<br />
<br />space-step sizes such as OJ, 0.5, and 2.5 cm were
<br />tested, It can readily be seen from Figure 8 that the
<br />smallest space-step size yields the best results. As the
<br />/Sf. size increases, oscillation is magnified and pro-
<br />longed, A best.fit line connecting the computed
<br />points for each ~ z size can be drawn as marked in
<br />broken line (Figure 8). It appears that the best-fit line
<br />for lJ;l = 0,] cm merges in the best-fit line for lJ;l = 05
<br />cm which subsequently merges in the best-fit line for
<br />!::;z = 2.5 cm and so forth, in essence becoming one
<br />single best-fit infiltration decay curve. In other words,
<br />if the infIltration rate computation could be reversed,
<br />starting at t = 00, a larger ~z could cause the earlier
<br />occurrence in oscillation and an eventual breakdown.
<br />From the accuracy point of view, it is apparently
<br />more accurate to use the smaller ~z in the computa-
<br />tion. Despite a large /:,Z used, a divergence problem
<br />did not seem to occur on many computer runs tested
<br />herein. Because the finite-difference computation
<br />with a very smalllY. is time consuming and expensive,
<br />use of a small b.z cannot be justified unless one is only
<br />interested in the accuracy of the solution. In view of
<br />the necessity of considering both accuracy and
<br />efficiency involved in a particular computer run, the
<br />range of interest associated with the computation of
<br />the infIltration rate versus time must be taken into
<br />account in the selection of a suitable ~z size.
<br />
<br />Figure 8 demonstrates the differences in the
<br />range of the computed infiltration rate for the three
<br />different /:,Z sizes used in the computation, given the
<br />various lengths of computer time (e.g" UNIVAC
<br />I 108 CPU time 95 seconds for /:,Z = 0.1 em up to t =
<br />0.02 hours, 155 seconds for lJ;l = 05 cm up to t = 0,8
<br />hours, and 64 seconds for /:,Z = 2,5 cm up to t = 10
<br />hours), Computer output for each /:,Z size used is
<br />
<br />It
<br />I
<br />s
<br />" .
<br />..
<br />l'
<br />~ z
<br />~
<br />,
<br />n
<br />;!
<br />.. 5
<br />~ ~
<br />:s
<br />
<br />labeled in different symbols, An inspection of Figure
<br />8 reveals that using a lJ;l size of either 0.1,0,5, or 25
<br />em could yield approximately the same inftltration
<br />decay curve at about half an hour after ponding.
<br />However, in terms of computer time involved in the
<br />computation, use of lJ;l = 2,5 em would be more than
<br />twenty times less expensive than that of lJ;l = 05 em
<br />as far as that range of the infiltration rate (Le., t>O,5
<br />hours) is concerned. In another case, if one is
<br />interested in the infiltration rate shortly after pond-
<br />ing, say t<0.002 hours (or 7.2 seconds), a/:,z size of
<br />0.1 em or less shouid be used. Use of a /:,z size of 0.5
<br />or 2.5 cm in the latter case is obviously not adequate.
<br />
<br />Effect of initial moisture content (00)
<br />
<br />Different initial moisture contents were tested
<br />to determine the effect of initial moisture content on
<br />the accuracy of the present finite-difference modeL
<br />For comparison, an equal ~z size, 2.5 cm, was used
<br />and values of all variables except 80 were kept
<br />constant in the computation of the same hypothetical
<br />conditions shown in Figure 8. Computed inftltration
<br />decay curves for 00 equal to 0.1,0.2,03, and 0.4, by
<br />using about the equal length of computer time were
<br />depicted in different symbols, as shown in Figure 9,
<br />As anticipated, the higher the initial moisture con-
<br />tent, the smaller the computational oscillation When
<br />the initial moisture content is high such as 0.4, close
<br />to saturation, the infIltration rate computed by
<br />means of Eq, 33 does not appear to have a large error
<br />for t>OJ hours, As the value of 00 decreases,
<br />computational oscillation, although damped out in
<br />the end, is amplified and prolonged, The present
<br />finite-difference model, was tested and shown to be
<br />valid up to 00 = om, but it broke down for 00 less
<br />
<br />
<br />z
<br />
<br />a%.:: 0.1 em
<br />- a%.:: 0.5 em
<br />.....:lo- az:: 2.5 em
<br />
<br />_______ _ __ _ ____ _ _ f_K!...:.~~:.m~~ __ __ _ _ ___ __ ___~________
<br />
<br />3 4 S6789
<br />I
<br />
<br />z
<br />
<br />3 4 56789
<br />'A
<br />
<br />10-4
<br />
<br />z
<br />
<br />3 4 56789 ,
<br />10'
<br />
<br />,
<br />
<br />3 4 5 6789
<br />10-2
<br />
<br />z
<br />
<br />3 4 56789
<br />10.1
<br />
<br />z
<br />
<br />Figure 8, Examples of the effect of space-step (Az) size on computed infdtration rate (f) under a hypothetical
<br />immediate ponding situation,
<br />
<br />Time - t - hour
<br />
<br />26
<br />
|