odeling I: A Generalized Simulator1 ady, Two-Dimensional Flow 4 -

C. R. Ameman MEMBER

M A E

PUTATIONAL scheme was Flowing soil water has been found The purposes of t . paper are to to remove fine particles from soil pro- briefly describe the model an$ to dis- files and has been studied as one of cuss a programming concept that

equation using finite dif- the agents contributing to soil and could be used to &generalize other The successive overrelax- landscape genesis (Vasilyev 1948; models. 11i later papers I will discuss

thod was used. The Bunting 1961; Smirnov 1%3). Flow- sensitivity of finite difference models developed for imple- ing soil water also carries many solu- to various factors.

e computational scheme ble substances (Buckman and ralized using a technique Brady 1969). MODEL DESCRIPTION

ng, which allows To improve our understanding of ne model is a ,digital rnmputer model, without in- such processes as surface and sub- program, writ.en in USASI porbran,

surface hydrology, soil genesis, land- which should run on most cbmputers scape morphology development, and 6 t h a of modacation,

- the distribution of chemical sub- statements likely vary from one stances through the landscape, we computer to another are flagged. must study Soil water and its move- model us= the successive over- ment. For some processes we can- relaxation (SOR) method (of finite

ature of Richards' equation not be satisfied simply to investigate differencing applied to th,e steady- ted in the overrelaxation inputs and outputs but must endeavor ,tatet two-dimeasional form of which had a maximum to define actual flow paths and ve- ~ i ~ h ~ ~ d ~ , an optimum value. Values locities. Model capabilities and limitations

an maximum caused insta- In most situations involving large- me as follows: scale and large-~01"me Soil water 1 movement, hydraulic head gtadimts

INTRODUCTION provide the major forces and, along from ib crucial role in the with soil hydraulic condudvhy de- at least

anae of organic lie, soil water ortant and active part of

, Moat ground- id areas are re- by soil water

; themfore, move-

, springflow, and the and 1972, Freeze 1971, Pinder and

the l a m is one of the con- ctors for dry weather 1%9, Ameman 1%9, RDstr 1970,

rice of gronndwater supplies. h i n d 1971, and Pinder 1973). Reirenauer et al. (1963) have devised tions a model that has been generalid in

generation of surface storm a d snowmelt

submitted for publication in h e w e d and appmed for put,-

y the Soil and Water Division of ASAE ln 19%. b n t e d as *ME Papr regions simply by changing input data

botion of the Watershed Research and (b) to provide sufficiently de- hydraulic central Reg.n, ARS, uSDA, in tailed documentation that others, with daries of with the ~ h u r i and wisc011~in only brief acquaintance with digital w e n t statiom, Cdarn&a, computer programming or with the as

d Madision, WI, mqectively. ox is: C. R. AMERMAN, Research

field of mathematics involved, could 7 Hydr llttghw2 ~ ~ t e r s h e d R ~ W U E ~ unit, use the model. The model, named a function of tral Region, ARS, USDA,. Columbia, STDY2, is fully documented in an- (h) may be

other publication CAmerman 1976). Alternatively,

TMs i '- ?e is reprinted Erom the TRBNSAONS of the AS- (Vol. 19, No. 3; pp. Published by the,iherican Society of Asidturd Ensin-. St. Joseuh. MIehiean s >! . - J .t 1

FIG. 1 Stencil for mte difference grid.

equation relating K and h, but this requires some programming ability on his part.

8 A given soil unit is considered isotropic.

9 Hysteresis in the K-h relation- ship is ignored.

10 Darcy-type flow in the proto- type system is assumed.

11 A surface-of-seepage problem is handled by trial and error and re- quires the user to interact with the model.

12 Convergence checking re- quires the user to interact with the model. An auxiliary program is pro- vided as an aid for this.

13 The finite-difference grid is specified by means of input data cards. The grid may be square, rec- tangular, or irregular with grid-size varying over the section.

14 An auxiliary program is pro- vided to change mesh spacings of an existing grid for which a solution al- ready exists.

Input to the model consists of model specifications and, except as provided in No. 7 above, a table of K as a function of h. The user may select one of two model-generated starting arrays of h, or he may submit an array as input data in either punchcard or magnetic tape form.

Output of the model may be di- vided into four groups:

1 Printed model identification and model specifications.

2 Printed intermediate data for use in checking convergence.

3 Printed array of soil-water hy- draulic heads.

4 Array of soil-water pressure heads in printed form and in either punchcard or magnetic tape form.

METHODS

Combining the equations of state and continuity with Darcy's law yields Richards' equation (Richards 1931) for transient, unsaturated flow. It may be written for the steady state and for two dimensions as

where H = hydraulic head = h + z (L) h = soil-water pressure head 0 z = position head, vertical dis-

tance above a datum (L) K = K(h), hydraulic conductivity

(LT- l) x = distance parallel to the x-

axis of the Cartesian coor- dinate system, positive to the right (L)

y = distance parallel to the y- axis, positive upward (L).

The Cartesian coordinate system may be rotated through an anglea to make the x-axis conform with the pre- dominant slope of the prototype system, i.e., the slope of a soil sur- face or bedrock. Because K is a func- tion of h, substituting h + z for H is convenient. Making this substitution and introducing a in equation [I] yields

FIG. 2 Example of complex c r o ~ w...

KZ;,~. . ~rnr-l: - + sin a J + c o s a ' I

a a h a a h aK a K - (K-) + - (K-) + sin a - + cos a - = 0 a x ax a y a y a x a y

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . [21 m-1 2 AY+.K~I ' , + Ay-.Ki, ,+%

+- ( Equation [2] is a nonlinear, elliptic- AY-+AY+ AY--AY+ type, partial differential equation. Description of a given problem is ' ' . ' ' ' . . . ' " " . ' ' ' . ' . ' ' complete when boundary geometrics and boundary hydraulic conditions where have been specified. There is no gen- m = values obtained during era1 analytical solution for all boun- the current iteration, dary geometrics, sequences of boun- m-1 = values obtained during dary conditions and K-h distributions. the preceding iteration, Numerical methods must usually be o = overrelaxation factor, applied.

The numerical method used by STDY2 is finite differencing with Ki-%,j = terms developed using the central 2

differencing technique. The result- ing system of equations is solved by

K ~ + ~ ~ the iterative scheme known as suc- 2 cessive overrelaxation (Forsythe and Wasow 1960, Smith 1965, Greenspan

K IJ-1 . + K. i j 1968). The finite-difference grid sten- =

cil is given in Fig. 1. 2

STDY2 solves the following SOR finite difference approximation to equation [2]

2

h!? = ( 1 - w ) hf7-l 'J 'J of node, i,j,

+ = mesh increment to right

2 Km-l Km-l hm-l of node, i,j, + w [ p ( 3 - 4 1 3 h f n + itf4,j Ay- = mesh increment above

Ax-+Ax+ Ax- 1-1 j i+l j)

Ax+ node, i,j,

- m -

COLUMNS COLUMNS

i = I 2 3 4 5 6 7 8 9 10 11 12 i = l 2 3 4 5 6 7 8 9 1 0 1 1 I

- ' 0 0 0000000 0 j = l 0 0 0000000 0 x . . . . . . . . . . 0 aL 0

. . . . . . . . . . . . . I 0 0 3 0 . . . . . . . 4 0 . . . . . . . . 0 4 0 . . . . . . . . . . . . . . 0 5 0 ; - 0 0 6 0 0 o o ; . . . . . . . . ]I0

u , 6 Q 0 00.. 0 . . . . . B 7 0 4 0 . . 0 0 3 7 . . . . 8 m . . 10 0 8 a , .

rn q a a a . . a 0 . . 9 v a 0 9 q a a a . . . . . . . . . . . 0 10 0 ' . 0 . . . . . . . 0

= O 0 . . . . . . . . . . 12 0 . $ 0 1 2 0 0

m: 13 0 q 808rna08 a 13 El 0 ElE3ElBOElGl E3

I I

I Scale b I i! 4 cm I - - 4

S c a l e : - cm 4 L e g e n d : . Calculation Node 0 1 2 3

I 0 Imaginary Node Legend: Calculation Node

I 0 Imaginary Node h -Specified Boundary Node

FIG. 3 Row s u b d m for Fig. 2. h-Specified Boundary Node

FIG. 4 Column subsections for Fig. 2. !y+ = mesh increment below

node, i,j. plex both in geometry and in the dis- at column 2 and ends at column 11. [31 rft tribution of hydraulic boundary con- Each row is impermeable on both

-Aes for which pressure head lS ditions. Figs. 3 and 4 illustrate how the left and right ends. These rows known. Otherwise* the section of Fig. 2 was subsectioned form a subsection in that, for all of requires that [31 be for in row and column directions, respec- them, one set of data can be given each node of the finite difference tively. The dots (Figs. 3 and 4) repre- the model to specify the end boundary grid in some sequence. The sent calculation nodes of the finite conditions and to govern the process-

begins at the end of difference grid. The circled dots, ing of the nodes. Row 6 has the same the top row and processes each row in labeled imaginary node in the figures' beginning and ending boundary con-

, sequence from left to right. Pressure- legends, were used to implement the ditions as the rows above it, but the head during an iter- impermeable condition of the boun- first node to be processed is on col- ation become immediately available daries to which they are adjacent. umn 6. Row 7 is different again, so for use in pressure heads As mentioned earlier, equation [3] that row 6 is unique and forms a for neighboring nodes. applies to each node of the finite dif- separate subsection. Rows 7, 8, and

I in equation [319 the nodes above ference grid that does not have an 9 are the same actual length as row I and the left of node i9j (nodes i-l,~ h-value specified for it. Equation [3] 6, but the beginning boundary con-

and i9j-1) have the same "perscrit does not apply outside the boundaries dition is one of pressure-head spe- 1 I as node i,j. of the flow region; imaginary nodes cified, so the node on the left end of I K-values with pres- are handled by special algorithms. the row is not to be processed. This

I sure head from the previous One purpose of subsectioning, then, calls for grouping them in a sub- Using new K-values at the was simply to control which nodes are section separate from row 6.

nodes above and the left of node processed by equation [3]. Row sub- Nodes are processed from left to i9j caused 'lower sectioning, in effect, specifies cross right and from top to bottom within since the overrelaxation section geometry. The other purpose, a subsection, and the subsections are factor (to be discussed later) was accomplished by both row and column processed in the order of their ap-

than when using K-values. subsectioning, is to apply boundary pearance in the input data deck. To conditions. maintain the orderly sequence of

SUBSECTIONING Rows 2 through 5 (Fig. 3) form a processing required by SOR, the row of the model was homogeneous set from the stand- subsections must be properly ordered

accomplished through the use of a point of both boundary conditions which means taking care that, for a I '

i technique called "subsectioning." and pattern of nodes to be processed given iteration, all nodes of a row have Fig. 2 shows a section somewhat com- by equation [3], i.e., each row begins been processed before processing the

i l l

H =O --4----*--&-4--L-a--LLe---- I '-r I S c a l e I I 2 I ; e m -25t 1 1 FIG. 5 Hvdraallc head .MLution for FP*. 2. - 8 -

FIG. 6 Canvergence rate PI influenced by the over-relaxation factor, o. next lower row. Thus, if the cross section of Fig. 2 had a vertical notch the water table boundary at the base was true for a similar equation. For or barrier halfway along its length and of the cross section toward the -30 cm values greater than omax, the ~ 0 1 ~ - extending to some depth from the top boundary in the notch- tion begins to diverge at some of the boundary, some of the upper rows nodes of the finite difference grid. in Fig. 3 would be bisected. This OVERRELAXATION For a Small range of o > omax, would mean that the two parts of A considerable body of theory is minor fluctuations may slowly grow these rows would form two sub-sec- available for applying SOR to linear about a generally converging solu- tions. The data for both these sub- problems. There is little theory appli- tion, and only a limited part of the sections must precede data for the cable to nonlinear problems, but ex- grid may be affected. But for o grossly subsection that includes the first perience shows that SOR can be suc- larger than w a x , the solution be- complete row of nodes below the notch cessfully applied to many of them. In comes unstable over the whole grid. or barrier. developing this model, the biggest The effects of small overestimations

A similar discussion can be given departure from extrapolation of li- of omax are sometimes not discern- for defining column subsections in near techniques to nonlinear was ible until considerable convergence Fig. 4. Column subsections are used encountered in applying the overre- has taken place. only to define boundary conditions at laxation factor, o. SOR with O< o < Using textbook methods based on the ends of columns, however, and do 2 is convergent for problems involv- linear theory to estimate wo t as a not affect the order of processing. ing linear partial differential q u a - first approximation to omax 7 or rec- Therefore, the order of column sub- tions regardless of boundary geometry tangular geometry proved unsuccess- section data can be arbitrary. and boundary hydraulic conditions. ful. The estimated values bore no dis-

The subsectioning concept, with The optimum value of o, wept, cernible relation to the omax values slight modifications, can be applied causes the most rapid convergence obtained by trial and error. Further- to other finite difference schemes. possible using SOR. For linear prob- more, for linear problems wept is It was applied to a steady-state model lems, Forsythe and Wasow (1960) usually larger than 1.0 (Forsythe and using an alternating direction impli- show that o slightly greater than Wasow 1960). But wmax often took cit (AD11 scheme, but the latter coopt causes more rapid convergence on values less than 1.0, sometimes proved to be inferior to SOR for cross than o slightly smaller than oopt. considerably less. sections that were not perfectly square In contrast with the linear case, a Fig. 6 shows that, like linear equa- or rectangular. Convergence for AD1 concept of optimum overrelaxation tions, convergence varies in a regular in these cases was many times slower factor is not quite accurate when manner. The upper curve extending than for SOR. applied to equation [3]. For this Erom 0 to 150 iterations was obtained

Fig, 5 shows the hydraulic head equation, there seems to be a maxi- at a representative node in a model distribution generated by the geo- mum w-value, omax, at which con- for which o was increased by 0.2 every metry and boundary conditions of vergence can be obtained. Reisenauer 30 iterations. The limits of each 30 Fig. 2. The direction of flow is from et al. (1963) apparently found this iterations are marked on the figure

,>- -- 1 . ------.: I' ' I .ell i by a tic mark above the curve. Inspec- are, again, matters of trial and error. the physical basis of soil water pheno ena tion of the curve reveals that it steep- Final convergence, however, should Wfie~-Inte"*encel New York. 493 P. f" ened every time o was increased; i.e., be obtained using o d omax. 6 Forsythe, George E., and W O ~

R. Wasow. 1960. Finite difference me* f9 convergence rate increased as w in- padal differential equations. ~ i l e y and ~ n n c

m t ~ e d . 'l'%!s was true when o = 1.80 New ~ o r k . 444 p. was used, even though this value was SUMMARY 7 Freeze, R. Allan. 1971. Three-d - greater than omax. The shorter curves demonstrate convergence rate for each o-value when starting at zero iterations. The maximum overrelax- ation factor for which all nodes con- verged lay between 1.65 and 1.70.

As implied in the preceding dis- cussion, the only known way to ob- tain a value for omax is by trial and error. STDY2 includes provision for trying different values of w while ad- vancing the solution. At the begin- ning of a computer run which models a situation, the user may speci- fy the o-values to be tested during that run and may specify the number of iterations for each test. He may also specify certain nodes for which h is to be printed at the end of each iteration;- after every other iteration or at some other frequency. When the run is finished, he may inspect the data for these nodes and determine whether fluctuation has begun or he may plot them in curves similar to the upper curve of Fig. 6. Restart data may be obtained at the end of each w test so that, after selecting a value for w, the solution may be con- tinued from the most advanced data set that was not preceded by excessive fluctuation.

Strictly speaking, convergence may be obtained only using od omax, and fastest true convergence apparently occurs using omax. However, the import of the w = 1.80 portion of the upper curve of Fig. 6, as verified by experience with other cases, is that highest initial convergence rates can sometimes be obtained with o-values that exceed omax. The magnitude by which w exceeds omax and the num- ber of iterations that can be run with- out excessively distorting the solution

A concept called "subsectioning" was used to generalize the applica- tion of a steady-state, two-dimen- sional. finite difference model of porous media flow. Subsectioning allows generalization of the model to the extent that it can be applied to a wide variety of prototypes simply by changing specifications in input data. Although the flow equation modeled is nonlinear, experience shows that finite differencing by suc- cessive overrelaxation yields .conver- gence. The primary departure from SOR linear theory appeared in con- nection with the overrelaxation fac- tor, o. This factor could not be esti- mated using the techniques of linear theory. For this nonlinear application, convergence rate increased as o in- creased until some maximum was reached. Values of w larger than this maximum led to instability in the solution. Experience indicates that the maximum o-value is about as likely to take on a value less than 1.0 as it is to take on a value greater than 1.0.

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ence, solutions of unsteady, two-dimensional, partially saturated porous media flow. Ph.D. Thesis, Purdue University, Lafayette, Indiana.

2 Amerman, C. R. 1976. Waterflow in soils: a generalized steady-state, two-dimension- al porous media flow model. U.S. Dept. of Agr. ARS-NC-30.

3 Buckman, Harry O., and Nyle C. Brady. 1%9. The nature and properties of soils. MacMillan, New York, Seventh Edition. 653 p.

4 Bunting, B. T. 1961. The role of seep- age moisture in soil formation, slope develop- ment, and stream initiation. Amer. Jour. Sci. 259:503-518.

5 Childs, E. C. 1969. An introduction to

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and Joseph L. H. Paulhus. 1958. Hydrology for engineers. McGraw-Hill, New York. 340 p.

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Soc. h e r . Proc. 36:399-401. tion from sources at arbitrary depth. Soil iSci.

15 Reisenauer, A. E., R. W. Nelson, p d C. N. Knudsen. 1%3. Steady Darcian t r y - port of fluids in heterogeneous partially satur- ated porous media. AEC Research and *I- opment Report No. HW-72335-PT2. 88 p. 16 Richards, L. A. 1931. Capillary condnc-

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two-dimensional, transient flow of water in bn- saturated and partly unsaturated soils. Soil Sci. Soc. Amer. Proc. 32:607-615. I 18 Smirnov, M. P. 1%3. The experietlce

of a study of subsurface runoff in the northern part of Western Sayan (under the conditionsipf the Ermakov Station). Pochvovedenie 4:93-1v. 19 Smith, G. D. 1%5. Numerical solut&n

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20 Taylor, G. S., and J. N. Luthin. 1969. Computer methods for transient anal$= (of water-table aquifers. Water Resources Res. 5144-152. 21 Vasilyev, I. S. 1948. The

a study of surface and subsurface runoff