates of coefficients.

Array of coefficient names.

Soil moisture diffusivity (D).

Maximum number of iterations.

Designates model type to be used in proqram:

•l: Three-parameter fit (9 , a, and n) (Mualem theory) r

•2: TWO-parameter fit (a, n) (Mualem theory)

•3: Three-parameter fit (9r' a, n) (Burdine theory).

Subroutine to calculate soil moisture content (9) from

pressure head (Eq. Al).

Number of cases considered.

Input data code:

=0: New data are read in

•l: Data from previous case are used.

Iteration number during program execution.

Number of observed data points (must not exceed 40).

Relative hydraulic conductivity (K ). r

Equals l-l/n for Mualem theory, l-2/n for Burdine theory.

Coefficient n in Eq. (Al) •

Dimensionless moisture content (El).

Hydraulic conductivity at saturation (Ks).

Residual sum of squares.

47

Inti · rmation Only

..

TABLE Al (CONTINUED):

VAl?IABLE

S'l'OPCR

TITLE(!)

we

WCR

wcs

X(I)

Y(I)

DEFINITION

Stop criterion. Iteration process stops when the

relative change in each coefficient becomes ~ess than

STOP CR.

Array containing information of title eards.

Volumetric moisture content (8).

Residual moisture content (8r).

Saturated moisture content (8s).

Array of observed pressure heads (values are assumed to

be positive).

Array of observed moisture contents. ·

48

Information nly

Tab'Le A2.

CARD COLUMNS

1

2

3

4

1-S

1-80

1-S 6-10

11-lS 16-20 21-30

31-40 41-SO

1-10

11-20

21-30

s 1-6 11-16 21-26

6,etc. 1-10

11-20

FOiiMAT

IS

20(A4)

IS IS IS IS

FlO.O

FlO.O FlO.O

FlO.O

FlO.O

FlO.O

A4,A2 A4,A2 A4,A2

FlO.O

FlO.O

VARIABLE

NC

TITLE

MODE NP NOB NDATA WCR

wcs SATK

B{l)

B{2)

B{3)

BI{l) BI{2) BI(J)

X{I)

Y{I)

49

COMMENT

Number of cases considered.

The following cards are repeated NC times. However, skip" cards 6, etc., if NDATA • 1 on third data card.

Defines model number (1, 2, or 3). Number of coefficients {2 or 3). Number of observations. Data input code. Residual moisture content. This information is only necessary when MODE • 2. • Saturated moisture content. Saturated hydraulic conductivity.

Initial value of 8r if NP • 3; Initial value of a. if NP • 2. Initial value of a. if NP • 3~ Initial value of n if NP • 2. Initial value of n if NP • 3.

Coefficient name of B{l). Coefficient name of B{2). Coefficient name of B(3) {only if NP • 3).

Value of observed pressure head (assumed to be positive). Value of observed moisture content.

Information Only

.-

:i'C!bZ.e .13. Input data for example 3 (Silt Loam G.E.3).

1 2 3 4- 5 Column: 12345678901234567890123456789012345678901234567890

Card

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

1 SILT LOAM G.E.3

1 3 13 0 0.180 0.002

WCR ALPHA 10.0 0.396 20.0 0.394 43.0 0.390 60.0 0.3855 80.0 0.379

111.0 0.370 190. o-, o. 340 285.0 0.300 400.0 0.260 600.0 0.220 800.0 0.200 900.0 0.194

1000.0 0.190

0.18 2.3 N

so

0.396 4.96

Information Only

• f

.·

Table A4. Output for example 3 (Silt Loam G.E.3).

51

Information Only

~ = ~ .., a = (llllto. ••• c = 0 = ~

"" N

K·.· . ·: . ~ ... : :::. CJ)_) @/::· I I

-~

(~.?/)

. ................................................................................ . • • • NCN-LINEAR LEAST SQUARES ANALYSIS • • •

• • SILT LOAM G.E.J • • ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

INPUT PARAMEYfRS •••••••••••••••a HODEL NUMBER••••••••••••••••••••••••••••••••••••• NUMBER OF COEFFICIENTS ••••••••••••••••••••••••••• NUMBER OF OBSERVATIONSe•••••••••••••••••••••••••• RESIDUAL MOISTURE CONTENT CFCR HODEL 2t••••••• SATURATED MOISTURE CtNTENT•••••••••••••••••••• SATURATED hYDRAULIC CONDUCTIVITY••••••••••••••

OBSERVED DATA ••••••••••••• oas. NO.

l 2 3 ~ 5 6 7 8 9

10 11 l2 13

PRESSURE HEAD 10. GO 20.00 le3.00 60.00 eo.oo

111.00 190.00 285.00 400.00 600.00 800.00 900.00

1000.00.

MOISTURE CONTENT 0.3960 0.3940 0.3900 0.3855 0.3790 0.3700 O.HOO 0.3000 0.2600 0.2200 0.2000 0.19~0 0.1900

: 1 3

13 0.1800 0.3960 o\.9600

• . ({/?)

. .

cL

(j}

Ci>

~~-.o ....... ..,.ONJ"\0001

zo..,..,..o.n.n 11\NOOOO • • • • • • NNNNNN

0..,.11\,._N,., c .., .:!:) ... ~~~ 'l: :tOONNN ~ ,...,., ................ ..J":):tOOOO 4":)00:>~:>

• • . • • • 000-.»":)~

:t .... ~ ........ '"' 2:-:I~NN ...... \j;:::)fO\fO\fO\f0\1"\ ..................... . . • • . • 000000

0 z z 0 -t-0-NI'f\.rll\ c :a: w ~ -

)( -a: ~ c ~

z 0

~ c ... "' a: CIC 0 v

., ~ ..J ~

"' w c ..J c z -~

C) C) "' C)

!"\ C) "' • ,.. .. _, c z c .,

o ... w 01"\ a: 0~ c

N 0~ ~ • • 0

... o "' I ~

"' c

"' Of0\.0 ..J o .... .o C) '1\ .... c

~Of0\01 c • • • w

.... oo z ... I

... NCO\ z C) ~

llt.:C,_.,,.. ~WN .... CD -CL.II\Oif'l ~:..-oN - ::1 ••• ~ OON

w u z w Q -II. Z ~ON ~«~.,.C) uwoocc

:a-ace "0 ••• .n ... oo­o-

• ~ l.&.4-ll\ !.&.1~00 00-:1"0 U0-:1':1 . . . . wooo • on

w ... c co a:: CUCL.Z -.:::a..J a: c c >

53

I .r ,._ II\ .1\ N ~ '" N ~ 0 ~ N II\ _ _,,.,NN:t-:t----NNI'f\1'1\ vtcooooooo:to-:~o:to :.u~o-:~ooo:tooooooo a: ..:l ••••••••••••••

0000000000000 IIIII III

w « ·:t':l:tOOOOO~OOOO :I 0 ·:) :t C) 0 0 0 0 0 0 C) 0 0 . "' ............ . "' ':1 ·~ II\ 0 0 0 0 .. 0 '"' 0 0 0 W :3' ·~ GCI Q - ~ N ... 0 .r .0 0 Q a:-ONO' ~ ........ .0 CL. ..

O,._..,CDN-•N.OO'f0\-#'011\ z .... ... - ....

I NN:lc:III\N .... II\GDNNII\,_ -..J-:1-NN_, ... ..,N ... IO\ ... ON vtcOO':IOOO~OOOOUO W~0-:1-,::lQOOO":l::tOOO « c • • • ••••••••••

00000~0':100000 I I I I I I I I

Information Only

((/.:.~:·: . . 0?::· CS\: .. (~ rf:}F:) I . . '•,•'•) " .... :·:_:.::::; . . ··.··.· . · .. : . ., .

·.• '· .

PRESSURE LOG P wt itEL K LOG RK AUS "' LOG KA DJFFUS LU~ D

o.o 0.3960 O.l.:JUE 01 O.lt'i6E \11

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o.zne 01 0.315 o.J96J 0.'195E 00 -0.001 o.~taoc ot o.u89 o. !.nc 06 5. 7J2

~ o.zazc 01 0.450 o. 3960 O.II82E 00 -o.ooa o.~117E Ul 0.6bl 0.4~t8E oo 5.651

O.HSE 01 0.525 0.3960 o.•HOE 00 -0.010 o.~td~c - u1 o.oo6 OolllE 06' 5.570

= O.l911E 01 0.600 0.3960 0. 971tE Ull -ll.01Z o.~taJc o1 0.681t U.lOUE U6 5. 'ttl d

o.~tne 01 0.675 0.3960 o.9oBE 00 -O.Ollt O.lt8Uc Ul 0.682 o.c:;5e 06 S.lt07

~ Oe562E 01 0.750 0.3959 Oo'iblc 00 -o. 01 r O.lt 17E 01 0.679 o. 2u.: 06 5.32~

Oe668E 01 0.825 0.3959 0.955E 00 -o.ozo O.lt l3E 01 o.u!i 0.1HE 0, 5.~42

0.7941: 01 0.900 o. 1959 O.'llt6E 00 -o. Ollt O.ltu9E 01 U.C)11 o.ltt~tE 06 S.l!i8 ., 0.91tltE 01 0.975 0.3958 o.9.35E 00 -0.029 O.ltoltE 01 0.6bb O.ll9E Oil 5.ullt

a o.uze 02 1.050 o.J957 o.922c 00 -0.035 o.lt.57i: 01 o.c.6o .J.97.5E llS 4.909

o.1.n1: 02 1.125 0.3'15b O.'I01E 00 -0.04) O.lt~UE 01 o.uSJ o.aoot: 05 4.901

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·, = 0.126;.:; 01 2.100 0.3650 o.28&E 00 -o.SS1 o.u9c: 01 O.lltlt U.JlOE Olt l.SlH

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0.2fi9E Ol l.ltl5 o.l.'ll6 o. J ne-o1 -1.426 O.llt6E ou -0.130 o.~zor: 03 2.121

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o.1oae 03 2.850 0.2100 O.l35E-02 -2.869 U.670E-OZ -2.11't Oeb29E oz lel9H

) O.BielE 03 2.925 0.197cJ o.ulE-OJ -3.196 0.3l6E-Ol -2.500 O.ie\l!iE 02 1.608

- OelOOE Ole 3.000 o.1an o.Z96E-03 -3.529 o.11t1e-o2 -Z.U3 0.2601: Ol lelt16

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o • .r.~tlE o5 4.650 o.u21 O.b92E-11 -11.161) O.l'tlE-111 -1il.~65 Oe11t1E-&»2 -2.8!H

= o.511E o5 ... 125 0.1322 Oe110E-l1 -11.509 0.1~4E-10 -10.U1) o.9u2E-03 -3.045

0.631E OS ~t.8oo 0.1320 O.ll~E-11 -11.S!J7 O.b89E-11 -11.162 0.571£-0l -3.lJ9

o.rsoe o5 't.B15 o. U19 OebllE-12 -12.206 o.lo9E-11 -11.511 u•lc.9c-Ol -1.433

0 O.B91E 05 lt.950 o. uu 0.21~E-1Z -12.555 0.13dE-11 -11.859 0.216E-lll -3.c,2l '

0.1UcaE 06 5.0l5 0.1111 o.12~e-12 -12.901 0.620E-l2 -12.l&>8 o.tsU:-oJ -.3.bl2

o.126E 06 5.lu0 0.1311 ~.jbOE-13 -13.252 0.27dE~12 -12.556 0.9b'tE-Oit -4.016

= o.1soe 06 5.115 o.ll16 o.251c-13 -11.601 O.l21tE-1l -1.l.905 Oe6l1E-04 -4.2l0

o.178t: 06 5.41!:50 0.1316 o.11ze-11 -13.9't9 o.557E-13 -13.254 o.J91tc-&>~t -'t.ltOit

~ o.zue o6 5.325 0.1315 &».504E-1't -1~t.298 o.z5oe-11 -1.3.602 0.252E-Oit -4.~98

~ o.c:51E 06 5elt00 0.1315 o.z26E-1~t -14.6'tJ 0.112E-13 -13.951 0.161£-0it -'t.7'12

o.Z911E 06 5.415 o.uu Oel01E-11t -11te99S 0.502E-11t -11t.liJO 0.1 03~-04 . -lt.987

o.3~·SE o6 5.550 o.u11t O.'t5lE-lS -1S.3~1t o.2l5£-14 -llt.6't& 0.659&:-05 -5.1bl

o.lt22E 06 5.6~5 0.1314 0.203£-15 -15.692 O.l01E-11t -llt.997 o.~tzz~-o5 -5.315

0.501E 06 ·s."K•)o 0.1314 0.910E-16 -16.01tl Oe't51E-15 -1S.Jit6 o.ztoe-os -s.sc.9

Tab~e AS. Fortran listinq of SOBYP.

56

Information Only

(8:·.

c c c c c c c c c c

c c

c

.................................................................. • • • • •

NCN-LINEAR LEAST-SQUARES ANALYSIS OF SOIL HYDRAULIC PROPERTIES

SOHYP * APRIL 1980 *

• ..................................................................

OI~ENSICN Xl401 9Y(40J 9R(40),FC40J,OELZC40tlti,LSORTl40),8(3JtBIC6), 1El~J,P(3J,PHtl3J,Ql3),T8l3J,Al3t3J,Ol3t31tTITLEl20J,THl3J

DATA STOPC~/.0010/,MlT/20/

----- REAC NUMBER Of CASES CONSIDERED ----­REAOCSrlOOOJ NC DC 144 IC•ltNC REACl5tlOC2J TITLE WR1TEC6,1004J TITLE

C ----- READ INPUT PARAMETERS READC5tl0QOJ MCDE,hP,NOB,NDATA,WCR,WCS,SATK WRITEl6tlOOSJ ~CDE,NP,NOB,NCR,WCS,SATK

c C ----- READ INITIAL ESTIMATES ----­

READC5,10C6J (Btllrl•l,NPJ c c

c c

c c

----- REAC COEFFICIENTS NAM5S NBt•2*NP REAO(S,l007J lBlllJtl•ltNBIJ

----- REAC AND WRITE EXPERIMENTAL DATA ----­WRITEl6, 10081 lFlNCATA.GT.OJ GO TO l DO 4 t•1tNOB

4 REACl5tlOC6J XCIJ,YCIJ 8 00 10 1•1 ,NCB

10 NRITEl6,10lll ltXCIJ,Yll)

00 12 l•l,NP 12 THU 1•81 U

1FtlNP-2)*lNP-3JI l4tl6t14 14 WR1TE(6, 1016)

GO TC 142 16 GA•0.02

CALL HCDELCTHtftNOBtXtWCS,~OOE,NP,WCRl SSC:•O. DO 32 l•leNOB R l IJ •Y (I)-FlU

32 SS,•SSQ+RliJ•ACIJ NlT•O WRITEl6tlOlOJ lflHOOE.EQ.21 WRITEl6el026) NtT,WCRtBllJ,Bl21tSSQ,MODE

57

Information Only

.·

MAIN

1 FC MOOE.NE.2J WRITEl6,1026J Nl T, 8Ult8Uit8&31 ,SSQ,MOOE c C ----- BEGIN OF ITERATION -----

c

34 NIT•NIT+l GA•O.l*GA DO 38 J•ltNP TE'-P•TH(J) THCJJ•l.Ol*TH(J) QCJJ•O CALL MOOELtTH,OELZCl,J),NQB,X,WCStMODE,NP,WCRl DO 36 l•l,NOB DELZCI,JJ•DELZCI,Jl-FII)

36 Q(JJ•QCJJ+DELZCl,JI*Rlil QlJJ•lOO.•~tJI/THCJJ

C ----- STEEPEST CSSCENT -----

c

38 THCJJ•TEMP DO ltlt I•ltNP DO 42 J•ltl SUJII•O DO ItO K•ltNOB

ItO SU,•SUM+DElZ(~,IJ*OELZCK,Jl DCI,JJ•lCOOO.•SUH/CTHtlJ•THCJII

lt2 0 CJ, IJ•DU,Jl

C ----- 0 • MOMENT MATRIX 44 ECtJ•SQRTCOtt,IJI SO 00 52 I•ltNP

00 52 J•l,NP 52 ACI,JJ•Dti,JJ/CECII*ECJII

t -, t --· --- A IS THE SCALED MOMENT ~ATRIX

DO 54 I•l,NP PCI )•QU J/EUJ PHil U•P( U

54 Atl,l)•A(l,IJ+GA tALL MATINVCA,NP,PJ

t C ----- P/E IS THE CORRECTION VECTOR -----

STEP•l.O 56 DO 58 I•ltNP 58 T8llt•PCII*STEP/Etll+THCIJ

DO 62 I•l,NP 1FlTHCII*TBCllt66,66t62

62 CONTINUE SUMB•O.O CALL MOOELlTB,F tNOB,X,WCS,MOCE ,NP,WCRI 00 64 I•ltNCB RllJ•YCIJ-Fll)

64 SUMB•SUMB+RCIJ•Rtll 66 SUMl•O.O

SUfiiZ•O.O SUJ113•0.0 DO 68 I•ltNP

58

Information Only •••• ·-·- • • ...... _.! ••

.. ' r ;. &

.· c c

MAIN

SUMl•SUMl+PCIJ*PHICIJ SUMZ•SUM2+PCli*PliJ

68 SU~3•SUM3+PHICII*PHICII ANGLE•57.29578•ARCDSCSUM1/SQRTCSUM2*SUM3JJ

DO 72 I•ltNP lFCTHCtt•TBClJ)l~t1~t72

72 CC~TlNUE IFCSUMB/S~Q-l.OJ80t80t74

7~ IFCA~GLE-~0.01?6,76,71 76 SlF.P•STEP/2.0

GO TO 56 78 GA•lO.*GA

GO TO 50 c C --~- PRl~T COEFFICIENTS AFTER EACH ITERATION

c

80 CONTINUE DO 82 I•ltNP

82 THCl)•TBCU IFCMOOE.EQ.21 W~tTEC6t1026J NtT,WCR,THC1JtTHC2JtSUM8,MODE I FC I'OOE.NE .z) '-RITEl6tl026J ·NIT t THt U, THC2), THU t ,SUMB, MODE 1FCMODE.EC.2J GO TO 90 1F(TH(1J.GT.0.005t GO TO 90 WRI'T'El6t1028) GO TO 144

90 DO 92 I•ltNP t FUBS (P( II •STEP/Ell U/ U.OE-ZO+A8SlTHC I U J-STOPCiU 92,92,94

92 CONTINUE . GO TO 96

94 SSQ•SUMB IFCNIT.LE.MtTI GO TO 3~

C ----- END OF ITERATION LOOP 96 CONTINUE

CALL MATINVCO,NP,PJ c C _ _.. ___ WRITE CORR:LATIOH MATRIX

DO 98 I•l ,NP 98 Elii•SQRTCDCltlll

WRITE(6,1C~~~ Ct,I•l,NPI DO 102 l•l.NP DO 100 J•l,I

100 ACJ,II•DCJ,IJ/CECIJ•ECJIJ 102 WRITEl6tlC48) l,(A(J,t),J•ltll

c C ----- CALCULATE 95~ CONFIDENCE INTERVAL -----

Z•l.IFLOATC~OB-~PJ SDEV•SQRTtz•suMBJ WRITEC6, 1052) TVAR•l.96+Z*CZ.3779+Z*CZ.713S+Z*C3.187936+2.466666*Z**Zllt DO 108 I•l,NP SECOEF• Elii•SOEV TVALUE• fhCl)/SECCEF

59

Information Only

r.::-: ~sM·

..

TSf:C•TVAR*SECOEF TMCO!•THl I 1-TSEC TPCCE•THl U+TSEC K•2*I

MAIN

J•K-1 108 WR tTEl6tlOSBI 8 IlJt.BilKit THCI t,SECOEF, TVALUE, TMCOE, TPCOE

c C ----- PREPARE FINAL OUTPUT

LSC:RT( U•l DO 116 .1•2,NCB TEMP•RlJI K•J-1 DO 111 L•ltK. LL•LSORTlLI tFCTEMP-RlLLII 112t112tlll

111 CCNTINUE . LSCRTlJI•J GO TO 116

112 KK•J 113 KK•KK-1

LSCRT fKK+U•LSORTIKKI tF(KK-L) ll!t1l5tl13

115 LSCltTlL)•J 116 CONTINUE

WR. ITE l 6 ,1066) DO 118 I•1,N08 J•LSCRTlNC8+1-I J

118 WRITEl6tl0681 ltXCIJtYllltFCtJ,Rll),J,XlJJ,YlJJ,FlJJ,RlJI c C ----- WRITE SOil HYDRAULIC PROPERTIES -----

IIIR!TEl6r1069) PRESS.a1.18850 RN1•0.0 ~KlN•leO WR!TEC6r1072) RNl,~CS,RKlN,SATK DO 140 1•1 , 7 S tFlRKLN.LT.C-16 .. 11 GO TO 142 PRESS•l.l8S50*PRESS. IFlMOOE-21 120,122,120

120 WCR•THllJ AlPHI.•THl 2) RN•THlll GO TO 124

122 AlPHA•THl U RN•THC21

124 RM•l.-1./RN 1FCMODE.EQ.3) RM•l.-Z./RN RNl•RM*RN RWC•l./tl.+lAlPHA*PRESSI**RNI**RM WC•kCR+(kCS-wCRJ•RwC TERM•l.-RwC•lALPHA•PRESSJ**RN1 tF((TERMeLT.S.:-o5t.OR.lRWC.LT•0•06)) TERM • ~~•R.wC••C1e/RMJ tF(MOUEeEC.31· RK=RWC*RwC*TERM 1Fl~OO~.NE.3J RK•SQRTlRWCJ•TE~M•TERM

60

Information Only ........... ~ .·. ;· ,--. ··.··· .. ,., ...

'· . '

•

• .•

.. . c:· ..

c

MAIN

TERM•ALPHA*RNl*CWCS-WCRJ*RWC*RWC**Cl./RMJ*CALPHA*PRESSI**CRN-t.J

AK•SATK*"" DtFFUS•AK/TERM PRLN•ALOGlOCPRESS) AKLN•ALOGlOUKJ RKLN•ALOGlOCRK) DtFLN•ALOGlO&DtFFUS)

1~0 ~R1TEC6tl0701 PR~SS,PRLN,WCtRKtRKLNeAK,AKLN,DJFFUStDIFLN

l't2 CONTINUE l't4 CONTINUE

C ----- END OF PRCBLEH -----1000 FORMATl415t5FlO.OJ 1002 FORMATC20A4t 100~ FOQMAT(1Hle10Xt82ClH*I/11Xt1H•,eox,1H*/11XtlH*e 9Xt 1 NON-LlNEAR LEA

1ST SQUARES ANALYSlS 1 t38XtlH*/l1XtlH*t80XtlH*/11XtlH*t20A4tlH*/11Xt

21H* ,aox, lH*/UXt82UH*J) 1005 FORMATC//11Xt 1 1NPUT PARAMETERS 1 /11Xt16C1H•J/

211X,'M00EL NUMBER•••••••••••••••••••••••••••••••••••••'tl3/ 311X, '·NUMBER CF COEFFICIENTS •• ••••••• •••••••••• ••••••••' tl3/ 'tll.X, 1 NUMBER OF OBSERVAT10NS ••••••••••••••••••••••••••• •,t3/

Sll.X,•R:StDUAL MOISTURE CONTENT CFOR MODEL 2)•••••••'tfl0e4/

611.X, 'SATURATED MOISTURE CONTENT • .• ••• ••• •••• •• •••••• 1 tf10e4/ 711Xe 1 SATURATED HYDRAULIC CONOUCT1VlTY •••••••••••••• •,Fl0e4J

1006 FOr:lMAT(4FlO.OJ . 1007 FORMATC~CA~tA2,4XJJ 1008 F~r:lMATC//11Xe 1 0BSERVED DATA 1 ,/11X,13C1H•I/l1Xt 1 0BS. N0. 1 e4le 1 PRESS

lURE ~fAD 1 ,2.X, 1 MC1STURE CONTENT') 1011 FORMATCllX,15tSX,Fl2.2t4.X,F12.4J 1016 FORMATC//5X,l0llM*J•' eRROR: INCORRECT NUMBER ~F COEFFICIENTS')

1026 FCRMATC15X,I2,10XtF8.~t3.X,FlOe6tZX,F10.~t,SX,F12.7t4Xtl~)

1028 FOPMAT(//11Xe'WCR IS LESS THAN 0.005, USE TWO-PARAMETER HODEL WITH

1 WCR • 0.0 1 )

1030 FO~HATllHle10Xt 1 ITERATlON N0 1 t8Xt'WCR 1 t8Xt'ALPHA 1 t10X,'N 1 tl3X,'SSQ

l',SX,'MODEL'J . 1044 FORMATl//ll.X, 1 CORRELATION MATRIX 1 /l1Xtl8ClH•J/14Xt10(4X,I2,SlJ)

1048 FORMATC1LX,I3,10lZX,F7.~,2XJJ . 1052 FCRMATC//11X,'NCN-LINEAR LEAST-SQUARES ANALYSIS: FINAL RESULTS'/

lllXt 48 UH• J/64X,' ~51 CDNFI DENCE L 1H1TS' /llX, 'VAR1A8L.E 1 ,ax,' VALUE' i

27X, 1 S.E.CCEFF.•,3x,•T-VALUE 1 ,6Xt 1 LOWER',lOX,'UPPER'I 1058 FORHATC13X,A~,A2e4XtflO.S,SX,F9.4eSX,F6.2t4X,F9.~,5ltF9.4J

1066 FOPHATl//lOX,BClH-Je 1 0ROERED BY COMPUTER INPUT•, 8C1H-Je 7l,10ClH­

U, 'ORDERED eY RESIDUALS', 10C 1H-J/26X, 'HOISTU.RE CONTENT' ,3.X, 1 RESI-•

1t24X,'MOISTUAE CONTENT',3X,'RESI-'/lOX,'ND 1 r3X,'PRESSUR:•,5Xt 1 0BS'

2,4l, 1 FITTED't4X,'DUAL'• 9Xe 1N0',3lt'PRESSURE'tSX,'OBS•,~x,•FlTTED'

3e4Xe'DUAL 1 )

1068 FOAMAT&lOX,I2,Fl0.2tlX,3F9.4,8Xtl2rFlOe2tlX,3F9.4J lC69 FORMATllHltlO.X, 1 PRESSURE 1 ,4Xt 1 LDG P1 t6Xe 1 WC 1 ,7Xt 1 REL K1 t5Xr'LOG RK

11 t6X, 1 ABS K',4X, 1 LOG KA•,sx,•DIFFUS'e5Xe'L0G 0 1 1

1070 FOAMATllOX,El0.3,F8.3,FlOe4r3lE13.3tF8.3JJ .1072 FOAMATllOX,El0.3,8X,F10.4tE13.3t8X,E13.3J

STOP END

61

Information Only • • • • • 0 • • ,.. 0 • ' • • • . ..... ... : · · · .r :-· • • /' .. ......

• .•

. . ~·.

(b

MA TINY

.SUBROUTINE MATI NVU t NP, 8' DIMENSION A(3,3J,al31,JNDEX(3,2l 00 2,J•ltlt

2 JNOEX(J,U•O 1•0

~ AMAX••1.0 DOt 10 J•1 ,NP IFliNDEX&J,lJI 1~eoe10

6 DO 10 K•l,NP 1FCJNDEX(K,1J) lOtitlO

I P•ABSUCJ rl<.)) IFlP.LE.AMAXJ GO TO lw IR•J IC•K. AMAX•P

10 CONTINUE Jf(AMAX) 30,30e1~

14 lNDEXCICtli•IR lfll~.EQ.JCI GO TO 18 DO 16 L•l,NP P•AC IRt~l AUR,LJ•A& lCtLI

16 ACJC,LJ•P P•BCIR J BURI•BUC) BliCJ•P 1•1+1 1NDEXII,2J•IC

18 P•l./ACIC,ICI AC IC,ICJ•l.ll 00 20 ~·l,NP

20 ACIC,LI•ACICeLI•P 8UCJ•6UCJ•P DO Zit K•1tNP IF!K..EQ.ICJ GO Tu Zit P•ACKelC) A&K.tiC J•O. 0 DO 22 L•leNP

22 A(K.,L)•ACKrLJ-AllC,~J•P 8CK.J•8CKJ-atiCJ•P

2~ CONTINUE GO TO It

26 lC•INOEX& I r2J IR•lNDEXl lCtU DO 28 K•l.tNP P•AC Kt IRI .' CK, IRJ•AUriCI

28 ACKt lCJ•P 1•1-1

30 IFCIJ 26t3Zr26 32 RETURN

END

62

Information Only

• • .

• •

. .:··~· ..

(;})

. •

\: ·:··.·. ~

c c c c c

MODEL

$U8ROUfiNe.MOOE~CatFYtN08tXtWCS,HQDEtNP,WCRJ DIMENSION 8C3JeFY(It01tX(It0J .

MODE•l : MUALEM THEORY Mil TH THREE COEFFICIENTS MODE•~ : MUALc~ TriEORY MIITH T•O COEFFlClENTS MODE•l : BURDINe THSORr WlTH THREE COEFFlClENTS

IFCMODE-21 lo,zo,JO 10 CONTINUE

DO 12 J•ltN08 12 FYCJI•8&1J+(WCS-8CliJ/Cl.+ldl21•XCJJI••BlliJ••Cl.-l./8(3J)

RETURN 20 CONTINUE

DO 22 J•ltNOB ZZ FY&JI•WCR+CWCS-~CAJ/(1.+(8&1J*X(JJJ••BC2JJ••ll.-1.18(2JJ

RETURN 30 CONTINUE

DO 32 J•l,NOB 32 FYlJJ•B&lJ+CWCS-oC1JJ/ll.+l8l2l*XlJJJ•*Bllll**lle-2e/Bl3JI

RETURN EHD

63

Information On y