Determination of the capillary conductivity of soils at low moisture tensions

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1 Determinatin f the apillary ndutivity f sils at lw misture tensins J. BUTIJN and J. WESSELING Institute fr Land and Water Management Researh, Wageningen, The Netherlands REPRINTED FRM NETHERLANDS JURNAL F AGRICULTURAL SCIENCE VL. 7 N 2 MAY 1959

3 DETERMINATIN F THE CAPILLARY CNDUC TIVITY F SILS AT LW MISTURE TENSINS l ) J. BUTIJN and J. WESSELING Institute fr Land and Water Management Researh, Wageningen, The Netherlands SUMMARY A methd is desribed t alulate the apillary ndutivity data frm utflw measurements btained with a sutin apparatus als in use fr the determinatin f the lwer part f the misture harateristi f sils. Measurements were made n different sil prbes frm the same prfile. An apprximate slutin fr the influene f the prus plate was intrdued. The results f the measurements fr a sandy sil and a sandy lam are given. 1 INTRDUCTIN During the last deades the knwledge n the relatin between misture nsumptin and yield f rps is rapidly inreasing. Frm investigatins n this relatin it beame evident that besides the ttal uptake f water sil misture tensin is a very imprtant fatr in rp grwth. This means, that sil misture tensin shuld be kept lw, in rder t get maximum yields. Exept in the ase f sub-irrigatin this an nly be realized by water transprt frm the water table int the rt zne. Misture transprt nly takes plae, hwever, when the water table is kept high r évaptranspiratin rates are small. This is due t the fat that the apillary ndutivity f the sil is lw at high and mderately high misture tensins. S steep misture gradients are neessary fr misture flw f sme imprtane (WESSELING, 1957). The steady state apillary rise f water frm a water table an be alulated when the relatin between apillary ndutivity and sil misture tensin is knwn. Suh alulatins are given by WIND (1955), WESSELING (1957) and GARDNER (1958). These authrs either used ndutivity data fund in literature r determined frm field experiments, as in the ase f WIND. Rather few ndutivity data are available, hwever, despite the fat that a methd fr the determinatin has been perfrmed already mre than 25 years ag by RICHARDS (1931). During the last few years several new methds were prpsed but mst f them make use f disturbed sil samples (MRE, 1939, STAPLE and LEHANE, 1954, BRUCE and KLUTE, 1956, CHILDS and CLLIS-GERGE, 1950). When ndutivity data f a disturbed sil sample are used fr the alulatin f misture transprt in a well-defined sil prfile with its typial sequene f layers, eah f them with a prnuned struture, appreiable differenes with what atually happens in the prfile will be fund. The nly way ut in this ase is, t determine the apillary ndutivity f eah layer f the prfile n an undisturbed sil sample. A theretial mputatin f the ndutivity data frm the pre size distributin i.e. frm the misture harateristi as prpsed by CHILDS and CLLIS- GERGE (1950) has ertain advantages. Reently MARSHALL (1958) fund a reasn- J ) Reeived fr publiatin Marh 11,

4 able agreement between this theretial methd and experimental results btained by ther authrs. The mputatin takes a lt f time, hwever, and the s-alled mathing-fatr remains undetermined. GARDNER (1956) develped a methd fr the alulatin f the ndutivity data frm pressure plate utflw data. The advantage f this methd is that it an be mbined with the determinatin f the higher part f the misture harateristi f the sil, fr whih pressure plates are used. A disadvantage is, hwever, that the methd an nly be applied at high misture tensins. As is pinted ut abve, in flw prblems under nditins prevailing in nature, the apillary ndutivity f the sil at lw misture tensins is far mre imprtant. A methd therefre was develped t mpute the ndutivity at lw misture tensins frm utflw data btained with a sutin apparatus that is mmnly used fr the determinatin f the lwer part f the misture harateristi. The methd, and the results btained with it, are desribed in the next setins. 2 DESCRIPTIN F THE APPARATUS The apparatus, used fr the determinatin f the lwer part f the misture harateristi was mdified in suh a way that the flw f water ut f the sil sample uld be measured. The apparatus whih was used is given in fig. 1. Undisturbed sil samples (M) are taken in metal rings with an area f 20 m 2 and 5 m length. After saturatin during tw days, the sample was plaed n a prus plate (P) whih is emented in a glass-vessel (G), n- 156 FIG. 1 SUCTIN APPARATUS USED FR THE UTFLW MEASUREMENTS. FR DESCRIPTIN SEE THE TEXT.

5 neted with a water reservir (W.) The glass-tube ends in a alibrated hri zntal apillary tube (C). The glass-vessel is vered with a ap (K), that is nneted with the vessel by means f a rubber-ring (R), in rder t prevent evapratin frm the sample. A small apillary tube was munted in rder t maintain atmsferi pressure at the upper end f the sil sample. When equilibrium has been attained between the misture in the sil sample and a ertain sutinfre, the glass-vessel is raised ver a small distane, s that the sutin inreases with a small amunt Due t this inrease in sutin, water will mve t the right hand side, the distane depending n the magnitude f the water remval. Ry measuring the displaement f the menisus at different times, the water remval an be determined and frm these data the apillary ndutivity f the sil may be alulated. The measurements are suseptible t temperature differenes. When rm temperature is nt nstant the menisus n the apillary tube may beme displaed withut any hange in the misture ntent f the sil sample. While the measurements are arried ut, with relatively shrt time intervals, differenes in temperature during the experiment an be negleted. The mputed values fr the ndutivity must then, hwever, be redued t standard temperature. This standard temperature was taken t be 15 C whih is the temperature, uring in the rtzne f the sil during the greatest part f the summer in the Netherlands. 3 SLUTIN F THE FLW EQUATIN When equilibrium has been attained at a ertain sutin f h m, the misture ntent at the upper side f the sil sample will differ nly little frm that at the lwer end f the sample, when the height f the sample is relatively small. Withut intrduing large errrs the misture ntent may be nsidered t be nstant thrughut the sample. Furthermre, the sutinfre thrughut the sample may be taken as a nstant, having a value f h = H -f- M I, where H is the distane between the water surfae in the reservir and the upper bundary f the prus plate and I is the length f the sil sample. When the sutin is inreased frm h t h -\- Aha. lwer misture ntent will be attained after a ertain time thrughut the sample. When the inrease in sutin A h is small, there will be nly a small differene between the initial and final misture ntent in the sil sample and the apillary ndutivity is nsidered t be nstant. Mrever =- =, fr small hanges in misture ntent <9 and sutin h. Fr the flw f misture, the differential equatin (1) will then hld true (GARDNER, 1958) ôt' ~ 'ôz 2 where (9 = the part f the vlume upied by water, i.e. the vlumetri misture ntent f the sil t = time k = apillary ndutivity z = length dq = the slpe f the misture harateristi = dh [ j 157

6 There will be n flw arss the upper bundary f the sil sample and r, à(-),., henefrth - = at this plane. z Treating the sample as having a length 2 I with its imaginary upper bundary at a nstant misture ntent (-) at t >, the initial and bundary nditins will be : (-) ~ = n 0 - = - <9 1<Z<+1 z=±l t = 0 1 > 0 The slutin f eq. 1, subjet t 2, is analgus with that f the equatin f heat ndutin in a lng islated plate with the ends kept at a nstant temperature. This slutin is (f. CABLSLAW and JAEGER, p. 79 eq. 6). (2) k ) -~r 2 ' s - a 4 y- (-1) < 2?i + l,,, e a FT r, s ' i"" nz 3 The ttal quantity f water that will have passed thrugh the lwer surfae f the sil sample at eah time t an be given by z = l Q l = A/(0-0 )dz (4) r A being the area f the sil sample. The final amunt whih an be remved frm the sil sample between a sutin h and h -\- Ah is With the aid f eq. (3) the relatin Qr A I ( n,,) (5) U1 k (2n -V I) 2 ^J_ Qr Ti 2 «=» (2n -+- // K ' an be derived frm eqs. (4) and (5). This is the same relatin whih was used by GARDNER (1956) fr the alulatin f the apillary ndutivity frm pressure plate utflw data. Fr reasns dealt with in setin 5 it is smetimes easier t alulate the ndutivity frm the slpe f the urve Q t versus t, whih gives the utflw velity at eah time t. This quantity is derived frm eq. 6 r eq. 3 and is Substituting nw eq. 5 in eq. 7 we get y -L(* 2 +,?* 4l, k h A H e < "- (8) I n ="

7 4 AN APPRXIMATE SLUTIN Fr the early stages f flw an apprximate slutin f the prblem under disussin, may be btained by nsidering the sil sample as being infinite. The bundary and initial nditins are then given by = «z> t = a f) = (") z = I > These nditins are the same as thse fr infiltratin int a semi-infinite hrizntal sil lumn (KLUTE, 1952) and fr drainage t a anal frm a semiinfinite strip f sil (EDELMAN, 1947). The slutin f (1) subjet t (9) is 9 - * - - P r (,.*=...) (10) 0" f) where P F is the prbability integral taken frm the tabulated frm given by FISHER (Statistial Methds fr Researh Wrkers, 1948, p. 77). The amunt f water whih has flwn ut f the sample after a ertain time t will be Q t =-. (f) H - > ) A / P F f? z _.) dz (11) whih an be represented by Qt = J-14 A (f> W JI<y7 (12) During the first stages f flw there will be little r n differene between the results f eqs. 3 and 12, with respet t the misture prfile in the sil sample. As eq. (12) hlds nly true fr a limited length f time, this length f time has t be determined. This is dne by using the therv f KHAYENIIFF VAN 2 DE LEUR (1958). Sine 1.14 = -_- and h = f) f), differentiating eq. 12 with respet t t gives ' n qt -~ y- A h } k S (13) y 7t ' If we take nw the largest length f time fr whih eq. 13 may be used in suh a way that the value f q t frm eq. 13 differs less then 1 % frm the value f q t arding t eq. 8, we get U k i2 2 ' -*=]<E> f e' C ' : " " 7 ' 2 (H) Taking nw / = 4 l 2 2 T e l- ^ bemes 159

8 y. 4 X 0,99 ' UI > e and fr a given mbinatin f, k and /, t max an be alulated. (15) 5 THE INFLUENCE F THE PRUS PLATE The eqs. 8 and 13 derived in setins 3 and 4 are exat, when the prus plate used in the experiments, has n resistane with respet t the flwf water. Generally this is nt true and the influene f the sinter may nt be ignred, espeially when relatively large amunts f water are displaed, thus at large ndutivity values f the sil sample, therefre, withut intrduing the effet f the prus plate, the alulated ndutivity data will be t small. n the ther hand we failed t find an exat slutin in whih the effet f the prus plate was aunted fr 2 ). The utflw data, usedfr the alulatin f the apillary ndutivity are, hwever, taken during very shrt time intervals. Therefre an apprximate slutin may be used in whih the lss f head in the sinter is related in a linear way with the utflw arding t DAECY'S law, ne an therefre put : where kp = the hydrauli ndutivity f the sinter hi = lss f head in the sinter lp = height f the sinter. q t = k P y (16) The hydrauli ndutivity f the sinter was btained by measuring the flw f water thrugh the sinter withut sil sample. The area f the plate used fr this measurement was equal t that f the sil sample. In rder t avid variatins in hydrauli ndutivity during the measurement the determinatins were made with as shrt a lapse f time between them as pssible. Frm the knwn q t, the lss f head due t the sinter was alulated arding t eq. 16 and intrdued in eq. 8 and 13 s the hydrauli ndutivity was alulated frm the equatins and G 1' = l k 4k-k,)All e " (8a) C I C " =0 Q, = 1.14 (& n e 0 e v ) A i^t (12a) where@ r, = h v.. Fr q t and Q t, the utflw after 1 minute was hsen. 2 ) After this artile was written MILLER and ELRICK (Sil Sei. S. Amer. Tr. 22, 1958 : ) gave a methd t take int aunt the influene f the separating medium. An ther slutin will be given in the near future by RIJTEMA, als wrking at ur Institute. 160

9 6 RESULTS F THE MEASUBEMENTS The measurements were arried ut fr a dune sand ntaining 1 % humus and 99 % fratin > 100 //, and fr a sandy-lam sil ("lihte zavel") ntaining 1 % humus and 1 % fratin > 100,,, 50 % frm fi and 16 % < 16 a (abut 10 % < 2 /() All utflw measurements gave straight lines fr small values f t, when pltted against t. Due t sme inertia in the system and the influene f the prus plate this line did nt always interset the t-axis at t. Nevertheless t was taken in suh a way that t = was at the pint f intersetin f the line with the f-axis. In rder t simplify the alulatin, this frm was alulated arding t eq. 8a fr varius values f -, (I = 5 m ; t = 1 minute ; h h l 1 m ; X h 10'! 111 t i i t r I e FIG. 2 VALUES F </.; PLTTED AGAINST FR I = 5 CM, t = 1 MIN. I h A = 1 m 2 ) giving a straight line when pltted n lg : lg sales (fig. 2). The results f the measurements are given in table 1. Fr eah sutin step new sil prbes were used taken frm the same prfile. In the ase f the sandy sil, srptin and desrptin measurements were arried ut with the same prbe. Fr the lamy sil different prbes were used fr srptin and desrptin. The values arding t eq. 12a are smewhat higher than thse arding t eq. 8a but the differene is small. This is due t the fat, that the time was taken s shrt that the finite length f the prbe did nt influene the slutin arding t eq. 12a. Beause the ndutivity values are highly dependent n differenes in sil struture as the results shw, the errr intrdued by the apprximate slutin f the influene f the prus plate will nt be very imprtant. As uld be expeted, a ertain hysteresis effet ured between srptin and desrptin values. 161

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11 ACKNWLEDGEMENT The authrs thank Mr. D. A. KRAYENHFF VAN DE LEUR fr his assistane in the slutin f the prblem, Prf. F. HELLINGA fr his stimulating ritiism and Mrs. A. VAA fr the areful exeutin f the measurements and alulatins. BlBLIGBAPHY BRUCE, R. R. and A. KLUTE : The measurement f sil misture diffusivity. Pr. Sil Sei. S. 20 (1956) CARSLAW, H. S. and J. C. JAEGER : Cndutin f Heat in Slids. xfrd University Press. xfrd, CHILDS, E. C. and N. CLLIS GERGE : The permeability f prus materials. Pr. Ry. S, Lndn A 201 (1950) CHRISTENSEN, H. R. : Permeability-apillary ptential urves fr three prairy sils. Sil Sei. 57 (1944) EDELMAN, J. H. : ver de Berekening van Grndwaterstrmingen. Thesis Delft GARDNER, W. R. : Calulatin f apillary ndutivity frm pressure plate utflw data. Pr. Sil Sei. S. Am. 20 (1956) : Sme steady state slutins f the unsaturated misture flw equatin. Sil Sei. 85 (1958) KLUTE, A. : A numerial methd fr slving the flw equatin fr water in unsaturated materials. Sil Siene 73 (1962) KRAYENHFF VAN DE LEUR, D. A. : A study f nn steady grundwaterflw with speial referene t a reservir effiient. De Ingenieur 70 (1958) MARSHALL, T. J. : Relatin between permeability and size distributin f pres. Jurn. Sil Siene 9 (1958) MRE, R. E. : Water ndutin frm shallw water tables. Hilgardia 12 (1939) RICHARDS, L. A. : Capillary ndutin f liquids thrugh prus media. Physis I (1931) STAPLE, W. J. and J. J. LEHANE : Mvement f misture in unsatured sils. Canadian }. j Agr. Sei. 34 (1954) WESSELING, J. : Enige aspeten van de waterbeheersing in landbuwgrnden. Versl. L (1957). WIND, G. P. : A field experiment nerning apillary rise f misture in a heavy lay sil. Netherlands Jurn. f Agr. Sei. 3 (1955)

Content 1. Introduction 2. The Field s Configuration 3. The Lorentz Force 4. The Ampere Force 5. Discussion References

Content 1. Introduction 2. The Field s Configuration 3. The Lorentz Force 4. The Ampere Force 5. Discussion References Khmelnik. I. Lrentz Fre, Ampere Fre and Mmentum Cnservatin Law Quantitative. Analysis and Crllaries. Abstrat It is knwn that Lrentz Fre and Ampere fre ntradits the Third Newtn Law, but it des nt ntradit